{"id":97,"date":"2025-02-26T19:55:00","date_gmt":"2025-02-26T11:55:00","guid":{"rendered":"http:\/\/www.triode.cc\/?p=97"},"modified":"2025-09-28T11:15:08","modified_gmt":"2025-09-28T03:15:08","slug":"isogeny1-sidh","status":"publish","type":"post","link":"https:\/\/www.triode.cc\/index.php\/2025\/02\/26\/isogeny1-sidh\/","title":{"rendered":"\u540c\u6e90\uff081\uff09\u2014\u2014SIDH"},"content":{"rendered":"\n

\u53c2\u8003\u8d44\u6599<\/h2>\n\n\n\n
    \n
  1. Mathematics of Isogeny Based Cryptography<\/a><\/li>\n\n\n\n
  2. Supersingular isogeny key exchange for beginners<\/a><\/li>\n\n\n\n
  3. Elliptic Curves | SpringerLink<\/a><\/li>\n\n\n\n
  4. Isogeny | \u7cd6\u918b\u5c0f\u9e21\u5757\u7684blog<\/a><\/li>\n\n\n\n
  5. isogeny | languag3<\/a><\/li>\n\n\n\n
  6. ECCNotes4 – huangx607087’s Blog<\/a><\/li>\n<\/ol>\n\n\n\n

    \u524d\u7f6e\u77e5\u8bc6<\/h2>\n\n\n\n

    \u8d85\u5947\u5f02\uff08supersingular\uff09\u692d\u5706\u66f2\u7ebf<\/h3>\n\n\n\n

    \u5bf9\u4e8e\\(GF(p^r)\\)\u4e0b\u7684\u692d\u5706\u66f2\u7ebf\\(E\\)\uff0c\u82e5\\(|E|\\equiv 1\\pmod{p}\\)\uff0c\u5219\u79f0\u8be5\u66f2\u7ebf\u4e3a\u8d85\u5947\u5f02\u692d\u5706\u66f2\u7ebf<\/p>\n\n\n\n

    j-\u4e0d\u53d8\u91cf\uff08j-invariant\uff09<\/h3>\n\n\n\n

    \u5bf9\u4e8e\u4e00\u6761\u7531\u65b9\u7a0b\\(y^2=x^3+ax+b\\)\u5b9a\u4e49\u7684\u692d\u5706\u66f2\u7ebf\\(E\\)\uff0c\u5176\\(j\\)-\u4e0d\u53d8\u91cf\u7684\u5b9a\u4e49\u4e3a\uff1a<\/p>\n\n\n\n

    $$
    j(E)= 1728\\cdot\\frac{4a^3}{4a^3+27b^2}
    $$<\/p>\n\n\n\n

    \u5f53\u4e14\u4ec5\u5f53\u4e24\u6761\u5b9a\u4e49\u5728\u4ee3\u6570\u95ed\u5305\\(\\overline{k}\\)\u4e0a\u7684\u66f2\u7ebf\\(E\\)\u4e0e\\(E’\\)\u7684\\(j\\)-\u4e0d\u53d8\u91cf\u76f8\u540c\u65f6\u8fd9\u4e24\u6761\u66f2\u7ebf\u540c\u6784\uff0c\u5373\u5b58\u5728\u70b9\u5230\u70b9\u4e4b\u95f4\u7684\u53cc\u5c04\\(\\phi\\)\u4f7f\u5f97\uff1a<\/p>\n\n\n\n

    $$
    \\phi:E\\mapsto E’
    $$<\/p>\n\n\n\n

    \u901a\u8fc7\u53c2\u8003\u8d44\u65992\u6211\u4eec\u53ef\u4ee5\u77e5\u9053\u5bf9\u4e8eSIDH\uff0c\u5176\u4e3b\u8981\u4f5c\u7528\u4e8e\u6709\u9650\u57df\\(GF(p)\\)\u7684\u4e8c\u6b21\u6269\u5c55\uff08\u5373\\(GF(p^2)\\)\uff0c\u5176\u4e2d\\(p\\equiv 3\\pmod{4}\\)\uff09\uff0c\u65b9\u4fbf\u8d77\u89c1\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u8fd9\u4e2a\u4e8c\u6b21\u6269\u5c55\u57df\u4e2d\u7684\u5143\u7d20\u8868\u793a\u4e3a\\(u+vi\\)\uff08\u5176\u4e2d\\(u,v\\in GF(p)\\)\uff0c\\(i^2\\equiv-1\\pmod{p}\\)\uff09\uff0c\u5b9a\u4e49\u5728\u8fd9\u4e2a\u4e8c\u6b21\u6269\u5c55\u57df\u4e0a\u7684\u8d85\u5947\u5f02\\(j\\)-\u4e0d\u53d8\u91cf\u7684\u4e2a\u6570\u4e3a\\(\\lfloor p\/12\\rfloor+z\\)\uff0c\u8fd9\u4e9b\u8d85\u5947\u5f02\\(j\\)-\u4e0d\u53d8\u91cf\u6bcf\u4e00\u4e2a\u90fd\u5bf9\u5e94\u4e00\u6761\u540c\u6784\u610f\u4e49\u4e0b\u7684\u8d85\u5947\u5f02\u66f2\u7ebf\uff0c\u5176\u4e2d\\(z\\in\\{0,1,2\\}\\)\uff0c\\(z\\)\u7684\u53d6\u503c\u53d6\u51b3\u4e8e\\(p\\mod 12\\)\uff0c\u5177\u4f53\u5982\u4e0b\uff08\u53c2\u8003\u8d44\u65993\uff0cP264\uff09\uff1a<\/p>\n\n\n\n

    $$
    z=\\begin{cases} 0&,p\\equiv1\\pmod{12}\\\\ 1&,p\\equiv5\\pmod{12}\\\\ 1&,p\\equiv7\\pmod{12}\\\\ 2&,p\\equiv11\\pmod{12}\\\\ \\end{cases}
    $$<\/p>\n\n\n\n

    \u4f8b\u5982\u5bf9\u4e8e\u53c2\u8003\u8d44\u65992\u4e2d\u7684\u4f8b\u5b50\uff08\u5373\\(p=431\\)\uff09\uff0c\\(GF(p^2)\\)\u4e2d\u7684\\(\\lfloor p\/12\\rfloor+2=37\\)\u4e2a\\(j\\)-\u4e0d\u53d8\u91cf\u5206\u522b\u5bf9\u5e94\u4e00\u6761\u540c\u6784\u610f\u4e49\u4e0b\u7684\u8d85\u5947\u5f02\u66f2\u7ebf\uff0c\u8fd9\u4e9b\\(j\\)-\u4e0d\u53d8\u91cf\u5982\u4e0b\u56fe\u6240\u793a\uff1a<\/p>\n\n\n

    \n
    \"\"<\/div><\/a><\/figure>\n<\/div>\n\n\n

    \u5173\u4e8e\\(j\\)-\u4e0d\u53d8\u91cf\u4e0e\u66f2\u7ebf\u95f4\u540c\u6784\u7684\u5173\u7cfb\uff0c\u53c2\u8003\u8d44\u65992\u7ed9\u51fa\u4e86\u4e00\u4e2a\u4f8b\u5b50\uff1a\u5bf9\u4e8e\\(GF(431^2)\\)\u4e0b\u7684\u4e24\u6761\u66f2\u7ebf\uff1a<\/p>\n\n\n\n

    $$
    \\begin{aligned} E_1:y^2=x^3+(208i+161)x^2+x\\\\ E_2:y^2=x^3+(172i+162)x^2+x\\\\ \\end{aligned}
    $$<\/p>\n\n\n\n

    \u53ef\u4ee5\u8ba1\u7b97\u51fa\\(j(E_1)=j(E_2)=364i+304\\)\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5982\u4e0b\u4ee3\u7801\u6765\u6c42\u51fa\\(E_1\\)\u5230\\(E_2\\)\u7684\u540c\u6784\u6620\u5c04\uff1a<\/p>\n\n\n\n

    <\/circle><\/circle><\/circle><\/g><\/svg><\/span>