{"id":244,"date":"2025-08-10T22:33:00","date_gmt":"2025-08-10T14:33:00","guid":{"rendered":"http:\/\/www.triode.cc\/?p=244"},"modified":"2025-10-02T14:59:57","modified_gmt":"2025-10-02T06:59:57","slug":"acd-problem","status":"publish","type":"post","link":"https:\/\/www.triode.cc\/index.php\/2025\/08\/10\/acd-problem\/","title":{"rendered":"\u8fd1\u4f3c\u516c\u7ea6\u6570\u95ee\u9898 (ACD)"},"content":{"rendered":"\n
\n

\u53c2\u8003\u8d44\u6599\uff1aAlgorithms for the Approximate Common Divisor Problem<\/a><\/p>\n<\/blockquote>\n\n\n\n

\u4e8b\u5b9e\u4e0a\u5728\u7b2c\u4e00\u5c4aOpenHarmony CTF\u4e13\u9898\u8d5b\uff08\u7ebf\u4e0a\u9009\u62d4\u8d5b\uff09Crypto Write UP | Triode Field<\/a>\u7684\u65f6\u5019\u5c31\u5df2\u7ecf\u505a\u5230\u8fc7ACD\u95ee\u9898\u4e86\uff0c\u4f46\u662f\u5f53\u65f6\u662f\u73b0\u5b66\u73b0\u505a\u7684\uff0c\u5e76\u6ca1\u6709\u505a\u7cfb\u7edf\u6027\u7684\u5b66\u4e60\u548c\u5206\u6790\uff0c\u8d81\u7740\u6691\u5047\u6765\u8865\u4e60\u4e00\u4e0b\uff08\u7b11\uff09<\/p>\n\n\n\n

\u8fd1\u4f3c\u516c\u7ea6\u6570\u95ee\u9898\uff08ACD Problem\uff09<\/h2>\n\n\n\n

\u5bf9\u4e8e\u7ed9\u5b9a\u7684\\(\\gamma,\\eta,\\rho\\in\\mathbb{N}\\)\uff0c\u5bf9\u4e8e\u4e00\u4e2a\\(\\eta\\)-bit\u7684\u5947\u6570\\(p\\)\uff0c\u5b9a\u4e49\u5982\u4e0b\u9ad8\u6548\u91c7\u6837\u5206\u5e03\\(\\mathcal{D}_{\\gamma,\\rho}(p)\\)\uff1a <\/p>\n\n\n\n

$$
\\mathcal{D}_{\\gamma,\\rho}(p)=\\left\\{pq+r|q\\in\\mathbb{Z}\\cap\\left[0,\\frac{2^{\\gamma}}{p}\\right),r\\in\\mathbb{Z}\\cap\\left(2^{-\\rho},2^{\\rho}\\right)\\right\\}
$$<\/p>\n\n\n\n

\u5bf9\u4e8e\u8be5\u91c7\u6837\u5206\u5e03\uff0c\u6709\\(\\rho\\ll\\eta\\)\uff0c\u6240\u4ee5\u4efb\u53d6\\(x\\in\\mathcal{D}_{\\gamma,\\rho}(p)\\)\uff0c\u90fd\u6709\u6781\u5927\u6982\u7387\u6ee1\u8db3\\(x<2^{\\gamma}\\)\uff0c\u90a3\u4e48\u5bf9\u4e8e\u4e00\u4e2a\u5145\u5206\u5927\u7684\u6574\u6570\\(t\\)\uff0c\u4ece\\(\\mathcal{D}_{\\gamma,\\rho}(p)\\)\u4e2d\u53d6\\(t\\)\u4e2a\u6837\u672c\\(x_1,x_2,\\cdots,x_{t}\\)\uff0c\u90a3\u4e48\u6240\u8c13\u8fd1\u4f3c\u516c\u7ea6\u6570\u95ee\u9898\uff08Approximate Common Divisor Problem, ACD\uff09\u5373\u4e3a\u4ece\u7ed9\u5b9a\u7684\\(t\\)\u4e2a\u6837\u672c\\({x_1,x_2,\\cdots,x_{t}}\\)\u4e2d\u6062\u590d\u51fa\\(p\\).<\/p>\n\n\n\n

\u90e8\u5206\u8fd1\u4f3c\u516c\u7ea6\u6570\u95ee\u9898\uff08PACD Problem\uff09<\/h3>\n\n\n\n

\u6240\u8c13\u90e8\u5206\u8fd1\u4f3c\u516c\u7ea6\u6570\u95ee\u9898\uff08Partial Approximate Common Divisor Problem, PACD\uff09\uff0c\u5373\u4e3a\u5728ACD\u95ee\u9898\u7684\u6837\u672c\u57fa\u7840\u4e0a\u52a0\u4e0a\u4e86\u4e00\u4e2a\u7279\u6b8a\u6837\u672c\\(x_0=pq_0\\)\uff08\u5176\u4e2d\\(q_0\\)\u4e0e\\(\\mathcal{D}_{\\gamma,\\rho}(p)\\)\u4e2d\\(q\\)\u7684\u53d6\u503c\u8303\u56f4\u4e00\u81f4\uff09\uff0c\u4ece\u7ed9\u5b9a\u7684\\(t+1\\)\u4e2a\u6837\u672c\\({x_0,x_1,x_2,\\cdots,x_{t}}\\)\u4e2d\u6062\u590d\u51fa\\(p\\).<\/p>\n\n\n\n

\u8fd1\u4f3c\u516c\u7ea6\u6570\u95ee\u9898\u7684\u6c42\u89e3\u65b9\u6cd5<\/h2>\n\n\n\n

\u6c42\u89e3ACD\u4e3b\u6d41\u7684\u65b9\u6cd5\u6709\u4e09\u79cd\uff1aSDA(Simultaneous Diophantine approximation approach), OL(Orthogonal based approach)\u4ee5\u53caMP(Multivariate polynomial approach)\uff0c\u5728\u6b64\u4ec5\u4ecb\u7ecdSDA.<\/p>\n\n\n\n

\u540c\u6b65\u4e22\u756a\u56fe\u903c\u8fd1\u65b9\u6cd5\uff08SDA\uff09<\/h3>\n\n\n\n

\u8fd9\u662f\u6c42\u89e3ACD\u7684\u65b9\u6cd5\u4e2d\u6700\u7b80\u5355\u7684\u4e00\u79cd\uff0c\u5927\u81f4\u601d\u60f3\u5c31\u662f\u5bf9\u4e8e\u4ece\\(\\mathcal{D}_{\\gamma,\\rho}(p)\\)\u91c7\u6837\u51fa\u7684\\({x_0,x_1,\\cdots,x_t}\\)\uff08\u6b64\u5904\\(x_0\\)\u4e3a\u4e00\u822c\u6837\u672c\uff09\uff0c\u5f53\\(\\rho\\)\u5f88\u5c0f\u7684\u65f6\u5019\uff08\u4e5f\u5c31\u662f\\(r_0,r_1,\\cdots,r_t\\)\u90fd\u5f88\u5c0f\u7684\u65f6\u5019\uff09\uff0c\u5bf9\u4e8e\\(1\\le i\\le t\\)\uff0c\u603b\u6709\uff1a<\/p>\n\n\n\n

$$
\\frac{x_i}{x_0}=\\frac{pq_i+r_i}{pq_0+r_0}\\approx\\frac{q_i}{q_0}
$$<\/p>\n\n\n\n

\u5f88\u663e\u7136\uff0c\\(q_i\/q_0\\)\u662f\\(x_i\/x_0\\)\u7684\u540c\u6b65\u4e22\u756a\u56fe\u8fd1\u4f3c\uff08simultaneous Diophantine approximation\uff09\uff0c\u4e8b\u5b9e\u4e0a\u6709\uff1a<\/p>\n\n\n\n

$$
q_ix_0-q_0x_i=q_i(pq_0+r_0)-q_0(pq_i+r_i)=pq_0q_i+q_ir_0-pq_0q_i-q_0r_i=q_ir_0-q_0r_i
$$<\/p>\n\n\n\n

\u6240\u4ee5\u5fc5\u7136\u6709\\(|q_ix_0-q_0x_i|\\approx2^{\\rho+\\gamma-\\eta+1}\\)\uff0c\u90a3\u4e48\u53ef\u4ee5\u901a\u8fc7\u8fd9\u4e2a\u5173\u7cfb\u6784\u9020\u683c\u57fa\uff1a<\/p>\n\n\n\n

$$
\\pmb{B}=\\left(\\begin{matrix}
2^{\\rho+1}&x_1&x_2&\\cdots&x_t\\\\
&-x_0&&&\\\\
&&-x_0&&\\\\
&&&\\ddots&\\\\
&&&&-x_0
\\end{matrix}\\right)
$$<\/p>\n\n\n\n

\u5bf9\u4e8e\u8fd9\u4e2a\u683c\u6709\u5982\u4e0b\u5173\u7cfb\uff1a<\/p>\n\n\n\n

$$
\\begin{aligned}
(q_0,q_1,\\cdots,q_t)\\pmb{B}&=(2^{\\rho+1}p_0,q_0x_1-q_1x_0,\\cdots,q_0x_t-q_tx_0)\\\\
&=(2^{\\rho+1}p_0,q_0r_1-q_1r_0,\\cdots,q_0r_t-q_tr_0)
\\end{aligned}
$$<\/p>\n\n\n\n

\u663e\u7136\u5bf9\u76ee\u6807\u5411\u91cf\\((2^{\\rho+1}p_0,q_0r_1-q_1r_0,\\cdots,q_0r_t-q_tr_0)\\)\u6709\uff1a<\/p>\n\n\n\n

$$
||(2^{\\rho+1}p_0,q_0r_1-q_1r_0,\\cdots,q_0r_t-q_tr_0)||\\approx2^{\\rho+\\gamma-\\eta+1}\\sqrt{t+1}
$$<\/p>\n\n\n\n

\u53c8\u5bf9\u4e8e\u683c\u57fa\u77e9\u9635\uff0c\u6709\uff1a<\/p>\n\n\n\n

$$
|\\det(\\pmb{B})|=2^{\\rho+1}x_0^t\\approx2^{\\rho+\\gamma t+1}
$$<\/p>\n\n\n\n

\u90a3\u4e48\u7531\u9ad8\u65af\u542f\u53d1\u5f0f\u53ef\u77e5\u5f53<\/p>\n\n\n\n

$$
2^{\\rho+\\gamma-\\eta+1}\\sqrt{t+1}\\le\\sqrt{\\frac{t+1}{2\\pi e}}\\cdot|\\det(\\pmb{B})|^{1\/(t+1)}\\approx2^{(\\rho+\\gamma t+1)\/(t+1)}\\sqrt{t+1}
$$<\/p>\n\n\n\n

\u6210\u7acb\uff0c\u5373\uff1a<\/p>\n\n\n\n

$$
\\begin{aligned}
&\\rho+\\gamma-\\eta+1\\le \\frac{\\rho+\\gamma t+1}{t+1}\\\\
\\Rightarrow&\\rho-\\eta+1\\le\\frac{\\rho+\\gamma t+1-\\gamma t-\\gamma}{t+1}=\\frac{\\rho-\\gamma+1}{t+1}\\\\
\\Rightarrow&t+1\\le\\frac{\\gamma-\\rho-1}{\\eta-\\rho-1}<\\frac{\\gamma-\\rho}{\\eta-\\rho}\\\\
\\Rightarrow&t+1<\\frac{\\gamma-\\rho}{\\eta-\\rho}
\\end{aligned}
$$<\/p>\n\n\n\n

\u7684\u65f6\u5019\uff0c\u53ef\u4ee5\u4ece\u8fd9\u4e2a\u683c\u4e2d\u89c4\u7ea6\u51fa\u76ee\u6807\u5411\u91cf\\((2^{\\rho+1}p_0,q_0r_1-q_1r_0,\\cdots,q_0r_t-q_tr_0)\\)\uff0c\u6b64\u65f6\u6211\u4eec\u53ef\u4ee5\u4ece\u76ee\u6807\u5411\u91cf\u4e2d\u83b7\u5f97\\(q_0\\)\uff0c\u56e0\u4e3a\\(r_0\\)\u5f88\u5c0f\uff0c\u6240\u4ee5\u6709\\(x_0\\equiv pq_0+r_0\\equiv r_0\\pmod{q_0}\\)\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\\(r_0=x_0\\mod{q_0}\\)\uff0c\u4ece\u800c\u8ba1\u7b97\u51fa\\(p=\\frac{x_0-r_0}{q_0}\\).<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5173\u4e8e\u8fd1\u4f3c\u516c\u7ea6\u6570\u95ee\u9898<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[7,6,10],"class_list":["post-244","post","type-post","status-publish","format-standard","hentry","category-3","tag-crypto","tag-6","tag-10"],"_links":{"self":[{"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/posts\/244","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=244"}],"version-history":[{"count":3,"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/posts\/244\/revisions"}],"predecessor-version":[{"id":247,"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/posts\/244\/revisions\/247"}],"wp:attachment":[{"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/media?parent=244"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/categories?post=244"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.triode.cc\/index.php\/wp-json\/wp\/v2\/tags?post=244"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}