{"id":912,"date":"2026-02-07T09:58:47","date_gmt":"2026-02-07T09:58:47","guid":{"rendered":"http:\/\/www.selinuxplus.com\/?p=912"},"modified":"2026-02-07T10:05:29","modified_gmt":"2026-02-07T10:05:29","slug":"schnorr%e9%98%88%e5%80%bc%e7%ad%be%e5%90%8d%e6%96%b9%e6%a1%88%ef%bc%9afrost","status":"publish","type":"post","link":"http:\/\/www.selinuxplus.com\/?p=912","title":{"rendered":"Schnorr\u9608\u503c\u7b7e\u540d\u65b9\u6848\uff1aFROST"},"content":{"rendered":"\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88frost\"><\/a><\/p>\n\n\n\n<h2>\u4ecb\u7ecd<\/h2>\n\n\n\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#%E4%BB%8B%E7%BB%8D\"><\/a><\/p>\n\n\n\n<p>FROST(Flexible Round-Optimized Schnorr Threshold Signatures)\u662f\u4e00\u79cd\u7075\u6d3b\u7684\u5faa\u73af\u4f18\u5316Schnorr\u9608\u503c\u7b7e\u540d\u65b9\u6848\uff0c\u5b83\u51cf\u5c11\u4e86\u7b7e\u540d\u64cd\u4f5c\u4e2d\u7684\u7f51\u7edc\u5f00\u9500\uff0c\u540c\u65f6\u63d0\u9ad8\u4e86Schnorr\u9608\u503c\u56e0\u4e3a\u7b7e\u540d\u534f\u8bae\u7684\u6280\u672f\u6c34\u5e73\uff0c\u5b83\u53ef\u4ee5\u5728\u5355\u8f6e\u4e2d\u5b89\u5168\u5730\u6267\u884c\u7b7e\u540d\u64cd\u4f5c\u800c\u4e0d\u662f\u7b7e\u540d\u64cd\u4f5c\u7684\u901a\u7528\u6027\uff0c\u4f46\u53c8\u5141\u8bb8\u771f\u6b63\u7684\u7b7e\u540d\u64cd\u4f5c\uff0c\u56e0\u6b64\u7b7e\u540d\u64cd\u4f5c\u53ea\u9700\u8981\u9608\u503c\u6570\u91cf\u7684\u53c2\u4e0e\u3002<\/p>\n\n\n\n<p>FROST\u65e2\u53ef\u4ee5\u4f18\u5316\u4e3a\u5177\u6709\u5171\u4eab\u9636\u6bb5\u7684\uff08\u975e\u5e7f\u64ad\uff09\u5355\u8f6e\u7b7e\u7f72\u534f\u8bae\u3002FROST\u5728\u6ca1\u6709\u53c2\u4e0e\u8005\u884c\u4e3a\u4e0d\u5f53\u7684\u4e50\u89c2\u60c5\u51b5\u4e0b\uff0c\u5b9e\u73b0\u4e86\u6548\u7387\u7684\u63d0\u9ad8\u3002\u56e0\u4e3a\u8c08\u5224\u56de\u5408\u53ef\u4ee5\u4e0e\u7b7e\u540d\u56de\u5408\u8fdb\u884c\uff0c\u6240\u4ee5\u7b7e\u540d\u64cd\u4f5c\u53ef\u4ee5\u5f02\u6b65\u8fdb\u884c\uff1b\u4e00\u65e6\u56de\u5408\u5b8c\u6210\uff0c\u7b7e\u540d\u8005\u53ea\u8981\u9700\u8981\u63a5\u6536\u5e76\u6700\u7ec8\u56de\u590d\u4e00\u4e2a\u6d88\u606f\u5c31\u53ef\u4ee5\u521b\u5efa\u4e00\u4e2a\u7b7e\u540d\u3002<\/p>\n\n\n\n<p>FROST\u5b9e\u73b0\u884c\u4e3a\u5141\u8bb8\u6548\u7387\u6539\u8fdb\u7684\u90e8\u5206\u539f\u56e0\u662f\u534f\u8bae\u5728\u5b58\u5728\u4e0d\u7aef\u7684\u53c2\u4e0e\u8005\u65f6\u4e2d\u6b62\uff08\u8be5\u53c2\u4e0e\u8005\u961f\u5217\u88ab\u8bc6\u522b\u5e76\u88ab\u6392\u9664\u5728\u672a\u6765\u7684\u64cd\u4f5c\u4e4b\u5916\uff09&#8211;\u8fd9\u662f\u5b9e\u9645\u90e8\u7f72\u573a\u666f\u7684\u884c\u4e3a\u7684\u5408\u7406\u6a21\u5f0f\u3002\u5728\u534f\u8bae\u8fc7\u7a0b\u4e2d\uff0c\u5982\u679c\u884c\u4e3a\u6709\u4e0d\u7aef\u7684\u53c2\u4e0e\u8005\u63d0\u4f9b\u6784\u9020\u7684\u503c\uff0c\u8bda\u5b9e\u7684\u5404\u65b9\u53ef\u4ee5\u8bc6\u522b\u5e76\u4fee\u590d\u4e0d\u7aef\u7684\u53c2\u4e0e\u8005\uff0c\u5e76\u91cd\u65b0\u8fd0\u884c\u534f\u8bae\u3002FROST\u7684\u7075\u6d3b\u8bbe\u8ba1\u80fd\u591f\u5176\u5176\u6301\u9608\u503c\u7b7e\u540d\u7684\u8bb8\u591a\u5b9e\u9645\u4f7f\u7528\u60c5\u51b5\u3002\u6b64\u5916\uff0c\u867d\u7136\u4e00\u4e9b\u9608\u503c\u65b9\u6848\u8981\u6c42\u6240\u6709\u53c2\u4e0e\u8005\u5728\u64cd\u4f5c\u7b7e\u540d\u671f\u95f4\u90fd\u662f\u6d3b\u8dc3\u7684\uff0c\u5e76\u5c06\u534f\u8bae\u7684\u9608\u503c\u5c5e\u6027\u79f0\u4e3a\u5b89\u5168\u5c5e\u6027\uff0c\u4f46 FROST \u5141\u8bb8\u4efb\u4f55\u9608\u503c\u6570\u91cf\u7684\u53c2\u4e0e\u8005\u4ea7\u751f\u6709\u6548\u7b7e\u540d\u3002\u56e0\u6b64\uff0cFROST \u53ef\u4ee5\u652f\u6301\u53c2\u4e0e\u8005\uff08\u6216\u53c2\u4e0e\u8bbe\u5907\uff09\u7684\u4e00\u4e2a\u5b50\u96c6\u53ef\u4ee5\u4fdd\u6301\u79bb\u7ebf\u7684\u4f8b\u5b50\uff0c\u8fd9\u4e2a\u5c5e\u6027\u5728\u5b9e\u8df5\u4e2d\u5f80\u5f80\u662f\u6240\u9700\u8981\u7684\u3002<\/p>\n\n\n\n<h2>\u80cc\u666f<\/h2>\n\n\n\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#%E8%83%8C%E6%99%AF\"><\/a><\/p>\n\n\n\n<h3>\u9608\u503c\u7b7e\u540d\u65b9\u6848<\/h3>\n\n\n\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88\"><\/a><\/p>\n\n\n\n<p>\u9608\u503c\u7b7e\u540d\u65b9\u6848\u662f\u4e00\u79cd\u5bc6\u7801\u5b66\u57fa\u672c\u539f\u7406\uff0c\u7528\u4e8e\u4fc3\u8fdb\u4e00\u7ec4\u53c2\u4e0e\u8005\u5bf9\u79c1\u94a5\u7684\u5171\u540c\u7b7e\u540d\uff0c\u8fd9\u6837\u4e00\u6765\uff0c\u9608\u503c\u7b7e\u540d\u65b9\u6848\u4e2d\u7684\u4e00\u4e9b\u5b9e\u73b0\u9700\u8981\u5728\u5927\u89c4\u6a21\u548c\u91cd\u9a8c\u8bc1\u4e0b\u6267\u884c\u7b7e\u540d\u64cd\u4f5c\u3002\u4f8b\u5982\uff0c\u9608\u503c\u7b7e\u540d\u53ef\u4ee5\u88ab\u4e00\u7ec4\u7b7e\u540d\u8005\u7528\u4e8e\u52a0\u5bc6\u8d27\u5e01\u4e2d\u7684\u91d1\u878d\u4ea4\u6613\uff0c\u6216\u7531\u4e00\u7ec4\u53d7\u4fe1\u673a\u6784\u4ea7\u751f\u7684\u7f51\u7edc\u5171\u8bc6\u3002\u5728\u8fd9\u4e24\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u968f\u7740\u7b7e\u7ea6\u65b9\u6216\u7b7e\u7ea6\u64cd\u4f5c\u6570\u91cf\u7684\u589e\u52a0\uff0c\u9664\u4e86\u6bcf\u4e2a\u7b7e\u7ea6\u65b9\u6240\u7ecf\u5386\u7684\u8d1f\u8f7d\u589e\u52a0\u4e4b\u5916\uff0c\u53c2\u4e0e\u8005\u4e4b\u95f4\u4e3a\u4ea7\u751f\u8054\u5408\u7b7e\u540d\u6240\u9700\u7684\u8f6e\u6570\u4e5f\u6210\u4e3a\u6027\u80fd\u7ec8\u70b9\u3002\u5f53\u7b7e\u540d\u8005\u5229\u7528\u7f51\u7edc\u6709\u9650\u7684\u8bbe\u5907\u6216\u4e0d\u53ef\u9760\u7684\u7f51\u7edc\u8fdb\u884c\u4f20\u8f93\uff0c\u6216\u8005\u5e0c\u671b\u5141\u8bb8\u7b7e\u540d\u8005\u53c2\u4e0e\u7b7e\u540d\u64cd\u4f5c\u7684\u534f\u8bae\u65f6\uff0c\u8fd9\u4e2a\u95ee\u9898\u4f1a\u56e0\u6b64\u8fdb\u4e00\u6b65\u6076\u5316\u3002\uff0c\u4f18\u5316\u7b7e\u540d\u64cd\u4f5c\u7684\u7f51\u7edc\u5bf9\u7b7e\u540d\u64cd\u4f5c\u7684\u5b9e\u9645\u5e94\u7528\u975e\u5e38\u5f15\u4eba\u6ce8\u76ee\u3002<\/p>\n\n\n\n<h3>\u6c99\u7c73\u5c14\u00b7\u8c22\u5171\u4eab<\/h3>\n\n\n\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#shamir%E5%AF%86%E9%92%A5%E5%85%B1%E4%BA%AB\"><\/a><\/p>\n\n\n\n<p>\u8bb8\u591a\u9608\u503c\u65b9\u6848\u5efa\u7acb\u5728Shamir\u79d8\u5bc6\u5171\u4eab\u7684\u57fa\u7840\u4e0a\uff0c\u8fd9\u662f\u4e00\u4e2a(t,n)\u7684\u9608\u503c\u7b7e\u540d\u65b9\u6848\uff0c\u4f9d\u9760\u62c9\u683c\u6717\u65e5\u63d2\u503c\u6765\u6062\u590d\u79d8\u5bc6\u3002\u5728Shamir\u79d8\u5bc6\u5171\u4eab\u4e2d\uff0c\u4e00\u4e2a\u53d7\u4fe1\u4efb\u7684\u4e2d\u592e\u7ecf\u9500\u5546\u5c06\u4e00\u4e2a\u79d8\u5bc6\u5206\u53d1\u7ed9\u6bcf\u4e2a\u53c2\u4e0e\u8005\uff0c\u5176\u65b9\u5f0f\u662f\u4efb\u4f55\u5408\u4f5c\u7684\u53c2\u4e0e\u8005\u7684\u5b50\u96c6\u90fd\u53ef\u4ee5\u6062\u590d\u8be5\u79d8\u5bc6\u3002\u4e3a\u4e86\u5206\u914d\u8fd9\u4e2a\u79d8\u5bc6\uff0c\u5e84\u5bb6\u9996\u5148\u968f\u673a\u9009\u62e9t-1\u4e2a\u7cfb\u6570a1\uff0c&#8230;\uff0cat-1\uff0c\u5e76\u4f7f\u7528\u968f\u673a\u9009\u62e9\u7684\u503c\u4f5c\u4e3a\u7cfb\u6570\u6765\u5b9a\u4e49\u4e00\u4e2a\u5ea6\u6570\u4e3at-1\u7684\u6295\u7968\u5f0ff\uff08x\uff09=s+SUM\uff08ai\uff09 xi\uff0ci=1&#8230;t-1\uff09\uff0c\u5176\u4e2df\uff080\uff09=s\uff0c\u53c2\u4e0e\u8005\u6bcf\u4e2aPi\u7684\u79d8\u5bc6\u8d21\u732e\u662f\uff08i\uff0cf\uff08i\uff09\uff09\uff0c\u5e84\u5bb6\u88ab\u4fe1\u4efb\u4e3a\u5f0f\u8bda\u5b9e\u5730\u5206\u914d\u7ed9\u6bcf\u4e2a\u53c2\u4e0e\u8005P1\uff0c&#8230;\uff0cPn\u3002\u4e3a\u4e86\u91cd\u5efa\u79d8\u5bc6\uff0c\u81f3\u5c11\u6709t\u4e2a\u53c2\u4e0e\u8005\u8fdb\u884c\u62c9\u683c\u6717\u65e5\u63d2\u503c\u6765\u91cd\u5efa\u5de5\u4f5c\uff0c\u4ece\u800c\u627e\u5230\u503cs=f(0)\u3002\u7136\u800c\uff0ct\u4e2a\u53c2\u4e0e\u8005\u7684\u5c0f\u7ec4\u4e0d\u80fd\u91cd\u5efa\u79d8\u5bc6\uff0c\u56e0\u4e3a\u81f3\u5c11\u9700\u8981t\u4e2a\u70b9\u6765\u91cd\u5efa1\u5ea6\u7684\u6539\u9020\u3002<\/p>\n\n\n\n<h3>Feldman \u7684\u53ef\u9a8c\u8bc1\u5bc6\u94a5\u5171\u4eab<\/h3>\n\n\n\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#feldman%E7%9A%84%E5%8F%AF%E9%AA%8C%E8%AF%81%E5%AF%86%E9%92%A5%E5%85%B1%E4%BA%AB\"><\/a><\/p>\n\n\n\n<p>Feldman\u7684\u53ef\u9a8c\u8bc1\u79d8\u5bc6\u5171\u4eab\uff08VSS\uff09\u65b9\u6848\u5efa\u7acb\u5728Shamir\u79d8\u5bc6\u5171\u4eab\u7684\u57fa\u7840\u4e0a\uff0c\u589e\u52a0\u4e86\u4e00\u4e2a\u9a8c\u8bc1\u6b65\u9aa4\uff0c\u4ee5\u8bc1\u660e\u53c2\u4e0e\u8005\u7684\u5171\u4eab\u4e0e\u516c\u5171\u627f\u8bfa\u7684\u4e00\u81f4\u6027\uff0c\u8be5\u627f\u8bfa\u88ab\u8ba4\u4e3a\u5bf9\u6240\u6709\u53c2\u4e0e\u8005\u662f\u6b63\u786e\u53ef\u89c1\u7684\u3002\u4e3a\u4e86\u9a8c\u8bc1\u4e00\u4e2a\u5171\u4eab\u7684\u5f62\u6210\uff0c\u6bcf\u4e2a\u53c2\u4e0e\u8005\u7528\u8fd9\u4e2a\u627f\u8bfa\u6765\u9a8c\u8bc1\u4ed6\u4eec\u7684\u5171\u4eab\u3002\u5982\u679c\u9a8c\u8bc1\u5931\u8d25\uff0c\u53c2\u4e0e\u8005\u53ef\u4ee5\u5bf9\u7ecf\u9500\u5546\u53d1\u5e03\u7533\u8bc9\uff0c\u5e76\u91c7\u53d6\u540c\u6837\u5411\u6240\u6709\u5176\u4ed6\u53c2\u4e0e\u8005\u5e7f\u64ad\u8fd9\u4e00\u7b7e\u540d\u7684\u884c\u52a8\u3002FROST\u4e5f\u540c\u6837\u4f7f\u7528\u4e86\u8fd9\u79cd\u6280\u672f\u3002\u5728Feldman\u7684\u65b9\u6848\u4e2d\u4ea7\u751f\u7684\u627f\u8bfa\u5982\u4e0b\u3002\u7c7b\u4f3c\u7684Shamir\u79d8\u5bc6\u5206\u4eab\u4e00\u6837\uff0c\u5e84\u5bb6\u5bf9t-1\u4e2a\u968f\u673a\u503c\uff08a1\uff0c&#8230;\uff0cat-1\uff09\u8fdb\u884c\u91c7\u6837\uff0c\u5c06\u8fd9\u4e9b\u503c\u4f5c\u4e3a\u7cfb\u6570\u6765\u5b9a\u4e49\u6570\u4e3at-1\u7684\u8282\u70b9\u5f0ffi\uff0c\u4f7f\u5f97f(0)=s\u3002\u7136\u800c\uff0c\u5728\u5411\u6bcf\u4e2a\u53c2\u4e0e\u8005Pi\u4e4b\u524d\u8fdb\u884c\u79c1\u4eba\u4ea4\u4e92\uff08i\uff0cf(i)\uff09\u7684\u540c\u65f6\uff0c\u5e84\u5bb6\u4e5f\u5236\u5b9a\u4e86\u516c\u5171\u627f\u8bfaC = &lt; A0, &#8230;, At-1 &gt;\uff0c\u5176\u4e2dA0 = gs\uff0cAj = g^{aj}\u3002 \u8bf7\u6ce8\u610f\uff0c\u5728\u5206\u6563\u8bbe\u7f6e\u4e2d\uff0c\u6bcf\u4e2a\u53c2\u4e0e\u8005Pi\u5fc5\u987b\u786e\u4fdd\u4e0e\u6240\u6709\u5176\u4ed6\u53c2\u4e0e\u8005\u5bf9C\u6709\u76f8\u540c\u7684\u770b\u6cd5\u3002\u5728\u5b9e\u8df5\u4e2d\uff0c\u5b9e\u73b0\u8005\u901a\u8fc7\u4f7f\u7528\u4e00\u4e9b\u6280\u672f\u6765\u4fdd\u8bc1\u5404\u65b9\u89c2\u70b9\u7684\u4e00\u81f4\u6027\uff0c\u4f8b\u5982\u5c06\u627f\u8bfa\u53d1\u5e03\u5230\u4e00\u4e2a\u96c6\u4e2d\u7684\u670d\u52a1\u5668\u4e0a\uff0c\u8be5\u670d\u52a1\u5668\u4fe1\u4efb\u4e3a\u6240\u6709\u53c2\u4e0e\u8005\u63d0\u4f9b\u5355\u4e00\u7684\u89c2\u70b9\uff0c\u6216\u8005\u589e\u52a0\u53e6\u4e00\u4e2a\u534f\u8bae\u5171\u8bc6\uff0c\u53c2\u4e0e\u8005\u6bd4\u8f83\u4ed6\u4eec\u6536\u5230\u7684\u627f\u8bfa\u503c\u4ee5\u786e\u4fdd\u5b83\u4eec\u662f\u76f8\u540c\u7684\u3002<\/p>\n\n\n\n<h3>\u798f\u514b\u65af\u751f\u6210<\/h3>\n\n\n\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#%E5%88%86%E5%B8%83%E5%BC%8F%E5%AF%86%E9%92%A5%E7%94%9F%E6%88%90\"><\/a><\/p>\n\n\n\n<p>\u4e0eShamir\u79d8\u5bc6\u5171\u4eab\u7b49\u4f9d\u8d56\u4e8e\u53ef\u4fe1\u4ea4\u6613\u5546\u7684\u9608\u503c\u65b9\u6848\u4e0d\u540c\uff0c\u5bc6\u94a5\u751f\u6210\uff08DKG\uff09\u786e\u4fdd\u6bcf\u4e2a\u53c2\u4e0e\u8005\u5bf9\u5171\u4eab\u79d8\u5bc6\u7684\u751f\u6210\u505a\u51fa\u540c\u7b49\u8d21\u732e\u3002\u5728\u534f\u8bae\u8fd0\u884c\u7ed3\u675f\u65f6\uff0c\u6240\u6709\u53c2\u4e0e\u8005\u5171\u4eab\u4e00\u4e2a\u8054\u5408\u53c2\u4e0e\u8005Y\uff0c\u4f46\u6bcf\u4e2a\u53c2\u4e0e\u8005\u53ea\u6301\u6709\u76f8\u5e94\u79d8\u5bc6\u7684\u4efd\u989d\uff0c\u56e0\u6b64\u6ca1\u6709\u4efb\u4f55\u4e00\u7ec4\u5c0f\u4e8e\u9608\u503c\u516c\u94a5\u751f\u6210\uff08DKG\uff09\u901a\u8fc7\u4f7f\u6bcf\u4e2a\u53c2\u4e0e\u8005\u5171\u4eab\u79d8\u5bc6\u7684\u751f\u6210\u540c\u7c7b\u505a\u51fa\u8d21\u732e\u6765\u652f\u6301\u8fd9\u79cd\u5a01\u80c1\u6a21\u578b\u3002\u5728\u534f\u8bae\u8fd0\u884c\u7ed3\u675f\u65f6\uff0c\u6240\u6709\u53c2\u4e0e\u8005\u5171\u4eab\u4e00\u4e2a\u5171\u540c\u53c2\u4e0e\u8005Y\uff0c\u4f46\u6bcf\u4e2a\u53c2\u4e0e\u8005\u53ea\u6301\u6709\u76f8\u5e94\u79d8\u5bc6\u7684\u4e00\u4e2a\u4efd\u989dsi\uff0c\u8fd9\u6837\u5c31\u6ca1\u6709\u5c0f\u4e8e\u9608\u503c\u7684\u53c2\u4e0e\u8005\u96c6\u5408\u77e5\u9053\u3002<\/p>\n\n\n\n<h3>\u65bd\u8bfa\u5c14\u7b7e\u540d<\/h3>\n\n\n\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#schnorr%E7%AD%BE%E5%90%8D\"><\/a><\/p>\n\n\n\n<p>\u901a\u5e38\uff0c\u7531\u9608\u503c\u7b7e\u540d\u64cd\u4f5c\u4ea7\u751f\u7684\u7b7e\u540d\u6700\u597d\u4e0e\u5355\u4e2a\u53c2\u4e0e\u8005\u4ea7\u751f\u7684\u7b7e\u540d\u65e0\u6cd5\u533a\u5206\uff0c\u4ece\u800c\u4e0e\u73b0\u6709\u7cfb\u7edf\u4fdd\u6301\u521a\u6027\u517c\u5bb9\uff0c\u5e76\u9632\u6b62\u4e2a\u522b\u7b7e\u540d\u8005\u7684\u9690\u79c1\u6cc4\u9732\u3002\u7531FROST\u7b7e\u540d\u64cd\u4f5c\u4ea7\u751f\u7684\u7b7e\u540d\u4e0eSchnorr\u7b7e\u540d\u533a\u5206\u4e0d\u53ef\uff0c\u56e0\u6b64\u53ef\u4f7f\u7528\u6807\u51c6\u7684Schnorr\u9a8c\u8bc1\u64cd\u4f5c\u8fdb\u884c\u9a8c\u8bc1\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u73b0\u5728\u63cf\u8ff0\u5355\u7b7e\u540d\u8005\u8bbe\u7f6e\u4e2d\u7684Schnorr\u7b7e\u540d\u548c\u9a8c\u8bc1\u64cd\u4f5c[Sch89]\u3002<\/p>\n\n\n\n<p>Schnorr \u7b7e\u540d\u751f\u6210\uff1a<\/p>\n\n\n\n<p><a target=\"_blank\" rel=\"noreferrer noopener\" href=\"https:\/\/camo.githubusercontent.com\/9940dc069bcc7cd53108bef16835c7b392ccc4e847e66573cd9bc4632616948f\/68747470733a2f2f63646e2e6a7364656c6976722e6e65742f67682f68616370792f50696374757265426564406d61737465722f446f63756d656e742f313632393138353031313839332d313632393138353031313839302e706e67\"><\/a>\uff09<\/p>\n\n\n\n<p>\u7b2c\u4e00\u6b65\u751f\u6210\u968f\u673a\u6570k\uff0c\u7136\u540e\u7528k\u751f\u6210\u968f\u673a\u70b9R<\/p>\n\n\n\n<p>\u7b2c\u4e8c\u6b65\u901a\u8fc7\u54c8\u5e0c\u968f\u673a\u70b9R\u548c\u5168\u5c40Y\u548c\u6d88\u606fm\u751f\u6210\u6311\u6218c<\/p>\n\n\n\n<p>\u7b2c\u4e09\u6b65\u7528\u79c1\u94a5s\u901a\u8fc7k+s*c\u8ba1\u7b97\u51faz<\/p>\n\n\n\n<p>\u7b2c\u56db\u6b65\u5c06R\u548cz\u7ec4\u5408\u751f\u6210\u7b7e\u540d<\/p>\n\n\n\n<p>Schnorr \u7b7e\u540d\u9a8c\u8bc1\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image\"><a href=\"https:\/\/camo.githubusercontent.com\/5b0190756660b3b7820499b0547887ab6a7539162531b27d1ab02cdbc21563aa\/68747470733a2f2f63646e2e6a7364656c6976722e6e65742f67682f68616370792f50696374757265426564406d61737465722f446f63756d656e742f313632393138353033373135362d313632393138353033373135342e706e67\" target=\"_blank\" rel=\"noreferrer noopener\"><img src=\"https:\/\/camo.githubusercontent.com\/5b0190756660b3b7820499b0547887ab6a7539162531b27d1ab02cdbc21563aa\/68747470733a2f2f63646e2e6a7364656c6976722e6e65742f67682f68616370792f50696374757265426564406d61737465722f446f63756d656e742f313632393138353033373135362d313632393138353033373135342e706e67\" alt=\"Schnorr \u9a8c\u8bc1\"\/><\/a><\/figure>\n\n\n\n<p>\u7b2c\u4e00\u6b65\u4ece\u7b7e\u540d\u4e2d\u89e3\u6790\u51faR\u548cz\uff0c\u7136\u540e\u8ba1\u7b97\u51fac<\/p>\n\n\n\n<p>\u7b2c\u4e8c\u6b65\u7531Z = R + Y*c\u8ba1\u7b97\u51faR&#8217;\uff0cZ\u662f\u7531z\u751f\u6210\u7684\u70b9<\/p>\n\n\n\n<p>\u7b2c\u4e09\u6b65\u5224\u65ad\u8ba1\u7b97\u51fa\u7684R&#8217;\u4e0e\u89e3\u6790\u51fa\u7684R\u662f\u5426\u4e00\u81f4<\/p>\n\n\n\n<h3>\u52a0\u6cd5\u79d8\u5bc6\u5171\u4eab<\/h3>\n\n\n\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#%E5%8A%A0%E6%B3%95%E7%A7%98%E5%AF%86%E5%85%B1%E4%BA%AB\"><\/a><\/p>\n\n\n\n<p>\u4e0e\u4e0a\u8ff0\u7684\u5355\u6807\u7c7b\u4f3c\uff0cFROST\u8981\u6c42\u4e3a\u6bcf\u4e2a\u7b7e\u540d\u64cd\u4f5c\u751f\u6210\u4e00\u4e2a\u968f\u673a\u6570k\u3002\u7136\u800c\uff0c\u5728\u83b7\u53d6\u503c\u7b7e\u540d\u4e2d\uff0c\u5e94\u8be5\u662f\u6bcf\u4e2a\u53c2\u4e0e\u8005\u90fd\u8981\u53c2\u4e0ek\u7684\u751f\u6210\uff0c\u4f46\u4e0d\u77e5\u9053\u5b83\u7684\u7ed3\u679c\u3002\u867d\u7136Shamir\u79d8\u5bc6\u5171\u4eab\u53ca\u5176\u6d3e\u751f\u7ed3\u6784\u8981\u6c42\u5728\u79d8\u5bc6\u79d8\u5bc6\u65b9\u5f0ff\u4e0a\u5171\u4eab\uff0c\u5176\u4e2df(0)=s\uff0c\u4f46\u52a0\u6cd5\u79d8\u5bc6\u5171\u4eab\u65b9\u6848\u5141\u8bb8\u6bcf\u4e2a\u53c2\u4e0e\u8005\u5171\u540c\u4e00\u4e2a\u8ba1\u7b97\u5171\u4eab\u79d8\u5bc6\uff0c\u6bcf\u4e2a\u53c2\u4e0e\u8005Pi\u8d21\u732e\u4e00\u4e2a\u503csi\uff0c\u8fd9\u6837\u5f97\u5230\u7684\u5171\u4eab\u79d8\u5bc6\u662fs=SUM(si, i=1&#8230;t)\uff0c\u5373\u6bcf\u4e2a\u53c2\u4e0e\u8005\u7684\u8d21\u732e\u4e4b\u548c\u3002\u56e0\u6b64\uff0c\u8fd9\u79cdt-out-of-t\u7684\u79d8\u5bc6\u5171\u4eab\u4e0d\u80fd\u8fdb\u884c\uff1b\u6bcf\u4e2a\u53c2\u4e0e\u8005\u76f4\u63a5\u9009\u62e9\u81ea\u5df1\u7684si\u3002\u5176\u4e2d\uff0csi\u662fs\u7684\u52a0\u6027\u79d8\u5bc6\u5171\u4eab\uff0c\u90a3\u4e48s\u662fsi\u7684\u603b\u548c\uff0c\u90a3\u4e48(si)\/(Li ) \u662f\u76f8\u540c\u7684 Shamir \u79d8\u5bc6\u5171\u4eab\uff0c\u5176\u4e2d Li \u662f\u62c9\u683c\u6717\u65e5\u7cfb\u6570\u3002\u5728 FROST \u4e2d\uff0c\u53c2\u4e0e\u8005\u5728\u7b7e\u540d\u64cd\u4f5c\u4e2d\u4f7f\u7528\u8fd9\u79cd\u6280\u672f\u975e\u5438\u5f15\u5730\u751f\u6210\u4e00\u4e2a\u4e00\u6b21\u6027\u7684\u79d8\u5bc6\u968f\u673a\u6570\uff0c\u5b83\u662f\u6240\u6709\u7b7e\u540d\u53c2\u4e0e\u8005\u4e4b\u95f4\u5171\u4eab\u7684 Shamir \u79d8\u5bc6\u3002<\/p>\n\n\n\n<h2>\u971c\u51bb\u65b9\u6848<\/h2>\n\n\n\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#frost%E6%96%B9%E6%A1%88\"><\/a><\/p>\n\n\n\n<p>\u6211\u4eec\u73b0\u5728\u63cf\u8ff0 FROST \u534f\u8bae\uff0c\u8fd9\u662f\u4e00\u79cd\u7075\u6d3b\u7684\u8f6e\u6b21\u4f18\u5316 Schnorr \u9608\u503c\u7b7e\u540d\u65b9\u6848\uff0c\u5b83\u6700\u5927\u9650\u5ea6\u5730\u51cf\u5c11\u4e86\u5728\u9608\u503c\u8bbe\u7f6e\u4e2d\u751f\u6210 Schnorr \u7b7e\u540d\u7684\u7f51\u7edc\u5f00\u9500\uff0c\u540c\u65f6\u5141\u8bb8\u7b7e\u540d\u64cd\u4f5c\u7684\u4e0d\u53d7\u9650\u5236\u7684\u53c2\u4e0e\u6027\u548c\u7b7e\u540d\u53c2\u4e0e\u8005\u7684\u9608\u503c\u6570\u91cf\u3002<\/p>\n\n\n\n<h3>\u5bc6\u94a5\u751f\u6210<\/h3>\n\n\n\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#%E5%AF%86%E9%92%A5%E7%94%9F%E6%88%90\"><\/a><\/p>\n\n\n\n<p>\u7b2c\u4e00\u8f6e\uff1a<\/p>\n\n\n\n<p>1. \u6bcf\u4e2a\u53c2\u4e0e\u8005Pi\u5747\u53c2\u4e0e\u62bd\u53d6t\u4e2a\u968f\u673a\u503c\uff08ai0, &#8230;, ai(t-1)\uff09) &lt;-$-Zq\uff0c\u5e76\u4f7f\u7528\u8fd9\u4e9b\u503c\u4f5c\u4e3a\u7cfb\u6570\u6765\u5b9a\u4e49Zq\u4e0a\u5ea6\u6570\u4e3at-1\u7684\u704c\u6e89\u5f0ffi(x) = SUM(aij xj, j=0&#8230;t-1)\u3002<\/p>\n\n\n\n<p>2. \u6bcf\u4e2a Pi \u901a\u8fc7\u4f7f\u7528 ai0 \u4f5c\u4e3a\u5bc6\u94a5\u8ba1\u7b97 Schnorr \u7b7e\u540d SIGi = (wi, ci) \u6765\u8ba1\u7b97\u5bf9\u5e94\u79d8\u5bc6 ai0 \u7684\u77e5\u8bc6\u8bc1\u660e\uff0c\u4f7f\u5f97 k &lt;-$- Zq, Ri = gk, ci = H(i, CTX, g^{ai0}, Ri), wi = k + ai0* ci\uff0c\u5176\u4e2d CTX \u662f\u9632\u6b62\u91cd\u653e\u653b\u51fb\u7684\u4e0a\u4e0b\u6587\u5b57\u7b26\u4e32\u3002<\/p>\n\n\n\n<p>3. \u6bcf\u4e2a\u53c2\u4e0e\u8005Pi\u8ba1\u7b97\u4e00\u4e2a\u516c\u5171\u627f\u8bfaCi = &lt; Ai0, &#8230;, Ai(t-1) &gt;\uff0c\u5176\u4e2dAij = g^{aij}, 0 &lt;= j &lt;= t-1<\/p>\n\n\n\n<p>4.\u6bcf\u4e2aPi\u5411\u6240\u6709\u5176\u4ed6\u53c2\u4e0e\u8005\u5e7f\u64adCi, SIGi\u3002 5.\u5728\u6536\u5230\u6765\u81ea\u53c2\u4e0e\u80051 &lt;= p &lt;= n, p != i\u7684Cp\u3001SIGp\u540e\uff0c\u53c2\u4e0e\u8005Pi\u9a8c\u8bc1SIGp = (wp, cp)\uff0c\u5931\u8d25\u65f6\u7ec8\u6b62\uff0c\u68c0\u67e5\uff1acp =?= H(p, CTX, Ap0, g^{wp} * Ap0^{ cp})<\/p>\n\n\n\n<p>\u7b2c\u4e8c\u8f6e\uff1a<\/p>\n\n\n\n<p>1. \u6bcf\u4e2aPi\u5411\u5176\u4ed6\u53c2\u4e0e\u8005Pp\u5b89\u5168\u5730\u8d21\u732e\u4e00\u4efd\u79d8\u5bc6\u8d21\u732e\uff08p\uff0cfi(p)\uff09\uff0c\u5e76\u4e3a\u81ea\u5df1\u4fdd\u7559\uff08i\uff0cfi(i)\uff09\u3002<\/p>\n\n\n\n<p>2. \u6bcf\u4e2a Pi \u901a\u8fc7\u8ba1\u7b97\uff1ag^{fp(i)} =?= PROD(Apk(i^k mod q),k=0&#8230;t-1) \u6765\u9a8c\u8bc1\u4ed6\u4eec\u7684\u8d21\u732e\uff0c\u5982\u679c\u68c0\u67e5\u5931\u8d25\u5219\u4e2d\u6b62\u3002<\/p>\n\n\n\n<p>3.\u6bcf\u4e2aPi\u901a\u8fc7\u8ba1\u7b97si = SUM(fp(i), p=1&#8230;n)\u6765\u8ba1\u7b97\u4ed6\u4eec\u957f\u671f\u5b58\u5728\u7684\u79c1\u4eba\u7b7e\u540d\u4efd\u989d\uff0c\u5e76\u5b89\u5168\u5730\u5b58\u50a8si\u3002<\/p>\n\n\n\n<p>4. \u53ef\u4ee5\u6bcf\u4e2a Pi \u8ba1\u7b97\u4ed6\u4eec\u7684\u516c\u5171\u9a8c\u8bc1\u8d21\u732e Yi = g^{si}\uff0c\u4ee5\u53ca\u8be5\u7ec4\u7684\u516c\u5f00\u9a8c\u8bc1\u8d21\u732e Y = PROD(Aj0, j=1&#8230;n)\u3002\u4efb\u4f55\u53c2\u4e0e\u8005\u90fd\u901a\u8fc7\u8ba1\u7b97 Yi = PROD( (Ajk)(i^k mod q), j=1&#8230;n, k=0&#8230;t-1) \u6765\u8ba1\u7b97\u4efb\u4f55\u5176\u4ed6\u53c2\u4e0e\u8005\u7684\u516c\u5f00\u9a8c\u8bc1\u8d21\u732e\u3002<\/p>\n\n\n\n<h3>\u8d1f\u62c5<\/h3>\n\n\n\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#%E9%A2%84%E5%A4%84%E7%90%86\"><\/a><\/p>\n\n\n\n<figure class=\"wp-block-image\"><a href=\"https:\/\/camo.githubusercontent.com\/df2aeccaa4f7465c182c805f2d1f1458442ac2cf3a5c53e8c330eb40f7174dac\/68747470733a2f2f63646e2e6a7364656c6976722e6e65742f67682f68616370792f50696374757265426564406d61737465722f446f63756d656e742f313632393236363238333437342d313632393236363238333437322e706e67\" target=\"_blank\" rel=\"noreferrer noopener\"><img src=\"https:\/\/camo.githubusercontent.com\/df2aeccaa4f7465c182c805f2d1f1458442ac2cf3a5c53e8c330eb40f7174dac\/68747470733a2f2f63646e2e6a7364656c6976722e6e65742f67682f68616370792f50696374757265426564406d61737465722f446f63756d656e742f313632393236363238333437342d313632393236363238333437322e706e67\" alt=\"\u9884\u5904\u7406\"\/><\/a><\/figure>\n\n\n\n<p>1.\u521b\u5efa\u4e00\u4e2a\u7a7a\u5217\u8868Li\u3002\u7136\u540e\uff0c\u5bf9\u4e8e1 &lt;= j &lt;= Q\uff0c\u6267\u884c\u4ee5\u4e0b\u64cd\u4f5c\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>- \u5355\u6b21\u4f7f\u7528\u7684nonces\u6837\u672c\uff08dij, eij\uff09&lt;-$- Zq* x Zq*\n- \u5f97\u51fa\u627f\u8bfa\u4efd\u989d\uff08Dij\uff0cEij\uff09=\uff08g^{dij}^\uff0cg^{eij}^\uff09\n- \u5c06\uff08Dij, Eij\uff09\u9644\u52a0\u5230Li\u3002\u5b58\u50a8((dij, Dij), (eij, Eij))\uff0c\u4ee5\u4fbf\u4ee5\u540e\u5728\u7b7e\u540d\u64cd\u4f5c\u4e2d\u4f7f\u7528\n<\/code><\/pre>\n\n\n\n<p>2.\u5c06(i, Li)\u53d1\u5e03\u5230\u4e00\u4e2a\u9884\u5b9a\u7684\u4f4d\u7f6e\uff0c\u8be5\u4f4d\u7f6e\u7531\u5b9e\u65bd\u8005\u6307\u5b9a\u3002<\/p>\n\n\n\n<h3>\u7b7e\u540d<\/h3>\n\n\n\n<p><a href=\"https:\/\/github.com\/chainx-org\/chainx-technical-archive\/blob\/main\/LiuBinXiao\/Taproot\/06_Schnorr%E9%98%88%E5%80%BC%E7%AD%BE%E5%90%8D%E6%96%B9%E6%A1%88FROST.md#%E7%AD%BE%E5%90%8D\"><\/a><\/p>\n\n\n\n<figure class=\"wp-block-image\"><a href=\"https:\/\/camo.githubusercontent.com\/7a736723bd7aa68573686136d452942d1bab5c3b85ae00f2e63809a40a2248c6\/68747470733a2f2f63646e2e6a7364656c6976722e6e65742f67682f68616370792f50696374757265426564406d61737465722f446f63756d656e742f313632393236363332373334322d313632393236363332373333382e706e67\" target=\"_blank\" rel=\"noreferrer noopener\"><img src=\"https:\/\/camo.githubusercontent.com\/7a736723bd7aa68573686136d452942d1bab5c3b85ae00f2e63809a40a2248c6\/68747470733a2f2f63646e2e6a7364656c6976722e6e65742f67682f68616370792f50696374757265426564406d61737465722f446f63756d656e742f313632393236363332373334322d313632393236363332373333382e706e67\" alt=\"\u7b26\u53f7\"\/><\/a><\/figure>\n\n\n\n<p>\u4ee4SA\u8868\u793a\u7b7e\u540d\u96c6\u5408\u8005\uff08\u4ed6\u81ea\u5df1\u53ef\u4ee5\u662f\u7b2ct\u4e2a\u7b7e\u540d\u53c2\u4e0e\u8005\u4e4b\u4e00\uff09\uff0cS\u88ab\u9009\u4e2d\u7528\u4e8e\u8be5\u7b7e\u540d\u64cd\u4f5c\u7684\u53c2\u4e0e\u8005\u7684\u96c6\u5408\uff0cB = &lt; (i, Dij, Eij) for i in S&gt;\u8868\u793a\u5bf9\u5e94\u4e8e\u6bcf\u4e2a\u53c2\u4e0e\u8005Pi\u7684\u53c2\u4e0e\u8005\u7d22\u5f15\u7684\u793e\u533a\u5217\u8868\uff0cLi\u662f\u5728\u5171\u4eab\u9636\u6bb5\u516c\u5e03\u7684Pi\u7684\u53ef\u7528\u627f\u8bfa\u503c\u7684\u96c6\u5408\u3002\u6bcf\u4e2a\u6807\u8bc6\u7b26i\u4e0ePi\u516c\u5e03\u7684\u7b2cj\u4e2a\u627f\u8bfa\uff08Dij\uff0cEij\uff09\u76f8\u8026\u5408\uff0c\u8fd9\u4e9b\u627f\u8bfa\u5c06\u7528\u4e8e\u8fd9\u4e2a\u7279\u5b9a\u7684\u7b7e\u540d\u64cd\u4f5c\u3002\u8ba9H1\u3001H2\u662f\u54c8\u5e0c\u51fd\u6570\uff0c\u5176\u8f93\u51fa\u5728Zq*\u3002<\/p>\n\n\n\n<p>1.SA\u9996\u5148\u4eceLi\u83b7\u53d6S\u4e2d\u6bcf\u4e2a\u53c2\u4e0e\u8005Pi\u7684\u4e0b\u4e00\u4e2a\u53ef\u7528\u627f\u8bfa\u5e76\u6784\u9020B\u3002<\/p>\n\n\n\n<p>2. \u5bf9\u4e8e S \u4e2d\u7684\u6bcf\u4e2a i\uff0cSA \u5411 Pi \u53d1\u9001\u5143\u7ec4 (m, B)\u3002<\/p>\n\n\n\n<p>3. \u6536\u5230 (m, B) \u540e\uff0c\u6bcf\u4e2a Pi \u9996\u5148\u9a8c\u8bc1\u6d88\u606f m\uff0c\u7136\u540e\u68c0\u67e5 G* \u4e2d\u7684 Dp j\u3001Ep j \u4ee5\u83b7\u53d6 B \u4e2d\u7684\u6bcf\u4e2a\u627f\u8bfa\uff0c\u5982\u679c\u5176\u4e2d\u4efb\u4f55\u4e00\u4e2a\u68c0\u67e5\u5931\u8d25\u5219\u4e2d\u6b62\u3002<\/p>\n\n\n\n<p>4.\u7136\u540e\uff0c\u6bcf\u4e2aPi\u8ba1\u7b97\u51fa\u7ed1\u5b9a\u503c\u7684\u96c6\u5408rp = H1(p, m, B), p\u5728S\u4e2d\u3002\u7136\u540e\u6bcf\u4e2aPi\u7ed3\u679c\u7ec4\u627f\u8bfaR = PROD(Dpj * (Epj)^{rp}, p\u5728S\u4e2d)\uff0c\u4ee5\u53ca\u6311\u6218c = H2(m, R)\u3002<\/p>\n\n\n\n<p>5. \u6bcf\u4e2aPi\u901a\u8fc7\u8ba1\u7b97zi = dij + (eij * ri) + Li * si * c\uff0c\u4f7f\u7528\u4ed6\u4eec\u7684\u957f\u671f\u79d8\u5bc6\u5171\u4eabsi\u8ba1\u7b97\u4ed6\u4eec\u7684\u54cd\u5e94\uff0c\u4f7f\u7528S\u6765\u786e\u5b9aLi\u3002<\/p>\n\n\n\n<p>6. \u6bcf\u4e2aPi\u4ece\u4ed6\u4eec\u7684\u672c\u5730\u5b58\u50a8\u4e2d\u5b89\u5168\u5730\u5220\u9664((dij\uff0cDij),(eij\uff0cEij))\uff0c\u7136\u540e\u5c06zi\u8fd4\u56de\u7ed9SA\u3002<\/p>\n\n\n\n<p>7.\u7b7e\u540d\u805a\u5408\u5668SA\u6267\u884c\u4ee5\u4e0b\u6b65\u9aa4\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u63a8\u5bfc\u51fari = H1(i,m,B)\uff0cRi = Dij * (Eij)^{ri}\uff0c\u7528\u4e8eS\u4e2d\u7684i\uff0c\u968f\u540eR=PROD(Ri, i in S)\uff0cc = H2(m,R)\n\u901a\u8fc7\u68c0\u67e5 g^{zi} =?= Ri * {Yi}^{c * Li} \u4e3a\u6bcf\u4e2a\u7b7e\u540d\u5171\u4eab zi, i \u5728 S \u4e2d\u9a8c\u8bc1\u6bcf\u4e2a\u54cd\u5e94\u7684\u6709\u6548\u6027\u3002\u5982\u679c\u4e0d\u76f8\u7b49\uff0c\u9996\u5148\u8bc6\u522b\u5e76\u62a5\u544a\u884c\u4e3a\u4e0d\u5f53\u53c2\u4e0e\u8005\uff0c\u7136\u540e\u4e2d\u6b62\u3002\u5426\u5219\uff0c\u7ee7\u7eed\n\u8ba1\u7b97\u7ec4\u7684\u54cd\u5e94 z = SUM(zi, i in S)\n\u5c06\u7b7e\u540dSIG = (z, c) \u4e0e\u4fe1\u606fm\u4e00\u8d77\u53d1\u5e03<\/code><\/pre>\n\n\n\n<p>SA\u6700\u540e\u68c0\u67e5\u6bcf\u4e2a\u53c2\u4e0e\u8005\u62a5\u544a\u7684zi\u4e0e\u4ed6\u4eec\u7684\u8d21\u732e\u8d21\u732e\uff08Dij\uff0cEij\uff09\u548c\u4ed6\u4eec\u7684\u6d88\u8017\u8d21\u732eYi\u662f\u5426\u4e00\u81f4\u3002\u5982\u679c\u6bcf\u4e2a\u53c2\u4e0e\u8005\u90fd\u53d1\u51fa\u4e86\u6b63\u786e\u7684zi\uff0c\u90a3\u4e48zi\u503c\u7684\u603b\u548c\u4e0ec\u4e00\u8d77\u6784\u6210\u4e86m\u4e0a\u7684Schnorr\u7b7e\u540d\u3002\u8fd9\u4e2a\u7b7e\u540d\u5c06\u88ab\u4e00\u4e2a\u4e0d\u77e5\u9053FROST\u88ab\u7528\u4e8e\u7b7e\u540d\u7684\u9a8c\u8bc1\u8005\u6b63\u786e\u751f\u6210\uff0c\u4ed6\u4f7f\u7528\u6807\u51c6\u7684\u5355\u65b9Schnorr\u9a8c\u8bc1\u65b9\u7a0b\u4e0eY\u4f5c\u4e3a\u7b7e\u540d\u8fdb\u884c\u9a8c\u8bc1\uff08\u7b2c2.4\u8282\uff09\u3002\u5904\u7406\u77ed\u6682\u7684\u672a\u51b3\u652f\u51fa\u3002\u7531\u4e8e\u5728\u4fee\u590d\u7b97\u6cd5\u4e2d\u63cf\u8ff0\u7684\u4fee\u590d\u9636\u6bb5\u4ea7\u751f\u6bcf\u4e2a\u968f\u673a\u6570\u548c\u627f\u8bfa\u8d21\u732e\u6700\u591a\u53ea\u80fd\u4f7f\u7528\u4e00\u6b21\uff0c\u53c2\u4e0e\u8005\u5728\u7b7e\u540d\u64cd\u4f5c\u4e2d\u4f7f\u7528\u8fd9\u4e9b\u503c\u540e\u5c06\u5176\u5220\u9664\uff0c\u5982\u7b7e\u540d\u7b97\u6cd5\u7684\u6b65\u9aa45\u6240\u793a\u3002\u4e00\u4e2a\u610f\u5916\u91cd\u7528\u7684(dij, eij)\u4f1a\u5bfc\u81f4\u53c2\u4e0e\u8005\u7684\u957f\u671f\u79d8\u5bc6si\u7684\u53c2\u4e0e\uff0c\u6240\u4ee5\u53c2\u4e0e\u8005\u5fc5\u987b\u5b89\u5168\u5730\u5220\u9664\u5b83\u4eec\uff0c\u5e76\u50cf\u4efb\u4f55Schnorr\u7b7e\u540d\u7684\u73b0\u5b9e\u4e00\u6837\u9632\u5fa1\u5feb\u7167\u56de\u6eda\u653b\u51fb\u3002\u7136\u800c\uff0c\u5982\u679cSA\u5728\u7b7e\u540d\u534f\u8bae\u671f\u95f4\u9009\u62e9\u91cd\u65b0\u4f7f\u7528\u4e00\u4e2a\u627f\u8bfa\u96c6\uff08Dij\uff0cEij\uff09\uff0c\u8fd9\u6837\u505a\u53ea\u4f1a\u5bfc\u81f4\u53c2\u4e0e\u8005Pi\u4e2d\u7ec8\u6b62\u534f\u8bae\uff0c\u56e0\u6b64\u4e0d\u4f1a\u589e\u52a0SA\u7684\u6743\u529b\u3002<\/p>\n\n\n\n<h2>\u603b\u7ed3<\/h2>\n\n\n\n<p><a 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