{"id":321,"date":"2025-10-22T09:22:50","date_gmt":"2025-10-22T01:22:50","guid":{"rendered":"http:\/\/www.triode.cc\/?p=321"},"modified":"2025-10-31T14:22:33","modified_gmt":"2025-10-31T06:22:33","slug":"qwb-s9-crypto-write-up","status":"publish","type":"post","link":"https:\/\/www.triode.cc\/index.php\/2025\/10\/22\/qwb-s9-crypto-write-up\/","title":{"rendered":"\u5f3a\u7f51\u676f S9 Crypto\u65b9\u5411 Write Up"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">check-little<\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u9898\u5514\u7cfb\u5649\u51fa\u35ce\u5927\u4f6c\u2026\u2026<\/p>\n<\/blockquote>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>from Crypto.Util.number import *  \nfrom Crypto.Util.Padding import pad  \nfrom Crypto.Cipher import AES  \nimport os  \n  \nflag, key = open('secret').read().split('\\n')  \n  \ne = 3  \n  \nwhile 1:  \n    p = getPrime(1024)  \n    q = getPrime(1024)  \n    phi = (p - 1) * (q - 1)  \n    if phi % e != 0:  \n        break  \nN = p * q  \nc = pow(key, e, N)  \n  \niv = os.urandom(16)  \nciphertext = AES.new(key = long_to_bytes(key)&#91;:16&#93;, iv = iv, mode = AES.MODE_CBC).encrypt(pad(flag.encode(),16)).hex()  \n  \nf = open('output.txt', 'w')  \nf.write(f'N = {N}\\n')  \nf.write(f'c = {c}\\n')  \nf.write(f'iv = {iv}\\n')  \nf.write(f'ciphertext = {ciphertext}\\n')<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Util<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">number <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Util<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Padding <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> pad  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Cipher <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> AES  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> os  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">flag<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> key <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">open<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">secret<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">read<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">split<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #EBCB8B\">\\n<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">e <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">while<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">getPrime<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1024<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">getPrime<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1024<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    phi <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">q <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> phi <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> e <\/span><span style=\"color: #81A1C1\">!=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">break<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">N <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> q  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">c <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">pow<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">key<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> e<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> N<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">iv <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> os<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">urandom<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">16<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">ciphertext <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> AES<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">new<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">key<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">long_to_bytes<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">key<\/span><span style=\"color: #ECEFF4\">)&#91;:<\/span><span style=\"color: #B48EAD\">16<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">iv<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> iv<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">mode<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> AES<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">MODE_CBC<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">encrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">pad<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">flag<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">encode<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #B48EAD\">16<\/span><span style=\"color: #ECEFF4\">)).<\/span><span style=\"color: #88C0D0\">hex<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">open<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">output.txt<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">w<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">write<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&#39;N = <\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">N<\/span><span style=\"color: #EBCB8B\">}\\n<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">write<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&#39;c = <\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">c<\/span><span style=\"color: #EBCB8B\">}\\n<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">write<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&#39;iv = <\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">iv<\/span><span style=\"color: #EBCB8B\">}\\n<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">write<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&#39;ciphertext = <\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">ciphertext<\/span><span style=\"color: #EBCB8B\">}\\n<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u4e00\u773c\u5c0f\u516c\u94a5RSA\uff0c\u5c1d\u8bd5\u5f00\u6839\u6ca1\u6709\u529e\u6cd5\u5f97\u5230\\(key\\)\uff0c\u540e\u6765\u6ce8\u610f\u5230\u52a0\u5bc6\u7528\u7684\u662f\\(key\\)\u7684\u9ad816\u5b57\u8282\uff08\u5373\u9ad8128\u4f4d\uff09\uff0c\u6240\u4ee5\u5728\u60f3\u662f\u5426\u53ef\u80fd\u53ea\u6062\u590d\u9ad8128\u4f4d\uff0c\u4f46\u662f\u4e5f\u6ca1\u627e\u5230\u65b9\u6cd5\uff0c\u540e\u9762\u62ff\u5df2\u77e5\u6570\u636e\u9010\u4e2a\u548c\\(N\\)\u6c42gcd\uff0c\u53d1\u73b0\\(c\\)\u5c45\u7136\u548c\\(N\\)\u4e0d\u4e92\u8d28\uff0c\u90a3\u641e\u5230\u7684\u5e94\u8be5\u5c31\u662f\\(N\\)\u7684\u4e00\u4e2a\u8d28\u56e0\u6570\u4e86\uff0c\u76f4\u63a5\u89e3RSA\u5c31\u80fd\u5f97\u5230\\(key\\)\uff0c\u4ece\u800c\u6062\u590dflag\uff1a<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>from Crypto.Util.number import *  \nfrom Crypto.Cipher import AES  \nfrom gmpy2 import iroot, gcd  \n  \nN = ...  \nc = ...  \niv = b'\\x91\\x16\\x04\\xb9\\xf0RJ\\xdd\\xf7}\\x8cW\\xe7n\\x81\\x8d'  \nct = bytes.fromhex(\"bf87027bc63e69d3096365703a6d47b559e0364b1605092b6473ecde6babeff2\")  \n  \ne = 3  \np = gcd(N, c)  \nq = N \/\/ p  \n  \nphi = (p - 1) * (q - 1)  \nd = inverse(3, phi)  \nkey = pow(c, d, N)  \n  \nprint(AES.new(long_to_bytes(key)&#91;:16&#93;, AES.MODE_CBC, iv).decrypt(ct))<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Util<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">number <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Cipher <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> AES  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> gmpy2 <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> iroot<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> gcd  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">N <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> ...  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">c <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> ...  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">iv <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #EBCB8B\">\\x91\\x16\\x04\\xb9\\xf0<\/span><span style=\"color: #A3BE8C\">RJ<\/span><span style=\"color: #EBCB8B\">\\xdd\\xf7<\/span><span style=\"color: #A3BE8C\">}<\/span><span style=\"color: #EBCB8B\">\\x8c<\/span><span style=\"color: #A3BE8C\">W<\/span><span style=\"color: #EBCB8B\">\\xe7<\/span><span style=\"color: #A3BE8C\">n<\/span><span style=\"color: #EBCB8B\">\\x81\\x8d<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">ct <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">bytes<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">fromhex<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">bf87027bc63e69d3096365703a6d47b559e0364b1605092b6473ecde6babeff2<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">e <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gcd<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">N<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> c<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> N <\/span><span style=\"color: #81A1C1\">\/\/<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">phi <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">q <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">d <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">inverse<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> phi<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">key <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">pow<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">c<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> d<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> N<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">AES<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">new<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">long_to_bytes<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">key<\/span><span style=\"color: #ECEFF4\">)&#91;:<\/span><span style=\"color: #B48EAD\">16<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> AES<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">MODE_CBC<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> iv<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">decrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">ct<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u603b\u4e4b\u5c31\u662f\u5f88\u65e0\u8bed\u2026\u2026<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">ezran<\/h2>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>from Crypto.Util.number import *  \nfrom random import *  \n  \nf = open('flag.txt', 'r')  \nflag = f.read().encode()  \n  \ngift=b''  \nfor i in range(3108):  \n    r1 = getrandbits(8)  \n    r2 = getrandbits(16)  \n    x=(pow(r1, 2*i, 257) &amp; 0xff) ^ r2  \n    c=long_to_bytes(x, 2)  \n    gift+=c  \n  \nm = list(flag)  \nfor i in range(2025):  \n    shuffle(m)  \n  \nc = \"\".join(list(map(chr,m)))  \n  \nf = open('output.txt', 'w')  \nf.write(f\"gift = {bytes_to_long(gift)}\\n\")  \nf.write(f\"c = {c}\\n\")<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Util<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">number <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> random <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">open<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">flag.txt<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">r<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">flag <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> f<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">read<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">encode<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">gift<\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&#39;&#39;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3108<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    r1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">8<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    r2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">16<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    x<\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">pow<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">r1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">257<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">&amp;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">0x<\/span><span style=\"color: #B48EAD\">ff<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">^<\/span><span style=\"color: #D8DEE9FF\"> r2  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    c<\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #88C0D0\">long_to_bytes<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    gift<\/span><span style=\"color: #81A1C1\">+=<\/span><span style=\"color: #D8DEE9FF\">c  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">m <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">list<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">flag<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2025<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #88C0D0\">shuffle<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">m<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">c <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;&quot;<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">join<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">list<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">map<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">chr<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\">m<\/span><span style=\"color: #ECEFF4\">)))<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">open<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">output.txt<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">w<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">write<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&quot;gift = <\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #88C0D0\">bytes_to_long<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">gift<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #EBCB8B\">}\\n<\/span><span style=\"color: #A3BE8C\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">write<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&quot;c = <\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">c<\/span><span style=\"color: #EBCB8B\">}\\n<\/span><span style=\"color: #A3BE8C\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u53ef\u4ee5\u770b\u5230\u9898\u76ee\u9996\u5148\u751f\u6210\u4e863108\u7ec4\u6570\u636e\uff0c\u5206\u522b\u8bb0\u4e3a\\(r_{i,1},r_{i,2}\\)\uff08\u5176\u4e2d\\(0\\le i&lt;3108\\)\uff09\uff0c\u6709\uff1a<\/p>\n\n\n\n<p>$$<br>x_i=[(r_{i,1}^{2i}\\mod{257})\\&amp;255]\\oplus r_{i,2}<br>$$<\/p>\n\n\n\n<p>\u56e0\u4e3a\\(r_{i,2}\\)\u4e3a16-bit\u6574\u6570\uff0c\u800c\\([(r_{i,1}^{2i}\\mod{257})\\&amp;255]\\)\u5f88\u660e\u663e\u4e3a8-bit\u6574\u6570\uff0c\u6240\u4ee5\u53ef\u4ee5\u77e5\u9053\\(x_{i}\\)\u7684\u9ad88\u4f4d\u80af\u5b9a\u662f\\(r_{i,2}\\)\u7684\u9ad88\u4f4d\uff0c\u6240\u4ee5\u6211\u4eec\u5176\u5b9e\u53ef\u4ee5\u786e\u5b9a\\(3108\\times8=24864\\)\u4f4d\uff0c\u8fdc\u8fdc\u5927\u4e8e\\(19968\\)\u4f4d\uff0c\u90a3\u4e48\u7406\u8bba\u4e0a\u6211\u4eec\u662f\u53ef\u4ee5\u6062\u590d\u51faMT19937\u7684\u72b6\u6001\u7684\uff0c\u6240\u4ee5\u53ef\u4ee5\u901a\u8fc7gf2bv\u5199\u51fa\u5982\u4e0b\u6c42\u89e3\u4ee3\u7801\uff1a<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>from Crypto.Util.number import *\nfrom gf2bv import *\nfrom gf2bv.crypto.mt import *\nfrom tqdm import *\nimport sys\nsys.set_int_max_str_digits(10000000)\n\ngift = \"...\"\ngift = int(gift)\nc = \")9Lsu_4s_eb__otEli_nhe_tes5gii5sT@omamkn__ari{efm0__rmu_nt(0Eu3_En_og5rfoh}nkeoToy_bthguuEh7___u\"\ngiftstr = long_to_bytes(gift)\n\ns = [giftstr&#91;i: i+2&#93; for i in range(0, len(giftstr), 2)]\n\nLS = LinearSystem(&#91;32&#93; * 624)\nmt = LS.gens()\nRNG = MT19937(mt)\n\ntmp = []\nfor x in s:\n    RNG.getrandbits(8)\n    tmp.append((RNG.getrandbits(16) >> 8) ^ x&#91;0&#93;)\ntmp.append(mt&#91;0&#93; ^ 0x80000000)\n\nsols = LS.solve_all(tmp)\n\nfor sol in sols:\n    rng = MT19937(sol).to_python_random()\n    try:\n        for i in range(3108):\n            r1 = rng.getrandbits(8)\n            r2 = rng.getrandbits(16)\n            assert (pow(r1, 2*i, 257) &amp; 0xff) ^ r2 == bytes_to_long(s&#91;i&#93;)\n\n        x = &#91;i for i in range(len(c))&#93;\n        for i in range(2025):\n            rng.shuffle(x)\n\n        flag = \"\"\n        for i in range(len(c)):\n            flag += c&#91;x.index(i)&#93;\n        print(flag)\n        break\n    except:\n        pass<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Util<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">number <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> gf2bv <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> gf2bv<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">mt <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> tqdm <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> sys<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">sys<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">set_int_max_str_digits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">10000000<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">gift <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">...<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">gift <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">gift<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">c <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">)9Lsu_4s_eb__otEli_nhe_tes5gii5sT@omamkn__ari{efm0__rmu_nt(0Eu3_En_og5rfoh}nkeoToy_bthguuEh7___u<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">giftstr <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">long_to_bytes<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">gift<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">s <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #D8DEE9FF\">giftstr<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\"> i<\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">giftstr<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)]<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">LS <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">LinearSystem<\/span><span style=\"color: #ECEFF4\">(&#91;<\/span><span style=\"color: #B48EAD\">32<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">624<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">mt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> LS<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">gens<\/span><span style=\"color: #ECEFF4\">()<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">RNG <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">MT19937<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">mt<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">tmp <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> x <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> s<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    RNG<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">8<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    tmp<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">((<\/span><span style=\"color: #D8DEE9FF\">RNG<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">16<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">&gt;&gt;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">8<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">^<\/span><span style=\"color: #D8DEE9FF\"> x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">tmp<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">mt<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">^<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">0x<\/span><span style=\"color: #B48EAD\">80000000<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">sols <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> LS<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">solve_all<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">tmp<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> sol <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> sols<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    rng <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">MT19937<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">sol<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">to_python_random<\/span><span style=\"color: #ECEFF4\">()<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">try<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3108<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            r1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> rng<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">8<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            r2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> rng<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">16<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">assert<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">pow<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">r1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">257<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">&amp;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">0x<\/span><span style=\"color: #B48EAD\">ff<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">^<\/span><span style=\"color: #D8DEE9FF\"> r2 <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">bytes_to_long<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">s<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">c<\/span><span style=\"color: #ECEFF4\">))&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2025<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            rng<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">shuffle<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        flag <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;&quot;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">c<\/span><span style=\"color: #ECEFF4\">)):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            flag <\/span><span style=\"color: #81A1C1\">+=<\/span><span style=\"color: #D8DEE9FF\"> c<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">index<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">flag<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">break<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">except<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">pass<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u53ef\u4ee5\u6c42\u89e3\u51faflag\uff1a<code>flag{7hE_numbEr_0f_biT5_i5_Enou9h_@L5o_ThE_r4nk_must_3n0ugh}<\/code><br>\u5f53\u7136\uff0c\u6839\u636e\u7c7b\u4f3c<a href=\"https:\/\/tangcuxiaojikuai.xyz\/post\/69eaef2e.html#Random-game-3-Solves-376-500-pts\">\u540c\u6d4e\u5927\u5b66\u7b2c\u4e8c\u5c4a\u7f51\u7edc\u5b89\u5168\u65b0\u751f\u8d5bCatCTF\u7684Random game<\/a>\u4e00\u9898\u7684\u5206\u6790\u65b9\u6cd5\u53ef\u4ee5\u83b7\u5f97\u66f4\u591a\u5df2\u77e5\u4f4d\uff0c\u8fd9\u91cc\u4e0d\u8fc7\u591a\u8d58\u8ff0.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">sk\uff08\u8d5b\u540e\u590d\u73b0\uff09<\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u5f88\u9057\u61be\u6bd4\u8d5b\u7684\u65f6\u5019\u6ca1\u505a\u51fa\u6765<br>\u590d\u73b0docker\u955c\u50cf\uff1atriodelzx0916\/qwb2025sk<\/p>\n<\/blockquote>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>from random import randint  \nfrom Crypto.Util.number import getPrime, inverse, long_to_bytes, bytes_to_long  \nfrom math import gcd  \nimport signal  \nfrom secret import flag  \n  \ndef gen_coprime_num(pbits):  \n    lbits = 2 * pbits + 8  \n    lb = 2**lbits  \n    ub = 2**(lbits + 1)  \n    while True:  \n        r = randint(lb, ub)  \n        s = randint(lb, ub)  \n        if gcd(r, s) == 1:  \n            return r, s  \n  \ndef mult_mod(A, B, p):  \n    result = &#91;0&#93; * (len(A) + len(B) - 1)  \n    for i in range(len(A)):  \n        for j in range(len(B)):  \n            result&#91;i + j&#93; = (result&#91;i + j&#93; + A&#91;i&#93; * B&#91;j&#93;) % p  \n      \n    return result  \n  \ndef gen_key(p):  \n    f = &#91;randint(1, 2**128) for i in \":)\"&#93;  \n    h = &#91;randint(1, 2**128) for i in \":(\"&#93;  \n      \n    R1, S1 = gen_coprime_num(p.bit_length())  \n    R2, S2 = gen_coprime_num(p.bit_length())  \n  \n    B = [&#91;randint(1, p - 1) for i in \":(\"&#93; for j in \":)\"]  \n  \n    P = []  \n    for b in B:  \n        P.append(mult_mod(f, b, p))  \n      \n    Q = []  \n    for b in B:  \n        Q.append(mult_mod(h, b, p))  \n  \n    for i in range(len(P)):  \n        for j in range(len(P&#91;i&#93;)):  \n            P&#91;i&#93;&#91;j&#93; = P&#91;i&#93;&#91;j&#93; * R1 % S1  \n            Q&#91;i&#93;&#91;j&#93; = Q&#91;i&#93;&#91;j&#93; * R2 % S2  \n  \n    sk = &#91;(R1, S1), (R2, S2), f, h, p&#93;  \n    pk = &#91;P, Q, p&#93;  \n  \n    return sk, pk  \n  \ndef encrypt(pk, pt):  \n    P, Q, p = pk  \n    pt = bytes_to_long(pt)  \n  \n    PP = 0  \n    QQ = 0  \n  \n    for i in range(len(P)):  \n        u = randint(1, p)  \n        for j in range(len(P&#91;0&#93;)):  \n            PP = PP + P&#91;i&#93;&#91;j&#93; * (u * pt**j % p)  \n            QQ = QQ + Q&#91;i&#93;&#91;j&#93; * (u * pt**j % p)  \n  \n    return PP, QQ  \n  \ndef decrypt(sk, ct):  \n    RS1, RS2, f, h, p = sk  \n    R1, S1 = RS1  \n    R2, S2 = RS2  \n  \n    P, Q = ct  \n    invR1 = inverse(R1, S1)  \n    invR2 = inverse(R2, S2)  \n    P = (P * invR1 % S1) % p  \n    Q = (Q * invR2 % S2) % p  \n  \n    f0q = f&#91;0&#93; * Q % p  \n    f1q = f&#91;1&#93; * Q % p  \n    h0p = h&#91;0&#93; * P % p  \n    h1p = h&#91;1&#93; * P % p  \n  \n    a = f1q + p - h1p % p  \n    b = f0q + p - h0p % p  \n  \n    pt = -b * inverse(a, p) % p  \n    pt = long_to_bytes(pt)  \n  \n    return pt  \n  \nsignal.alarm(30)  \np = getPrime(256)  \nsk, pk = gen_key(p)  \nticket = long_to_bytes(randint(1, p))  \nenc_ticket = encrypt(pk, ticket)  \nprint(pk)  \nprint(enc_ticket)  \n  \nfor i in range(2):  \n    op = int(input(\"op>\").strip())  \n    if op == 1:  \n        msg = input(\"pt:\").strip().encode()  \n        ct = encrypt(pk, msg)  \n        print(f\"ct: {ct}\")  \n    elif op == 2:  \n        user_input = input(\"ct:\").strip().split(\" \")  \n        if len(user_input) == 2:  \n            ct = [int(user_input&#91;0&#93;), int(user_input&#91;1&#93;)]  \n        else:  \n            print(\"invalid ct.\")  \n            break  \n        user_input = input(\"your f:\").strip().split(\" \")  \n        if len(user_input) == 2:  \n            user_f = [int(user_input&#91;0&#93;), int(user_input&#91;1&#93;)]  \n        else:  \n            print(\"invalid f.\")  \n            break  \n  \n        user_input = input(\"your h:\").strip().split(\" \")  \n        if len(user_input) == 2:  \n            user_h = [int(user_input&#91;0&#93;), int(user_input&#91;1&#93;)]  \n        else:  \n            print(\"invalid h.\")  \n            break  \n  \n        user_input = input(\"your R1 S1:\").strip().split(\" \")  \n        if len(user_input) == 2:  \n            user_r1s1 = [int(user_input&#91;0&#93;), int(user_input&#91;1&#93;)]  \n        else:  \n            print(\"invalid R1 S1.\")  \n            break  \n  \n        user_input = input(\"your R2 S2:\").strip().split(\" \")  \n        if len(user_input) == 2:  \n            user_r2s2 = [int(user_input&#91;0&#93;), int(user_input&#91;1&#93;)]  \n        else:  \n            print(\"invalid R2 S2.\")  \n            break  \n  \n        pt = decrypt((user_r1s1, user_r2s2, user_f, user_h, p), ct)  \n        if pt == ticket:  \n            print(flag)  \n        else:  \n            print(pt.hex())  \n    else:  \n        print(\"invalid op.\")  \n        break  \n  \nprint(\"bye!\")<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> random <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> randint  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Util<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">number <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> getPrime<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> inverse<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> long_to_bytes<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> bytes_to_long  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> math <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> gcd  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> signal  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> secret <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> flag  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_coprime_num<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">pbits<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    lbits <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> pbits <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">8<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    lb <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\">lbits  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    ub <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">lbits <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">while<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">True<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        r <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">lb<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> ub<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        s <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">lb<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> ub<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gcd<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">r<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> s<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> r<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> s  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">mult_mod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">A<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">B<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">p<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    result <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">A<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">B<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">A<\/span><span style=\"color: #ECEFF4\">)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">B<\/span><span style=\"color: #ECEFF4\">)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            result<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">result<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> A<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> B<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">      <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> result  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_key<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">p<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    f <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #B48EAD\">128<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">:)<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    h <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #B48EAD\">128<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">:(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">      <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    R1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_coprime_num<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">bit_length<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    R2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_coprime_num<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">bit_length<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    B <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">:(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">:)<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    P <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> b <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> B<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        P<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">mult_mod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> b<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">      <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> b <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> B<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        Q<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">mult_mod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">h<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> b<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> R1 <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> S1  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            Q<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> R2 <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> S2  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    sk <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;(<\/span><span style=\"color: #D8DEE9FF\">R1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S1<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">R2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S2<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> f<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> h<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pk <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> sk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pk  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">encrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">pk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">pt<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    P<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> pk  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">bytes_to_long<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    PP <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    QQ <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        u <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            PP <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> PP <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">u <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> pt<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\">j <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            QQ <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> QQ <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">u <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> pt<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\">j <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> PP<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> QQ  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">decrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">sk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">ct<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    RS1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> RS2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> f<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> h<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> sk  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    R1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> RS1  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    R2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> RS2  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    P<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> ct  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    invR1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">inverse<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">R1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    invR2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">inverse<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">R2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S2<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    P <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> invR1 <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> S1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">Q <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> invR2 <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> S2<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    f0q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> f<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> Q <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    f1q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> f<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> Q <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    h0p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> h<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> P <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    h1p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> h<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> P <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    a <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> f1q <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> h1p <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    b <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> f0q <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> h0p <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\">b <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">inverse<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">a<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">long_to_bytes<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> pt  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">signal<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">alarm<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">30<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">getPrime<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">256<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">sk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pk <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_key<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">ticket <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">long_to_bytes<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">enc_ticket <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">encrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> ticket<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pk<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">enc_ticket<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    op <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">input<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">op&gt;<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">strip<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> op <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        msg <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">input<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">pt:<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">strip<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">encode<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        ct <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">encrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> msg<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&quot;ct: <\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">ct<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">elif<\/span><span style=\"color: #D8DEE9FF\"> op <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        user_input <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">input<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">ct:<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">strip<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">split<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            ct <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;)]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">else<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">invalid ct.<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">break<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        user_input <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">input<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">your f:<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">strip<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">split<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            user_f <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;)]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">else<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">invalid f.<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">break<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        user_input <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">input<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">your h:<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">strip<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">split<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            user_h <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;)]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">else<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">invalid h.<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">break<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        user_input <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">input<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">your R1 S1:<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">strip<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">split<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            user_r1s1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;)]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">else<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">invalid R1 S1.<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">break<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        user_input <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">input<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">your R2 S2:<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">strip<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">split<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            user_r2s2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">user_input<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;)]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">else<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">invalid R2 S2.<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">break<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        pt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">decrypt<\/span><span style=\"color: #ECEFF4\">((<\/span><span style=\"color: #D8DEE9FF\">user_r1s1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> user_r2s2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> user_f<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> user_h<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> ct<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> pt <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> ticket<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">flag<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">else<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">hex<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">else<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">invalid op.<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">break<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">bye!<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u56e0\u4e3a\u7ed9\u6211\u4eec\u7684\u53ea\u6709\u516c\u94a5\\(P,Q,p\\)\uff0c\u6211\u4eec\u8ddf\u8e2a\u4e00\u4e0b\u5bc6\u94a5\u751f\u6210\u7684\u6b65\u9aa4\uff1a<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>def mult_mod(A, B):\n\u00a0 \u00a0 result = &#91;0&#93; * (len(A) + len(B) - 1)\n\u00a0 \u00a0 for i in range(len(A)):\n\u00a0 \u00a0 \u00a0 \u00a0 for j in range(len(B)):\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 result&#91;i + j&#93; = (result&#91;i + j&#93; + A&#91;i&#93; * B&#91;j&#93;)\n\u00a0 \u00a0 return result\n\u00a0 \u00a0 \nf = &#91;var(f'f{i}') for i in range(2)&#93;\nh = &#91;var(f'h{i}') for i in range(2)&#93;\n\nR1, S1 = var('R1 S1')\nR2, S2 = var('R2 S2')\n\nB = [&#91;var(f'B{i}{j}') for j in range(2)&#93; for i in range(2)]\npt = var('pt')\nu = var('u')\n\nP = []\nfor b in B:\n\u00a0 \u00a0 P.append(mult_mod(f, b))\n\nQ = []\nfor b in B:\n\u00a0 \u00a0 Q.append(mult_mod(h, b))\n\nprint(P)\nprint(Q)<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">mult_mod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">A<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">B<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">\u00a0 \u00a0 result <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">A<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">B<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">\u00a0 \u00a0 <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">A<\/span><span style=\"color: #ECEFF4\">)):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">\u00a0 \u00a0 \u00a0 \u00a0 <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">B<\/span><span style=\"color: #ECEFF4\">)):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 result<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">result<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> A<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> B<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">\u00a0 \u00a0 <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> result<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">\u00a0 \u00a0 <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&#39;f<\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">h <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&#39;h<\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">R1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">R1 S1<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">R2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">R2 S2<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">B <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&#39;B<\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #EBCB8B\">}{<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">pt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">pt<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">u <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">u<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">P <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> b <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> B<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">\u00a0 \u00a0 P<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">mult_mod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> b<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> b <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> B<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">\u00a0 \u00a0 Q<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">mult_mod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">h<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> b<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">Q<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u53ef\u4ee5\u5f97\u5230\u5728\u8fd8\u672a\u4e58\u4e0a\\(R_1,R_2\\)\u7684\u65f6\u5019\u6709\uff1a<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>P = [&#91;B00*f0, B01*f0 + B00*f1, B01*f1&#93;, &#91;B10*f0, B11*f0 + B10*f1, B11*f1&#93;]\nQ = [&#91;B00*h0, B01*h0 + B00*h1, B01*h1&#93;, &#91;B10*h0, B11*h0 + B10*h1, B11*h1&#93;]<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #D8DEE9FF\">P <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #D8DEE9FF\">B00<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">f0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> B01<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">f0 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> B00<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">f1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> B01<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">f1<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">B10<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">f0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> B11<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">f0 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> B10<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">f1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> B11<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">f1<\/span><span style=\"color: #ECEFF4\">&#93;]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #D8DEE9FF\">B00<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">h0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> B01<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">h0 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> B00<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">h1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> B01<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">h1<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">B10<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">h0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> B11<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">h0 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> B10<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">h1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> B11<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">h1<\/span><span style=\"color: #ECEFF4\">&#93;]<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u56e0\u4e3a\\(R_1,S_1,R_2,S_2\\)\u672a\u77e5\uff0c\u6240\u4ee5\u6211\u4eec\u53ea\u77e5\u9053\\(P,Q\\)\u6700\u7ec8\u662f\u4e00\u4e2a\\(2\\times3\\)\u7684\u77e9\u9635\uff08\u5176\u5b9e\u770b\u8f93\u51fa\u4e5f\u80fd\u770b\u51fa\u6765\uff09\uff0c\u518d\u901a\u8fc7\u5982\u4e0b\u4ee3\u7801\u8ddf\u8e2a\u52a0\u5bc6\u51fd\u6570\uff1a<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>P = [&#91;var(f\"P{i}{j}\") for j in range(3)&#93; for i in range(2)]\nQ = [&#91;var(f\"Q{i}{j}\") for j in range(3)&#93; for i in range(2)]\n\npt = var(\"pt\")\nu = var(\"u\")\n\nPP = 0\nQQ = 0\n\nfor i in range(len(P)):\n\u00a0 \u00a0 # u = randint(1, p)\n\u00a0 \u00a0 for j in range(len(P&#91;0&#93;)):\n\u00a0 \u00a0 \u00a0 \u00a0 PP = PP + P&#91;i&#93;&#91;j&#93; * (u * pt**j)\n\u00a0 \u00a0 \u00a0 \u00a0 QQ = QQ + Q&#91;i&#93;&#91;j&#93; * (u * pt**j)\n\nprint(PP)\nprint(QQ)<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #D8DEE9FF\">P <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&quot;P<\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #EBCB8B\">}{<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&quot;Q<\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #EBCB8B\">}{<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)]<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">pt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">pt<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">u <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">u<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">PP <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">QQ <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">)):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">\u00a0 \u00a0 <\/span><span style=\"color: #616E88\"># u = randint(1, p)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">\u00a0 \u00a0 <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;)):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">\u00a0 \u00a0 \u00a0 \u00a0 PP <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> PP <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">u <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> pt<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">\u00a0 \u00a0 \u00a0 \u00a0 QQ <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> QQ <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">u <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> pt<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">PP<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">QQ<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>PP = P02*pt^2*u0 + P12*pt^2*u1 + P01*pt*u0 + P11*pt*u1 + P00*u0 + P10*u1\nQQ = Q02*pt^2*u0 + Q12*pt^2*u1 + Q01*pt*u0 + Q11*pt*u1 + Q00*u0 + Q10*u1<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #D8DEE9FF\">PP <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> P02<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #81A1C1\">^<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">u0 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> P12<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #81A1C1\">^<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">u1 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> P01<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">u0 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> P11<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">u1 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> P00<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">u0 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> P10<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">u1<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">QQ <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> Q02<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #81A1C1\">^<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">u0 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> Q12<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #81A1C1\">^<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">u1 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> Q01<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">u0 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> Q11<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">u1 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> Q00<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">u0 <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> Q10<\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">u1<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u6309\u539f\u4ee3\u7801\u6574\u7406\u53ef\u5f97\uff1a<\/p>\n\n\n\n<p>$$<br>\\begin{aligned}<br>PP&amp;=P_{02}\\cdot (pt^2\\cdot u_0\\mod{p})+P_{12}\\cdot (pt^2\\cdot u_1 \\mod{p})+P_{01}\\cdot (pt\\cdot u_0\\mod{p})+P_{11} \\cdot (pt \\cdot u_1\\mod{p}) + P_{00} \\cdot (u_0\\mod{p})+P_{10}\\cdot (u_1\\mod{p})\\\\<br>QQ&amp;=Q_{02}\\cdot (pt^2\\cdot u_0\\mod{p})+Q_{12}\\cdot (pt^2\\cdot u_1 \\mod{p})+Q_{01}\\cdot (pt\\cdot u_0\\mod{p})+Q_{11} \\cdot (pt \\cdot u_1\\mod{p}) + Q_{00} \\cdot (u_0\\mod{p})+Q_{10}\\cdot (u_1\\mod{p})<br>\\end{aligned}<br>$$<\/p>\n\n\n\n<p>\u5f88\u663e\u7136\u5b58\u5728\u7ebf\u6027\u5173\u7cfb\uff0c\u53ef\u4ee5\u6784\u9020\u5982\u4e0b\u65b9\u7a0b\u7ec4\uff1a<\/p>\n\n\n\n<p>$$<br>\\begin{aligned}<br>PP&amp;=P_{02}x_{02}+P_{12}x_{12}+P_{01}x_{01}+P_{11}x_{11}+P_{00}x_{00}+P_{10}x_{10}\\\\<br>QQ&amp;=Q_{02}x_{02}+Q_{12}x_{12}+Q_{01}x_{01}+Q_{11}x_{11}+Q_{00}x_{00}+Q_{10}x_{10}<br>\\end{aligned}<br>$$<\/p>\n\n\n\n<p>\u9884\u671f\u6709\\(x_{ij}=u_{i}pt^{j}\\mod{p}\\)\uff0c\u5c1d\u8bd5\u6784\u9020\u6570\u636e\u7528cuso\u6c42\u89e3\uff1a<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>from random import *  \nfrom Crypto.Util.number import *  \nfrom math import *  \nfrom sage.all import *  \nimport cuso  \n  \n  \ndef gen_coprime_num(pbits):  \n    lbits = 2 * pbits + 8  \n    lb = 2 ** lbits  \n    ub = 2 ** (lbits + 1)  \n    while True:  \n        r = randint(lb, ub)  \n        s = randint(lb, ub)  \n        if gcd(r, s) == 1:  \n            return r, s  \n  \n  \ndef mult_mod(A, B, p):  \n    result = &#91;0&#93; * (len(A) + len(B) - 1)  \n    for i in range(len(A)):  \n        for j in range(len(B)):  \n            result&#91;i + j&#93; = (result&#91;i + j&#93; + A&#91;i&#93; * B&#91;j&#93;) % p  \n  \n    return result  \n  \n  \ndef gen_key(p):  \n    f = &#91;randint(1, 2 ** 128) for i in \":)\"&#93;  \n    h = &#91;randint(1, 2 ** 128) for i in \":(\"&#93;  \n  \n    R1, S1 = gen_coprime_num(p.bit_length())  \n    R2, S2 = gen_coprime_num(p.bit_length())  \n  \n    B = [&#91;randint(1, p - 1) for i in \":(\"&#93; for j in \":)\"]  \n  \n    P = []  \n    for b in B:  \n        P.append(mult_mod(f, b, p))  \n  \n    Q = []  \n    for b in B:  \n        Q.append(mult_mod(h, b, p))  \n  \n    for i in range(len(P)):  \n        for j in range(len(P&#91;i&#93;)):  \n            P&#91;i&#93;&#91;j&#93; = P&#91;i&#93;&#91;j&#93; * R1 % S1  \n            Q&#91;i&#93;&#91;j&#93; = Q&#91;i&#93;&#91;j&#93; * R2 % S2  \n  \n    sk = &#91;(R1, S1), (R2, S2), f, h, p&#93;  \n    pk = &#91;P, Q, p&#93;  \n  \n    return sk, pk  \n  \n  \ndef encrypt(pk, pt):  \n    P, Q, p = pk  \n    pt = bytes_to_long(pt)  \n  \n    PP = 0  \n    QQ = 0  \n  \n    for i in range(len(P)):  \n        u = randint(1, p)  \n        for j in range(len(P&#91;0&#93;)):  \n            PP = PP + P&#91;i&#93;&#91;j&#93; * (u * pt ** j % p)  \n            QQ = QQ + Q&#91;i&#93;&#91;j&#93; * (u * pt ** j % p)  \n  \n    return PP, QQ  \n  \n  \ndef decrypt(sk, ct):  \n    RS1, RS2, f, h, p = sk  \n    R1, S1 = RS1  \n    R2, S2 = RS2  \n  \n    P, Q = ct  \n    invR1 = inverse(R1, S1)  \n    invR2 = inverse(R2, S2)  \n    P = (P * invR1 % S1) % p  \n    Q = (Q * invR2 % S2) % p  \n  \n    f0q = f&#91;0&#93; * Q % p  \n    f1q = f&#91;1&#93; * Q % p  \n    h0p = h&#91;0&#93; * P % p  \n    h1p = h&#91;1&#93; * P % p  \n  \n    a = f1q + p - h1p % p  \n    b = f0q + p - h0p % p  \n  \n    pt = -b * inverse(a, p) % p  \n    pt = long_to_bytes(pt)  \n  \n    return pt  \n  \n  \np = getPrime(256)  \nsk, pk = gen_key(p)  \nticket = long_to_bytes(randint(1, p))  \nenc_ticket = encrypt(pk, ticket)  \n  \nx = [&#91;var(f\"x{i}{j}\") for j in range(3)&#93; for i in range(2)]  \n  \nPP, QQ = enc_ticket  \nP, Q, p = pk  \n  \nf1 = sum([P&#91;i&#93;&#91;j&#93; * x&#91;i&#93;&#91;j&#93; for i in range(2) for j in range(3)]) - PP  \nf2 = sum([Q&#91;i&#93;&#91;j&#93; * x&#91;i&#93;&#91;j&#93; for i in range(2) for j in range(3)]) - QQ  \n  \nrelations = &#91;f1, f2&#93;  \nbounds = {x&#91;i&#93;&#91;j&#93;: (0, p - 1) for i in range(2) for j in range(3)}  \n  \nroots = cuso.find_small_roots(relations, bounds)  \n# print(roots)  \n  \nfor root in roots:  \n    u0 = root[x&#91;0&#93;&#91;0&#93;]  \n    pt_u0 = root[x&#91;0&#93;&#91;1&#93;]  \n    u1 = root[x&#91;1&#93;&#91;0&#93;]  \n    pt_u1 = root[x&#91;1&#93;&#91;1&#93;]  \n  \n    pt0 = long_to_bytes(pt_u0 * inverse(u0, p) % p)  \n    pt1 = long_to_bytes(pt_u1 * inverse(u1, p) % p)  \n    if pt0 == pt1:  \n        pt = pt0  \n        print(pt == ticket)  \n        break<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> random <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Util<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">number <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> math <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> sage<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">all <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> cuso  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_coprime_num<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">pbits<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    lbits <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> pbits <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">8<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    lb <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> lbits  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    ub <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">lbits <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">while<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">True<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        r <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">lb<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> ub<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        s <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">lb<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> ub<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gcd<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">r<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> s<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> r<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> s  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">mult_mod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">A<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">B<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">p<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    result <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">A<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">B<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">A<\/span><span style=\"color: #ECEFF4\">)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">B<\/span><span style=\"color: #ECEFF4\">)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            result<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">result<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> A<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> B<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> result  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_key<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">p<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    f <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">128<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">:)<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    h <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">128<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">:(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    R1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_coprime_num<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">bit_length<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    R2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_coprime_num<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">bit_length<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    B <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">:(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">:)<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    P <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> b <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> B<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        P<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">mult_mod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> b<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> b <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> B<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        Q<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">mult_mod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">h<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> b<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> R1 <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> S1  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            Q<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> R2 <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> S2  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    sk <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;(<\/span><span style=\"color: #D8DEE9FF\">R1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S1<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">R2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S2<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> f<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> h<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pk <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> sk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pk  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">encrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">pk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">pt<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    P<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> pk  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">bytes_to_long<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    PP <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    QQ <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        u <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            PP <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> PP <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">u <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> pt <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            QQ <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> QQ <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">u <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> pt <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> PP<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> QQ  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">decrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">sk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">ct<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    RS1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> RS2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> f<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> h<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> sk  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    R1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> RS1  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    R2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> RS2  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    P<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> ct  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    invR1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">inverse<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">R1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    invR2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">inverse<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">R2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S2<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    P <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> invR1 <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> S1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">Q <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> invR2 <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> S2<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    f0q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> f<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> Q <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    f1q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> f<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> Q <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    h0p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> h<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> P <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    h1p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> h<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> P <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    a <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> f1q <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> h1p <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    b <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> f0q <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> h0p <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\">b <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">inverse<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">a<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">long_to_bytes<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> pt  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">getPrime<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">256<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">sk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pk <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_key<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">ticket <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">long_to_bytes<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">enc_ticket <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">encrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> ticket<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&quot;x<\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #EBCB8B\">}{<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">PP<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> QQ <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> enc_ticket  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> pk  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">sum<\/span><span style=\"color: #ECEFF4\">([<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">)])<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> PP  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">sum<\/span><span style=\"color: #ECEFF4\">([<\/span><span style=\"color: #D8DEE9FF\">Q<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">)])<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> QQ  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">relations <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">f1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> f2<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">bounds <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">{<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;:<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">)}<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">roots <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> cuso<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">find_small_roots<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">relations<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> bounds<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #616E88\"># print(roots)  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> root <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> roots<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    u0 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> root<\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt_u0 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> root<\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    u1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> root<\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt_u1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> root<\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt0 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">long_to_bytes<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pt_u0 <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">inverse<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">u0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">long_to_bytes<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pt_u1 <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">inverse<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">u1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> pt0 <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> pt1<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        pt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> pt0  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pt <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> ticket<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">break<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u591a\u6b21\u8fd0\u884c\u53d1\u73b0\u786e\u5b9e\u53ef\u4ee5\u89e3\u51fa\u660e\u6587\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u5728\u83b7\u5f97\u660e\u6587\u4e4b\u540e\u81ea\u5df1\u6784\u9020\u5bc6\u94a5\u5bf9\u5bf9\u660e\u6587\u8fdb\u884c\u52a0\u5bc6\uff0c\u518d\u5c06\u5bc6\u6587\u548c\u81ea\u5df1\u6784\u9020\u7684\u79c1\u94a5\u6254\u56de\u9776\u673a\u5c31\u53ef\u4ee5\u4e86\uff1a<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>from random import *  \nfrom Crypto.Util.number import *  \nfrom math import *  \nfrom sage.all import *  \nimport cuso  \nfrom pwn import *  \n  \nio = remote(\"127.0.0.1\", 1145)  \n  \ndef gen_coprime_num(pbits):  \n    lbits = 2 * pbits + 8  \n    lb = 2 ** lbits  \n    ub = 2 ** (lbits + 1)  \n    while True:  \n        r = randint(lb, ub)  \n        s = randint(lb, ub)  \n        if gcd(r, s) == 1:  \n            return r, s  \n  \n  \ndef mult_mod(A, B, p):  \n    result = &#91;0&#93; * (len(A) + len(B) - 1)  \n    for i in range(len(A)):  \n        for j in range(len(B)):  \n            result&#91;i + j&#93; = (result&#91;i + j&#93; + A&#91;i&#93; * B&#91;j&#93;) % p  \n  \n    return result  \n  \n  \ndef gen_key(p):  \n    f = &#91;randint(1, 2 ** 128) for i in \":)\"&#93;  \n    h = &#91;randint(1, 2 ** 128) for i in \":(\"&#93;  \n  \n    R1, S1 = gen_coprime_num(p.bit_length())  \n    R2, S2 = gen_coprime_num(p.bit_length())  \n  \n    B = [&#91;randint(1, p - 1) for i in \":(\"&#93; for j in \":)\"]  \n  \n    P = []  \n    for b in B:  \n        P.append(mult_mod(f, b, p))  \n  \n    Q = []  \n    for b in B:  \n        Q.append(mult_mod(h, b, p))  \n  \n    for i in range(len(P)):  \n        for j in range(len(P&#91;i&#93;)):  \n            P&#91;i&#93;&#91;j&#93; = P&#91;i&#93;&#91;j&#93; * R1 % S1  \n            Q&#91;i&#93;&#91;j&#93; = Q&#91;i&#93;&#91;j&#93; * R2 % S2  \n  \n    sk = &#91;(R1, S1), (R2, S2), f, h, p&#93;  \n    pk = &#91;P, Q, p&#93;  \n  \n    return sk, pk  \n  \n  \ndef encrypt(pk, pt):  \n    P, Q, p = pk  \n    pt = bytes_to_long(pt)  \n  \n    PP = 0  \n    QQ = 0  \n  \n    for i in range(len(P)):  \n        u = randint(1, p)  \n        for j in range(len(P&#91;0&#93;)):  \n            PP = PP + P&#91;i&#93;&#91;j&#93; * (u * pt ** j % p)  \n            QQ = QQ + Q&#91;i&#93;&#91;j&#93; * (u * pt ** j % p)  \n  \n    return PP, QQ  \n  \n_pk = io.recvline().strip().decode()  \npk = eval(_pk)  \nP, Q, p = pk  \n_ct = io.recvline().strip().decode()  \nct = eval(_ct)  \nPP, QQ = ct  \n  \nx = [&#91;var(f\"x{i}{j}\") for j in range(3)&#93; for i in range(2)]  \n  \nf1 = sum([P&#91;i&#93;&#91;j&#93; * x&#91;i&#93;&#91;j&#93; for i in range(2) for j in range(3)]) - PP  \nf2 = sum([Q&#91;i&#93;&#91;j&#93; * x&#91;i&#93;&#91;j&#93; for i in range(2) for j in range(3)]) - QQ  \n  \nrelations = &#91;f1, f2&#93;  \nbounds = {x&#91;i&#93;&#91;j&#93;: (0, p - 1) for i in range(2) for j in range(3)}  \n  \nroots = cuso.find_small_roots(relations, bounds)  \n  \nfor root in roots:  \n    u0 = root[x&#91;0&#93;&#91;0&#93;]  \n    pt_u0 = root[x&#91;0&#93;&#91;1&#93;]  \n    u1 = root[x&#91;1&#93;&#91;0&#93;]  \n    pt_u1 = root[x&#91;1&#93;&#91;1&#93;]  \n  \n    pt0 = long_to_bytes(pt_u0 * inverse(u0, p) % p)  \n    pt1 = long_to_bytes(pt_u1 * inverse(u1, p) % p)  \n  \n    if pt0 == pt1:  \n        pt = pt0  \n        break  \n  \nsk, pk = gen_key(p)  \nnew_ct = encrypt(pk, pt)  \n  \nio.sendlineafter(b'op>', b\"2\")  \n  \n# &#91;(R1, S1), (R2, S2), f, h, p&#93;  \n  \nio.sendlineafter(b'ct:', f'{new_ct&#91;0&#93;} {new_ct&#91;1&#93;}'.encode())  \nio.sendlineafter(b'your f:', f'{sk&#91;2&#93;&#91;0&#93;} {sk&#91;2&#93;&#91;1&#93;}'.encode())  \nio.sendlineafter(b'your h:', f'{sk&#91;3&#93;&#91;0&#93;} {sk&#91;3&#93;&#91;1&#93;}'.encode())  \nio.sendlineafter(b'your R1 S1:', f'{sk&#91;0&#93;&#91;0&#93;} {sk&#91;0&#93;&#91;1&#93;}'.encode())  \nio.sendlineafter(b'your R2 S2:', f'{sk&#91;1&#93;&#91;0&#93;} {sk&#91;1&#93;&#91;1&#93;}'.encode())  \n  \nio.interactive()<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> random <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Util<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">number <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> math <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> sage<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">all <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> cuso  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> pwn <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">io <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">remote<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">127.0.0.1<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1145<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_coprime_num<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">pbits<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    lbits <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> pbits <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">8<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    lb <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> lbits  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    ub <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">lbits <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">while<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">True<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        r <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">lb<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> ub<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        s <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">lb<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> ub<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gcd<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">r<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> s<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> r<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> s  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">mult_mod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">A<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">B<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">p<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    result <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">A<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">B<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">A<\/span><span style=\"color: #ECEFF4\">)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">B<\/span><span style=\"color: #ECEFF4\">)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            result<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">result<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> A<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> B<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> result  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_key<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">p<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    f <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">128<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">:)<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    h <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">128<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">:(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    R1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_coprime_num<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">bit_length<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    R2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_coprime_num<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">bit_length<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    B <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">:(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">:)<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    P <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> b <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> B<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        P<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">mult_mod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> b<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> b <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> B<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        Q<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">mult_mod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">h<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> b<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> R1 <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> S1  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            Q<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> R2 <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> S2  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    sk <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;(<\/span><span style=\"color: #D8DEE9FF\">R1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S1<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">R2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> S2<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> f<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> h<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pk <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> sk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pk  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">encrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">pk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">pt<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    P<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> pk  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">bytes_to_long<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pt<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    PP <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    QQ <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        u <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">randint<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">len<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;)):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            PP <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> PP <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">u <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> pt <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            QQ <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> QQ <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">u <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> pt <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> PP<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> QQ  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">_pk <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> io<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">recvline<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">strip<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">decode<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">pk <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">eval<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">_pk<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> pk  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">_ct <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> io<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">recvline<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">strip<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">decode<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">ct <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">eval<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">_ct<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">PP<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> QQ <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> ct  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #88C0D0\">var<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&quot;x<\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #EBCB8B\">}{<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">sum<\/span><span style=\"color: #ECEFF4\">([<\/span><span style=\"color: #D8DEE9FF\">P<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">)])<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> PP  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">f2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">sum<\/span><span style=\"color: #ECEFF4\">([<\/span><span style=\"color: #D8DEE9FF\">Q<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">)])<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> QQ  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">relations <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">f1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> f2<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">bounds <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">{<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #D8DEE9FF\">i<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #D8DEE9FF\">j<\/span><span style=\"color: #ECEFF4\">&#93;:<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> i <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> j <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">)}<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">roots <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> cuso<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">find_small_roots<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">relations<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> bounds<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> root <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> roots<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    u0 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> root<\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt_u0 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> root<\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    u1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> root<\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt_u1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> root<\/span><span style=\"color: #ECEFF4\">[<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;]<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt0 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">long_to_bytes<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pt_u0 <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">inverse<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">u0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    pt1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">long_to_bytes<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pt_u1 <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">inverse<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">u1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">%<\/span><span style=\"color: #D8DEE9FF\"> p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> pt0 <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> pt1<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        pt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> pt0  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">break<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">sk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pk <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">gen_key<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">new_ct <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">encrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pk<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pt<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">io<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">sendlineafter<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">op&gt;<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">2<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #616E88\"># &#91;(R1, S1), (R2, S2), f, h, p&#93;  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">io<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">sendlineafter<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">ct:<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">new_ct<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\"> <\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">new_ct<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">encode<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">io<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">sendlineafter<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">your f:<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">sk<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\"> <\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">sk<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">encode<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">io<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">sendlineafter<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">your h:<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">sk<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\"> <\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">sk<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">3<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">encode<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">io<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">sendlineafter<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">your R1 S1:<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">sk<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\"> <\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">sk<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">encode<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">io<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">sendlineafter<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #A3BE8C\">your R2 S2:<\/span><span style=\"color: #ECEFF4\">&#39;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">f<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">sk<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\"> <\/span><span style=\"color: #EBCB8B\">{<\/span><span style=\"color: #D8DEE9FF\">sk<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #EBCB8B\">}<\/span><span style=\"color: #A3BE8C\">&#39;<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">encode<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">io<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">interactive<\/span><span style=\"color: #ECEFF4\">()<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u53ef\u4ee5\u5f97\u5230flag\uff08\u6709\u6982\u7387\u4e0d\u6210\u529f\uff09\uff1a<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><a href=\"http:\/\/www.triode.cc\/wp-content\/uploads\/2025\/10\/Pasted-image-20251021180507.png\"><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='http:\/\/www.triode.cc\/wp-content\/uploads\/2025\/10\/Pasted-image-20251021180507.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" width=\"255\" height=\"56\" data-original=\"http:\/\/www.triode.cc\/wp-content\/uploads\/2025\/10\/Pasted-image-20251021180507.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" class=\"wp-image-354\"\/><\/div><\/a><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\">Blurred\uff08\u8d5b\u540e\u590d\u73b0\uff09<\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u633a\u597d\u7684\u4e00\u9053\u6837\u672c\u533a\u5206\u9898<br>\u590d\u73b0docker\u955c\u50cf\uff1atriodelzx0916\/blurred<\/p>\n<\/blockquote>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>from sage.all import *  \nfrom sage.misc import prandom  \nimport random  \nfrom Crypto.Cipher import AES  \nfrom Crypto.Hash import SHA256  \nfrom Crypto.Util.Padding import pad  \n  \nq = 1342261  \nn = 1031  \nPR = PolynomialRing(Zmod(q), \"x\")  \nx = PR.gens()&#91;0&#93;  \nQ = PR.quotient(x**n + 1)  \nflag = b\"flag{*****************************}\"  \n  \ndef sample(rand):  \n    return (rand.getrandbits(1) - rand.getrandbits(1)) * (rand.getrandbits(1) * rand.getrandbits(1))  \n  \n  \ndef GenNTRU():  \n    f = &#91;sample(prandom) for _ in range(n)&#93;  \n    g1 = &#91;sample(prandom)for _ in range(n)&#93;  \n    g2 = &#91;sample(prandom) for _ in range(n)&#93;  \n    g1x = Q(g1)  \n    g2x = Q(g2)  \n  \n    while True:  \n        fx = Q(f)  \n        try:  \n            h1 = g1x \/ fx  \n            h2 = g2x \/ fx  \n            return (h1.lift(), h2.lift(), fx)  \n        except:  \n            prandom.shuffle(f)  \n            continue  \n  \nfor _ in range(20259):  \n    c = int(input(\"c :\"))  \n    if c == 1:  \n        coin = random.getrandbits(1)  \n        if coin == 0:  \n            pk1, pk2, fx = GenNTRU()  \n        else:  \n            pk1, pk2, fx = Q.random_element().lift(), Q.random_element().lift(), Q(&#91;sample(prandom) for _ in range(n)&#93;)  \n  \n        print(\"Hints:\", pk1.list(), pk2.list())  \n          \n    elif c == 2:  \n        SHA = SHA256.new()  \n        SHA.update(str(random.getrandbits(256)).encode())  \n        KEY = SHA.digest()  \n        cipher = AES.new(KEY, AES.MODE_ECB)  \n        flag = pad(flag, AES.block_size)  \n        print(\"Flag:\", cipher.encrypt(flag))  \n    else:  \n        break<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> sage<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">all <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> sage<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">misc <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> prandom  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> random  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Cipher <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> AES  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Hash <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> SHA256  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Util<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Padding <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> pad  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1342261<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">n <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1031<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">PR <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">PolynomialRing<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">Zmod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">q<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">x<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> PR<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">gens<\/span><span style=\"color: #ECEFF4\">()&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> PR<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">quotient<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\">n <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">flag <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">flag{*****************************}<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">rand<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">rand<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> rand<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">rand<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> rand<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">GenNTRU<\/span><span style=\"color: #ECEFF4\">():<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    f <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">prandom<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    g1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">prandom<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    g2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">prandom<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    g1x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">g1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    g2x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">g2<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">while<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">True<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        fx <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">try<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            h1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> g1x <\/span><span style=\"color: #81A1C1\">\/<\/span><span style=\"color: #D8DEE9FF\"> fx  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            h2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> g2x <\/span><span style=\"color: #81A1C1\">\/<\/span><span style=\"color: #D8DEE9FF\"> fx  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">h1<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #D8DEE9FF\"> h2<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #D8DEE9FF\"> fx<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">except<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            prandom<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">shuffle<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">continue<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">20259<\/span><span style=\"color: #ECEFF4\">):<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    c <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">int<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">input<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">c :<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> c <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        coin <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> random<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> coin <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            pk1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pk2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> fx <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">GenNTRU<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">else<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            pk1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pk2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> fx <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">random_element<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">random_element<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(&#91;<\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">prandom<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n<\/span><span style=\"color: #ECEFF4\">)&#93;)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">Hints:<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pk1<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">list<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #D8DEE9FF\"> pk2<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">list<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">          <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">elif<\/span><span style=\"color: #D8DEE9FF\"> c <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        SHA <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> SHA256<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">new<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        SHA<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">update<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">str<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">random<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">256<\/span><span style=\"color: #ECEFF4\">)).<\/span><span style=\"color: #88C0D0\">encode<\/span><span style=\"color: #ECEFF4\">())<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        KEY <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> SHA<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">digest<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        cipher <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> AES<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">new<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">KEY<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> AES<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">MODE_ECB<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        flag <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">pad<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">flag<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> AES<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">block_size<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">Flag:<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> cipher<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">encrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">flag<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">else<\/span><span style=\"color: #ECEFF4\">:<\/span><span style=\"color: #D8DEE9FF\">  <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">break<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u670920259\u6b21\u4ea4\u4e92\u673a\u4f1a\uff0c\u5bf9\u4e8e\u6bcf\u6b21\u4ea4\u4e92\uff0c\u82e5\u9009\u62e92\u4f1a\u901a\u8fc7<code>random.getrandbits<\/code>\u751f\u6210\u5bc6\u94a5\u6765\u52a0\u5bc6flag\u5e76\u53d1\u51fa\uff1b\u82e5\u9009\u62e91\uff0c\u5219\u4f1a\u901a\u8fc7<code>random.getrandbits<\/code>\u751f\u6210\u4e00\u4e2a1 bit\u7684\u968f\u673a\u6570\u4f5c\u4e3a\u786c\u5e01\u7684\u503c\uff0c\u82e5\u786c\u5e01\u503c\u4e3a0\u5219\u901a\u8fc7<code>GenNTRU<\/code>\u751f\u6210\\(pk_1, pk_2, f_x\\)\uff0c\u82e5\u786c\u5e01\u503c\u4e3a1\u5219\u76f4\u63a5\u751f\u6210\u4e24\u4e2a\u5546\u73af\\(\\frac{(\\mathbb{Z}\/q\\mathbb{Z})[x]}{x^n+1}\\)\u4e0a\u7684\u591a\u9879\u5f0f\u4f5c\u4e3a\\(pk_1,pk_2\\)\uff0c\u7136\u540e\u968f\u673a\u751f\u6210\u4e00\u4e2a\u4e09\u5143\u591a\u9879\u5f0f\\(f_x\\)\uff0c\u89c2\u5bdf<code>GenNTRU<\/code>\u53ef\u4ee5\u770b\u5230\u8be5\u51fd\u6570\u9996\u5148\u751f\u6210\u4e86\u4e09\u4e2a\u968f\u673a\u4e09\u5143\u591a\u9879\u5f0f\\(g_1,g_2,f_x\\)\uff0c\u7136\u540e\u8ba1\u7b97\\(h_1=g_1\/f_x,h_2=g_2\/f_x\\)\uff0c\u7ecf\u8fc7\u5904\u7406\u540e\u5206\u522b\u4f5c\u4e3a\\(pk_1,pk_2\\)\uff0c\u5f88\u660e\u663e\u8fd9\u6837\u751f\u6210\u7684\u662fNTRU\u7684\u6837\u672c\uff0c\u90a3\u4e48\u6211\u4eec\u7684\u4efb\u52a1\u5c31\u662f\u533a\u5206NTRU\u6837\u672c\u548c\u968f\u673a\u6837\u672c\uff0c\u521a\u5f00\u59cb\u6211\u7684\u601d\u8def\u5c31\u662f\u76f4\u63a5\u6784\u9020NTRU\u683c\u5c1d\u8bd5\u89c4\u7ea6\u770b\u770b\u662f\u5426\u80fd\u89c4\u7ea6\u51fa\\((f_x,g_1)\\)\uff0c\u4f46\u662f\u8fd9\u91cc\u7684\\(n=1031\\)\uff0c\u592a\u5927\u4e86\uff0c\u5728\u6709\u9650\u65f6\u95f4\u5185\u5f88\u96be\u89c4\u7ea6\u51fa\u76ee\u6807\u5411\u91cf\uff0c\u800c\u4e14\u8fd9\u6837\u6211\u4eec\u53ea\u80fd\u7528\u5230\u751f\u6210\u6837\u672c\\((pk_1,pk_2)\\)\u4e2d\u7684\u4e00\u4e2a\u591a\u9879\u5f0f\uff0c\u5e76\u4e0d\u4f1a\u7528\u5230\u53e6\u5916\u4e00\u4e2a\u591a\u9879\u5f0f\uff08\u5982\u679c\u7528\u5230\u7684\u8bdd\u8ba1\u7b97\u65f6\u95f4\u66f4\u957f\u4e86\uff09\uff0c\u6240\u4ee5\u53ea\u80fd\u60f3\u53e6\u5916\u4e00\u79cd\u65b9\u6cd5\uff0c\u800c\u6211\u4eec\u5f88\u5bb9\u6613\u53ef\u4ee5\u77e5\u9053\u5bf9\u4e8e\u4efb\u610f\u6b63\u5947\u6570\\(k\\) \uff0c\u6709\uff1a<\/p>\n\n\n\n<p>$$<br>x^{k}+1=(x+1)\\left[\\sum_{i=0}^{k-1}(-1)^{i}x^{i}\\right]<br>$$<\/p>\n\n\n\n<p>\u56e0\u4e3a\u6211\u4eec\u5e76\u4e0d\u9700\u8981\u5c06\\(f_x\\)\u6062\u590d\u51fa\u6765\uff0c\u53ea\u9700\u8981\u533a\u5206\u51fa\u591a\u9879\u5f0f\u6837\u672c\u662f\u6765\u81ea<code>GenNTRU<\/code>\u8fd8\u662f\u968f\u673a\u6837\u672c\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u5728\\(\\frac{(\\mathbb{Z}\/q\\mathbb{Z})[x]}{x+1}\\)\u4e0a\u8fdb\u884c\u533a\u5206\uff0c\u6211\u4eec\u77e5\u9053\uff0c\u5728\\(\\frac{(\\mathbb{Z}\/q\\mathbb{Z})[x]}{x+1}\\)\u4e2d\uff0c\u6709\\(x=-1\\)\uff0c\u90a3\u4e48\u5fc5\u7136\u7684\uff0c\u5bf9\u4e8e\\(\\frac{(\\mathbb{Z}\/q\\mathbb{Z})[x]}{x^n+1}\\)\u4e2d\u7684\u4e00\u4e2a\u591a\u9879\u5f0f\\(f\\)\uff0c\u5fc5\u7136\u6709\\(f\\mod{x+1}=f(-1)\\in \\frac{(\\mathbb{Z}\/q\\mathbb{Z})[x]}{x+1}\\)\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6620\u5c04\uff1a<\/p>\n\n\n\n<p>$$<br>\\phi:\\frac{(\\mathbb{Z}\/q\\mathbb{Z})[x]}{x^n+1}\\rightarrow\\frac{(\\mathbb{Z}\/q\\mathbb{Z})[x]}{x+1},f(x)\\mapsto f(-1)<br>$$<\/p>\n\n\n\n<p>\u5c06\u5546\u73af\\(\\frac{(\\mathbb{Z}\/q\\mathbb{Z})[x]}{x^n+1}\\)\u4e2d\u7684\u6240\u6709\u591a\u9879\u5f0f\u6620\u5c04\u5230\\(\\frac{(\\mathbb{Z}\/q\\mathbb{Z})[x]}{x+1}\\)\u4e2d\uff0c\u4ece\u800c\u5bf9\u4e8e<code>GenNTRU<\/code>\u751f\u6210\u7684\u6837\u672c\uff0c\u6709\uff1a<\/p>\n\n\n\n<p>$$<br>\\begin{cases}<br>g_1=h_1f_x\\\\<br>g_2=h_2f_x\\\\<br>\\end{cases}\\rightarrow<br>\\begin{cases}<br>g_1(-1)\\equiv h_1(-1)f_x(-1)\\pmod{q}\\\\<br>g_2(-1)\\equiv h_2(-1)f_x(-1)\\pmod{q}\\\\<br>\\end{cases}<br>$$<\/p>\n\n\n\n<p>\u6839\u636e\u4e0a\u5f0f\u53ef\u4ee5\u6784\u9020\u683c\uff1a<\/p>\n\n\n\n<p>$$<br>\\pmb{B}=\\left(\\begin{matrix}<br>1&amp;h_{1}(-1)&amp;h_{2}(-1)\\\\<br>&amp;q&amp;\\\\<br>&amp;&amp;q<br>\\end{matrix}\\right)<br>$$<\/p>\n\n\n\n<p>\u9884\u671f\u6709\\((f_x(-1),k_1,k_2)\\pmb{B}=(f_x(-1),g_{1}(-1),g_{2}(-1))\\)\uff0c\u6309\u7167<code>GenNTRU<\/code>\u7684\u751f\u6210\u89c4\u5219\u6211\u4eec\u53ef\u4ee5\u77e5\u9053\u82e5\\(f_x,h_1,h_2\\)\u662f<code>GenNTRU<\/code>\u4e2d\u7ed9\u51fa\u7684\u6837\u672c\uff0c\u90a3\u4e48\u5176\u5bf9\u5e94\u7684\\(f_{x},g_{1},g_{2}\\)\u90fd\u662f\u4e09\u5143\u591a\u9879\u5f0f\uff0c\u6b64\u65f6\u4e0a\u8ff0\u683c\u89c4\u7ea6\u51fa\u7684\u77ed\u5411\u91cf\\((f_x(-1),g_{1}(-1),g_{2}(-1))\\)\u4e5f\u4f1a\u76f8\u5bf9\u8f83\u77ed\uff0c\u800c\u82e5\\(f_x,h_1,h_2\\)\u662f\u968f\u673a\u591a\u9879\u5f0f\u6837\u672c\uff0c\u90a3\u4e48\u901a\u8fc7\u540c\u6837\u65b9\u6cd5\u89c4\u7ea6\u51fa\u7684\u77ed\u5411\u91cf\u4f1a\u5f88\u957f\uff0c\u53ef\u4ee5\u901a\u8fc7\u4e0b\u8ff0\u4ee3\u7801\u8fdb\u884c\u6d4b\u8bd5\uff1a<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>from sage.all import *\nfrom sage.misc import prandom\nimport random\nfrom Crypto.Cipher import AES\nfrom Crypto.Hash import SHA256\nfrom Crypto.Util.Padding import pad\n\nq = 1342261\nn = 1031\nPR = PolynomialRing(Zmod(q), \"x\")\nx = PR.gens()&#91;0&#93;\nQ = PR.quotient(x**n + 1)\n\ndef sample(rand):\n    return (rand.getrandbits(1) - rand.getrandbits(1)) * (rand.getrandbits(1) * rand.getrandbits(1))\n\ndef GenNTRU():\n    f = &#91;sample(prandom) for _ in range(n)&#93;\n    g1 = &#91;sample(prandom)for _ in range(n)&#93;\n    g2 = &#91;sample(prandom) for _ in range(n)&#93;\n    g1x = Q(g1)\n    g2x = Q(g2)\n\n    while True:\n        fx = Q(f)\n        try:\n            h1 = g1x \/ fx\n            h2 = g2x \/ fx\n            return (h1.lift(), h2.lift(), fx)\n        except:\n            prandom.shuffle(f)\n            continue\n\nn1 = []\nfor _ in range(1000):\n    h1, h2, f = GenNTRU()\n    H1 = ZZ(h1(-1))\n    H2 = ZZ(h2(-1))\n    L = matrix(ZZ, [&#91;1, H1, H2&#93;, &#91;0, q, 0&#93;, &#91;0, 0, q&#93;])\n    res = L.LLL()&#91;0&#93;\n    n1.append(res.norm()**2)\n\nprint(min(n1), max(n1))\n\nn2 = []\nfor _ in range(1000):\n    h1, h2 = Q.random_element().lift(), Q.random_element().lift()\n    H1 = ZZ(h1(-1))\n    H2 = ZZ(h2(-1))\n    L = matrix(ZZ, [&#91;1, H1, H2&#93;, &#91;0, q, 0&#93;, &#91;0, 0, q&#93;])\n    res = L.LLL()&#91;0&#93;\n    n2.append(res.norm()**2)\n\nprint(min(n2), max(n2))<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> sage<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">all <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> sage<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">misc <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> prandom<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> random<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Cipher <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> AES<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Hash <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> SHA256<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Util<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Padding <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> pad<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1342261<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">n <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1031<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">PR <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">PolynomialRing<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">Zmod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">q<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">x<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> PR<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">gens<\/span><span style=\"color: #ECEFF4\">()&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> PR<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">quotient<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\">n <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">rand<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">rand<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> rand<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">rand<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> rand<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">GenNTRU<\/span><span style=\"color: #ECEFF4\">():<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    f <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">prandom<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    g1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">prandom<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    g2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">prandom<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    g1x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">g1<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    g2x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">g2<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">while<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">True<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        fx <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">try<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            h1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> g1x <\/span><span style=\"color: #81A1C1\">\/<\/span><span style=\"color: #D8DEE9FF\"> fx<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            h2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> g2x <\/span><span style=\"color: #81A1C1\">\/<\/span><span style=\"color: #D8DEE9FF\"> fx<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">h1<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #D8DEE9FF\"> h2<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #D8DEE9FF\"> fx<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">except<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            prandom<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">shuffle<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">continue<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">n1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1000<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    h1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> h2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> f <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">GenNTRU<\/span><span style=\"color: #ECEFF4\">()<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    H1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">ZZ<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">h1<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    H2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">ZZ<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">h2<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    L <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">matrix<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">ZZ<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> H1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> H2<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> q<\/span><span style=\"color: #ECEFF4\">&#93;])<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    res <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> L<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">LLL<\/span><span style=\"color: #ECEFF4\">()&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    n1<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">res<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">norm<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">min<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n1<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">max<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n1<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">n2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1000<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    h1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> h2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">random_element<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">random_element<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">()<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    H1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">ZZ<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">h1<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    H2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">ZZ<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">h2<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    L <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">matrix<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">ZZ<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> H1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> H2<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> q<\/span><span style=\"color: #ECEFF4\">&#93;])<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    res <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> L<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">LLL<\/span><span style=\"color: #ECEFF4\">()&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    n2<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">res<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">norm<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">min<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n2<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">max<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n2<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u7ecf\u8fc7\u591a\u6b21\u6d4b\u8bd5\u6211\u4eec\u53ef\u4ee5\u77e5\u9053\uff0c\u968f\u673a\u6837\u672c\u89c4\u7ea6\u51fa\u6765\u7684\u77ed\u5411\u91cf\u8303\u6570\u7684\u5e73\u65b9\u6700\u5c0f\u503c\u4e5f\u8fdc\u8fdc\u5927\u4e8e<code>GenNTRU<\/code>\u4e2d\u7ed9\u51fa\u7684\u6837\u672c\u89c4\u7ea6\u51fa\u6765\u7684\u77ed\u5411\u91cf\u8303\u6570\u7684\u5e73\u65b9\u7684\u6700\u5927\u503c\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u7f16\u5199\u5982\u4e0b\u6837\u672c\u533a\u5206\u5668\uff1a<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>def diff(h1, h2, q, bound):\n    H1 = ZZ(h1(-1))\n    H2 = ZZ(h2(-1))\n    L = matrix(ZZ, [&#91;1, H1, H2&#93;, &#91;0, q, 0&#93;, &#91;0, 0, q&#93;])\n    res = L.LLL()&#91;0&#93;\n    \n    if res.norm() ** 2 &lt; bound:\n        return 0\n    return 1<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">diff<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">h1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">h2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">bound<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    H1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">ZZ<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">h1<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    H2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">ZZ<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">h2<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    L <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">matrix<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">ZZ<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> H1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> H2<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> q<\/span><span style=\"color: #ECEFF4\">&#93;])<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    res <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> L<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">LLL<\/span><span style=\"color: #ECEFF4\">()&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> res<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">norm<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">&lt;<\/span><span style=\"color: #D8DEE9FF\"> bound<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u901a\u8fc7\u5982\u4e0b\u4ee3\u7801\u53ef\u4ee5\u6d4b\u8bd5\u4e0a\u8ff0\u533a\u5206\u5668\u7684\u533a\u5206\u6210\u529f\u7387\uff1a<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>from sage.all import *\nfrom sage.misc import prandom\nimport random\nfrom tqdm import trange\n\nq = 1342261\nn = 1031\nPR = PolynomialRing(Zmod(q), \"x\")\nx = PR.gens()&#91;0&#93;\nQ = PR.quotient(x**n + 1)\n\ndef sample(rand):\n    return (rand.getrandbits(1) - rand.getrandbits(1)) * (rand.getrandbits(1) * rand.getrandbits(1))\n\ndef GenNTRU():\n    f = &#91;sample(prandom) for _ in range(n)&#93;\n    g1 = &#91;sample(prandom)for _ in range(n)&#93;\n    g2 = &#91;sample(prandom) for _ in range(n)&#93;\n    g1x = Q(g1)\n    g2x = Q(g2)\n\n    while True:\n        fx = Q(f)\n        try:\n            h1 = g1x \/ fx\n            h2 = g2x \/ fx\n            return (h1.lift(), h2.lift(), fx)\n        except:\n            prandom.shuffle(f)\n            continue\n\ndef diff(h1, h2, q, bound):\n    H1 = h1(-1)\n    H2 = h2(-1)\n\n    L = matrix(ZZ, [&#91;1, H1, H2&#93;, &#91;0, q, 0&#93;, &#91;0, 0, q&#93;])\n    res = L.LLL()&#91;0&#93;\n    \n    if res.norm() ** 2 &lt; bound:\n        return 0\n    \n    return 1\n\nsuccess = 0\n\nfor _ in trange(100000):\n    coin = random.getrandbits(1)\n\n    if coin == 0:\n        pk1, pk2, fx = GenNTRU()\n    else:\n        pk1, pk2, fx = Q.random_element().lift(), Q.random_element().lift(), Q(&#91;sample(prandom) for _ in range(n)&#93;)\n\n    if diff(pk1, pk2, q, 10000) == coin:\n        success += 1\n\nprint(success \/ 100000)<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> sage<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">all <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> sage<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">misc <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> prandom<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> random<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> tqdm <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> trange<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1342261<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">n <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1031<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">PR <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">PolynomialRing<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">Zmod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">q<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">x<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> PR<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">gens<\/span><span style=\"color: #ECEFF4\">()&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> PR<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">quotient<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\">n <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">rand<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">rand<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #D8DEE9FF\"> rand<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">))<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">rand<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> rand<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">GenNTRU<\/span><span style=\"color: #ECEFF4\">():<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    f <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">prandom<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    g1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">prandom<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    g2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">prandom<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n<\/span><span style=\"color: #ECEFF4\">)&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    g1x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">g1<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    g2x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">g2<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">while<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">True<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        fx <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">try<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            h1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> g1x <\/span><span style=\"color: #81A1C1\">\/<\/span><span style=\"color: #D8DEE9FF\"> fx<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            h2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> g2x <\/span><span style=\"color: #81A1C1\">\/<\/span><span style=\"color: #D8DEE9FF\"> fx<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">h1<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #D8DEE9FF\"> h2<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #D8DEE9FF\"> fx<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">except<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            prandom<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">shuffle<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">f<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            <\/span><span style=\"color: #81A1C1\">continue<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">diff<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">h1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">h2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">bound<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    H1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">h1<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    H2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">h2<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    L <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">matrix<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">ZZ<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> H1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> H2<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> q<\/span><span style=\"color: #ECEFF4\">&#93;])<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    res <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> L<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">LLL<\/span><span style=\"color: #ECEFF4\">()&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> res<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">norm<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">&lt;<\/span><span style=\"color: #D8DEE9FF\"> bound<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">success <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">trange<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">100000<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    coin <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> random<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> coin <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        pk1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pk2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> fx <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">GenNTRU<\/span><span style=\"color: #ECEFF4\">()<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">else<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        pk1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pk2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> fx <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">random_element<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #D8DEE9FF\"> Q<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">random_element<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">(),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(&#91;<\/span><span style=\"color: #88C0D0\">sample<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">prandom<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">range<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">n<\/span><span style=\"color: #ECEFF4\">)&#93;)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">diff<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">pk1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> pk2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">10000<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">==<\/span><span style=\"color: #D8DEE9FF\"> coin<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        success <\/span><span style=\"color: #81A1C1\">+=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">success <\/span><span style=\"color: #81A1C1\">\/<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">100000<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u6d4b\u8bd5\u53ef\u4ee5\u77e5\u9053\u5728<code>bound<\/code>\u8bbe\u7f6e\u4e3a10000\u7684\u65f6\u5019\u533a\u5206\u7387\u53ef\u4ee5\u8fbe\u5230100%\uff0c\u90a3\u4e48\u6211\u4eec\u5c31\u53ef\u4ee5\u7528\u8fd9\u4e2a\u533a\u5206\u5668\u8fbe\u5230\u533a\u5206\u6837\u672c\u7684\u6548\u679c\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u83b7\u5f97\u6700\u591a20258\u4e2acoin\u503c\uff0c\u7ed3\u5408gf2bv\u5c31\u53ef\u4ee5\u6062\u590d\u51faMT19937\u7684\u72b6\u6001\uff0c\u4ece\u800c\u9884\u6d4b\u51fa\u52a0\u5bc6flag\u7528\u7684\u5bc6\u94a5\uff0c\u7528\u9884\u6d4b\u51fa\u7684\u5bc6\u94a5\u6765\u89e3\u5bc6\u5c31\u53ef\u4ee5\u5f97\u5230flag\uff1a<\/p>\n\n\n\n<div class=\"wp-block-kevinbatdorf-code-block-pro\" data-code-block-pro-font-family=\"Code-Pro-JetBrains-Mono\" style=\"font-size:.875rem;font-family:Code-Pro-JetBrains-Mono,ui-monospace,SFMono-Regular,Menlo,Monaco,Consolas,monospace;line-height:1.25rem;--cbp-tab-width:2;tab-size:var(--cbp-tab-width, 2)\"><span style=\"display:block;padding:16px 0 0 16px;margin-bottom:-1px;width:100%;text-align:left;background-color:#2e3440ff\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"54\" height=\"14\" viewBox=\"0 0 54 14\"><g fill=\"none\" fill-rule=\"evenodd\" transform=\"translate(1 1)\"><circle cx=\"6\" cy=\"6\" r=\"6\" fill=\"#FF5F56\" stroke=\"#E0443E\" stroke-width=\".5\"><\/circle><circle cx=\"26\" cy=\"6\" r=\"6\" fill=\"#FFBD2E\" stroke=\"#DEA123\" stroke-width=\".5\"><\/circle><circle cx=\"46\" cy=\"6\" r=\"6\" fill=\"#27C93F\" stroke=\"#1AAB29\" stroke-width=\".5\"><\/circle><\/g><\/svg><\/span><span role=\"button\" tabindex=\"0\" style=\"color:#d8dee9ff;display:none\" aria-label=\"\u590d\u5236\" class=\"code-block-pro-copy-button\"><pre class=\"code-block-pro-copy-button-pre\" aria-hidden=\"true\"><textarea class=\"code-block-pro-copy-button-textarea\" tabindex=\"-1\" aria-hidden=\"true\" readonly>from sage.all import *\nfrom sage.misc import prandom\nfrom tqdm import trange\nfrom pwn import *\nfrom gf2bv import *\nfrom gf2bv.crypto.mt import *\nfrom Crypto.Cipher import AES\nfrom Crypto.Hash import SHA256\n\nq = 1342261\nn = 1031\nPR = PolynomialRing(Zmod(q), \"x\")\nx = PR.gens()&#91;0&#93;\nQ = PR.quotient(x**n + 1)\n\ndef diff(h1, h2, q, bound):\n    H1 = h1(-1)\n    H2 = h2(-1)\n\n    L = matrix(ZZ, [&#91;1, H1, H2&#93;, &#91;0, q, 0&#93;, &#91;0, 0, q&#93;])\n    res = L.LLL()&#91;0&#93;\n    \n    if res.norm() ** 2 &lt; bound:\n        return 0\n    \n    return 1\n\np = remote(\"127.0.0.1\", 12345)\n\ncoins = []\n\nfor _ in trange(19938):\n    p.sendlineafter(b\"c :\", b\"1\")\n\n    p.recvuntil(b\"Hints: \")\n    h1 = Q(eval(p.recvuntil(b\"]\").strip().decode())).lift()\n    h2 = Q(eval(p.recvuntil(b\"]\").strip().decode())).lift()\n\n    coins.append(diff(h1, h2, q, 10000))\n\np.sendlineafter(b\"c :\", b\"2\")\np.recvuntil(b\"Flag: \")\nct = eval(p.recvline().strip().decode())\n\n# print(coin)\n\nLS = LinearSystem(&#91;32&#93; * 624)\nmt = LS.gens()\n\nmt19937 = MT19937(mt)\ntmp = []\n\nfor coin in coins:\n    tmp.append(mt19937.getrandbits(1) ^ coin)\ntmp.append(mt&#91;0&#93; ^ 0x80000000)\n\nfor res in LS.solve_all(tmp):\n    _rng = MT19937(res)\n    rng = _rng.to_python_random()\n    try:\n        for _ in trange(19938):\n            trash = rng.getrandbits(1)\n\n        SHA = SHA256.new()\n        SHA.update(str(rng.getrandbits(256)).encode())\n        KEY = SHA.digest()\n        cipher = AES.new(KEY, AES.MODE_ECB)\n\n        print(cipher.decrypt(ct))\n    except:\n        pass<\/textarea><\/pre><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" style=\"width:24px;height:24px\" fill=\"none\" viewBox=\"0 0 24 24\" stroke=\"currentColor\" stroke-width=\"2\"><path class=\"with-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2m-6 9l2 2 4-4\"><\/path><path class=\"without-check\" stroke-linecap=\"round\" stroke-linejoin=\"round\" d=\"M9 5H7a2 2 0 00-2 2v12a2 2 0 002 2h10a2 2 0 002-2V7a2 2 0 00-2-2h-2M9 5a2 2 0 002 2h2a2 2 0 002-2M9 5a2 2 0 012-2h2a2 2 0 012 2\"><\/path><\/svg><\/span><pre class=\"shiki nord\" style=\"background-color: #2e3440ff\" tabindex=\"0\"><code><span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> sage<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">all <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> sage<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">misc <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> prandom<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> tqdm <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> trange<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> pwn <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> gf2bv <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> gf2bv<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">mt <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Cipher <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> AES<\/span><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">from<\/span><span style=\"color: #D8DEE9FF\"> Crypto<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">Hash <\/span><span style=\"color: #81A1C1\">import<\/span><span style=\"color: #D8DEE9FF\"> SHA256<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1342261<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">n <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1031<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">PR <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">PolynomialRing<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">Zmod<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">q<\/span><span style=\"color: #ECEFF4\">),<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">x<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">x <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> PR<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">gens<\/span><span style=\"color: #ECEFF4\">()&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">Q <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> PR<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">quotient<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">x<\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\">n <\/span><span style=\"color: #81A1C1\">+<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">def<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">diff<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9\">h1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">h2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #D8DEE9\">bound<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    H1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">h1<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    H2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">h2<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">-<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    L <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">matrix<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">ZZ<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[&#91;<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> H1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> H2<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> q<\/span><span style=\"color: #ECEFF4\">&#93;])<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    res <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> L<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">LLL<\/span><span style=\"color: #ECEFF4\">()&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">if<\/span><span style=\"color: #D8DEE9FF\"> res<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">norm<\/span><span style=\"color: #ECEFF4\">()<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">**<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">2<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">&lt;<\/span><span style=\"color: #D8DEE9FF\"> bound<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">0<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">return<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">1<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">p <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">remote<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">127.0.0.1<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">12345<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">coins <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">trange<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">19938<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">sendlineafter<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">c :<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">1<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">recvuntil<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">Hints: <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    h1 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">eval<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">recvuntil<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">]<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">strip<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">decode<\/span><span style=\"color: #ECEFF4\">())).<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">()<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    h2 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">Q<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">eval<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">recvuntil<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">]<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">).<\/span><span style=\"color: #88C0D0\">strip<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">decode<\/span><span style=\"color: #ECEFF4\">())).<\/span><span style=\"color: #88C0D0\">lift<\/span><span style=\"color: #ECEFF4\">()<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    coins<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">diff<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">h1<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> h2<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> q<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">10000<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">sendlineafter<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">c :<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">2<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">recvuntil<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #81A1C1\">b<\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #A3BE8C\">Flag: <\/span><span style=\"color: #ECEFF4\">&quot;<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">ct <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">eval<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">p<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">recvline<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">strip<\/span><span style=\"color: #ECEFF4\">().<\/span><span style=\"color: #88C0D0\">decode<\/span><span style=\"color: #ECEFF4\">())<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #616E88\"># print(coin)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">LS <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">LinearSystem<\/span><span style=\"color: #ECEFF4\">(&#91;<\/span><span style=\"color: #B48EAD\">32<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">*<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #B48EAD\">624<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">mt <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> LS<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">gens<\/span><span style=\"color: #ECEFF4\">()<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">mt19937 <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">MT19937<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">mt<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">tmp <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #ECEFF4\">[]<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> coin <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> coins<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    tmp<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">mt19937<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">^<\/span><span style=\"color: #D8DEE9FF\"> coin<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">tmp<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">append<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">mt<\/span><span style=\"color: #ECEFF4\">&#91;<\/span><span style=\"color: #B48EAD\">0<\/span><span style=\"color: #ECEFF4\">&#93;<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">^<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #81A1C1\">0x<\/span><span style=\"color: #B48EAD\">80000000<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> res <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> LS<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">solve_all<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">tmp<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    _rng <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">MT19937<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">res<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    rng <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> _rng<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">to_python_random<\/span><span style=\"color: #ECEFF4\">()<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">try<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">for<\/span><span style=\"color: #D8DEE9FF\"> _ <\/span><span style=\"color: #81A1C1\">in<\/span><span style=\"color: #D8DEE9FF\"> <\/span><span style=\"color: #88C0D0\">trange<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">19938<\/span><span style=\"color: #ECEFF4\">):<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">            trash <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> rng<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">1<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        SHA <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> SHA256<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">new<\/span><span style=\"color: #ECEFF4\">()<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        SHA<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">update<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #88C0D0\">str<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">rng<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">getrandbits<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #B48EAD\">256<\/span><span style=\"color: #ECEFF4\">)).<\/span><span style=\"color: #88C0D0\">encode<\/span><span style=\"color: #ECEFF4\">())<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        KEY <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> SHA<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">digest<\/span><span style=\"color: #ECEFF4\">()<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        cipher <\/span><span style=\"color: #81A1C1\">=<\/span><span style=\"color: #D8DEE9FF\"> AES<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">new<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">KEY<\/span><span style=\"color: #ECEFF4\">,<\/span><span style=\"color: #D8DEE9FF\"> AES<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #D8DEE9FF\">MODE_ECB<\/span><span style=\"color: #ECEFF4\">)<\/span><\/span>\n<span class=\"line\"><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #88C0D0\">print<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">cipher<\/span><span style=\"color: #ECEFF4\">.<\/span><span style=\"color: #88C0D0\">decrypt<\/span><span style=\"color: #ECEFF4\">(<\/span><span style=\"color: #D8DEE9FF\">ct<\/span><span style=\"color: #ECEFF4\">))<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">    <\/span><span style=\"color: #81A1C1\">except<\/span><span style=\"color: #ECEFF4\">:<\/span><\/span>\n<span class=\"line\"><span style=\"color: #D8DEE9FF\">        <\/span><span style=\"color: #81A1C1\">pass<\/span><\/span><\/code><\/pre><\/div>\n\n\n\n<p>\u6709\u6982\u7387\u6210\u529f\uff1a<\/p>\n\n\n<div class=\"wp-block-image is-style-default\">\n<figure class=\"aligncenter size-large\"><a href=\"http:\/\/www.triode.cc\/wp-content\/uploads\/2025\/10\/Pasted-image-20251024001941.png\"><div class='fancybox-wrapper lazyload-container-unload' data-fancybox='post-images' href='http:\/\/www.triode.cc\/wp-content\/uploads\/2025\/10\/Pasted-image-20251024001941-1024x158.png'><img class=\"lazyload lazyload-style-1\" src=\"data:image\/svg+xml;base64,PCEtLUFyZ29uTG9hZGluZy0tPgo8c3ZnIHdpZHRoPSIxIiBoZWlnaHQ9IjEiIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8yMDAwL3N2ZyIgc3Ryb2tlPSIjZmZmZmZmMDAiPjxnPjwvZz4KPC9zdmc+\"  loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"158\" data-original=\"http:\/\/www.triode.cc\/wp-content\/uploads\/2025\/10\/Pasted-image-20251024001941-1024x158.png\" src=\"data:image\/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAANSURBVBhXYzh8+PB\/AAffA0nNPuCLAAAAAElFTkSuQmCC\" alt=\"\" class=\"wp-image-355\"  sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/div><\/a><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>\u5f3a\u7f51\u676fS9 Crypto\u65b9\u5411\u975e\u96f6\u89e3\u9898Write Up<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[7],"class_list":["post-321","post","type-post","status-publish","format-standard","hentry","category-write-up","tag-crypto"],"_links":{"self":[{"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/posts\/321","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/comments?post=321"}],"version-history":[{"count":9,"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/posts\/321\/revisions"}],"predecessor-version":[{"id":372,"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/posts\/321\/revisions\/372"}],"wp:attachment":[{"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/media?parent=321"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/categories?post=321"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/tags?post=321"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}