{"id":202205,"date":"2024-05-10T03:11:49","date_gmt":"2024-05-09T19:11:49","guid":{"rendered":"https:\/\/docs.pingcode.com\/ask\/ask-ask\/202205.html"},"modified":"2024-05-10T03:11:52","modified_gmt":"2024-05-09T19:11:52","slug":"%e6%9c%89%e5%93%aa%e4%ba%9b-%e6%9c%89%e9%99%90%e5%9f%9f%e4%b8%8a-%e7%9a%84%e5%a4%9a%e9%a1%b9%e5%bc%8f%e5%9b%a0%e5%bc%8f%e5%88%86%e8%a7%a3-%e7%ae%97%e6%b3%95","status":"publish","type":"post","link":"https:\/\/docs.pingcode.com\/ask\/ask-ask\/202205.html","title":{"rendered":"\u6709\u54ea\u4e9b \u6709\u9650\u57df\u4e0a \u7684\u591a\u9879\u5f0f\u56e0\u5f0f\u5206\u89e3 \u7b97\u6cd5"},"content":{"rendered":"<p style=\"text-align:center\"><img decoding=\"async\" src=\"https:\/\/cdn-kb.worktile.com\/kb\/wp-content\/uploads\/2024\/04\/24141912\/a83918fc-c183-4b20-a881-b3e9a3a66eab.webp\" alt=\"\u6709\u54ea\u4e9b \u6709\u9650\u57df\u4e0a \u7684\u591a\u9879\u5f0f\u56e0\u5f0f\u5206\u89e3 \u7b97\u6cd5\" \/><\/p>\n<p><p>\u5728\u6709\u9650\u57df\u4e0a\u7684\u591a\u9879\u5f0f\u56e0\u5f0f\u5206\u89e3\u7b97\u6cd5\u4e3b\u8981\u6709\uff1a<strong>\u4f2f\u83b1\u574e\u666e-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5\uff08Berlekamp-Zassenhaus algorithm\uff09\u3001\u574e\u6258-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5\uff08Cantor-Zassenhaus algorithm\uff09\u3001\u4f2f\u83b1\u574e\u666e\u7b97\u6cd5\uff08Berlekamp&#039;s algorithm\uff09<\/strong>\u3002\u5176\u4e2d\uff0c<strong>\u4f2f\u83b1\u574e\u666e-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5<\/strong>\u5728\u5b9e\u9645\u64cd\u4f5c\u4e2d\u975e\u5e38\u91cd\u8981\uff0c\u5b83\u901a\u8fc7\u5c06\u56e0\u5f0f\u5206\u89e3\u8f6c\u5316\u4e3a\u7ebf\u6027\u4ee3\u6570\u95ee\u9898\uff0c\u5e76\u4f7f\u7528\u968f\u673a\u6027\u4ee5\u63d0\u9ad8\u6548\u7387\u3002\u9996\u5148\uff0c\u7b97\u6cd5\u5bf9\u591a\u9879\u5f0f\u6267\u884c\u5e73\u65b9\u81ea\u7531\u5206\u89e3\uff0c\u786e\u4fdd\u591a\u9879\u5f0f\u7684\u6bcf\u4e2a\u56e0\u5b50\u90fd\u662f\u552f\u4e00\u7684\uff0c\u63a5\u7740\u4f7f\u7528\u4f2f\u83b1\u574e\u666e\u7b97\u6cd5\u627e\u5230\u591a\u9879\u5f0f\u7684\u4e00\u4e2a\u975e\u5e73\u51e1\u56e0\u5b50\u3002\u4f2f\u83b1\u574e\u666e-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5\u53ef\u4ee5\u89c6\u4e3a\u8fd9\u4e00\u8fc7\u7a0b\u7684\u4f18\u5316\uff0c\u56e0\u4e3a\u5728\u627e\u5230\u56e0\u5b50\u4e4b\u540e\uff0c\u5b83\u901a\u8fc7\u7279\u5b9a\u7684\u968f\u673a\u9009\u62e9\u8fdb\u4e00\u6b65\u5c06\u95ee\u9898\u7b80\u5316\uff0c\u4ece\u800c\u6709\u6548\u5730\u5728\u591a\u9879\u5f0f\u73af\u4e2d\u5206\u89e3\u7ed9\u5b9a\u591a\u9879\u5f0f\u3002<\/p>\n<\/p>\n<p><h3>\u4e00\u3001\u4f2f\u83b1\u574e\u666e\u7b97\u6cd5<\/h3>\n<\/p>\n<p><p>\u4f2f\u83b1\u574e\u666e\u7b97\u6cd5\u662f\u4e00\u79cd\u5728\u6709\u9650\u57df\u4e2d\u5bf9\u591a\u9879\u5f0f\u8fdb\u884c\u56e0\u6570\u5206\u89e3\u7684\u7b97\u6cd5\u3002<strong>\u5b83\u7684\u6838\u5fc3\u601d\u60f3<\/strong>\u662f\u6784\u9020\u4e00\u4e2a\u77e9\u9635\uff0c\u7528\u4ee5\u523b\u753b\u591a\u9879\u5f0f\u6839\u636e\u6a21\u6570\u5206\u89e3\u7684\u6027\u8d28\uff0c\u5e76\u89e3\u4e00\u4e2a\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6765\u67e5\u627e\u975e\u5e73\u51e1\u7684\u56e0\u5b50\u3002\u7b97\u6cd5\u7684\u4e3b\u8981\u6b65\u9aa4\u5305\u62ec\u6784\u9020\u8fd9\u4e2a\u6240\u8c13\u7684\u4f2f\u83b1\u574e\u666e\u77e9\u9635\u5e76\u8ba1\u7b97\u5176\u884c\u7a7a\u95f4\u57fa\uff0c\u7136\u540e\u901a\u8fc7\u8ba1\u7b97\u77e9\u9635\u7684\u96f6\u7a7a\u95f4\u627e\u5230\u591a\u9879\u5f0f\u7684\u975e\u5e73\u51e1\u56e0\u5b50\u3002<\/p>\n<\/p>\n<p><p><strong>\u6784\u5efa\u4f2f\u83b1\u574e\u666e\u77e9\u9635\uff1a<\/strong><\/p>\n<\/p>\n<p><p>\u9996\u5148\uff0c\u7ed9\u5b9a\u4e00\u4e2a\u6709\u9650\u57df F_q \u4e0a\u7684\u591a\u9879\u5f0f f(x)\uff0c\u6211\u4eec\u5173\u5fc3\u8fd9\u4e2a\u591a\u9879\u5f0f\u5728 F_q \u4e2d\u7684\u56e0\u5f0f\u5206\u89e3\u3002\u5bf9\u8fd9\u4e2a\u591a\u9879\u5f0f\uff0c\u5e76\u53d6\u5176 x^q \u2212 x \u7684\u6700\u5927\u516c\u7ea6\u6570\u3002\u8be5\u6b65\u9aa4\u57fa\u4e8e\u8d39\u9a6c\u5c0f\u5b9a\u7406\uff0cx^q \u2212 x \u4f1a\u5728 F_q \u4e0a\u88ab f(x) \u7684\u4efb\u4f55\u56e0\u5b50\u6574\u9664\u3002\u7136\u540e\u6784\u9020\u4e00\u4e2a n\u00d7n \u7684\u77e9\u9635\uff0c\u5176\u4e2d n \u4e3a f(x) \u7684\u6b21\u6570\u3002\u77e9\u9635 B \u7684\u6bcf\u4e00\u4e2a\u5143\u7d20 b_ij \u7531 f(x) \u7684 x^q &#8211; x \u6a21 f(x) \u4e4b\u5546\u7684\u7cfb\u6570\u786e\u5b9a\u3002<\/p>\n<\/p>\n<p><p><strong>\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff1a<\/strong><\/p>\n<\/p>\n<p><p>\u8ba1\u7b97\u4f2f\u83b1\u574e\u666e\u77e9\u9635\u7684\u884c\u7ea6\u5316\u5f62\u5f0f\uff0c\u4ece\u800c\u5bf9\u77e9\u9635\u8fdb\u884c\u64cd\u4f5c\uff0c\u76f4\u5230\u5176\u53d8\u6210\u884c\u6700\u7b80\u5f62\u6216\u884c\u9636\u68af\u5f62\u3002\u8fd9\u6837\u505a\u80fd\u5e2e\u52a9\u786e\u8ba4\u77e9\u9635\u7684\u96f6\u7a7a\u95f4\u662f\u4ec0\u4e48\uff0c\u5373\u89e3\u7ebf\u6027\u65b9\u7a0b Bv = 0 \u7684\u975e\u5e73\u51e1\u89e3 v\u3002\u8fd9\u4e2a\u96f6\u7a7a\u95f4\u7684\u7ef4\u6570\u544a\u8bc9\u6211\u4eec\u591a\u9879\u5f0f\u7684\u4e0d\u53ef\u7ea6\u56e0\u5b50\u7684\u6570\u91cf\uff0c\u800c\u6bcf\u4e2a\u57fa\u5411\u91cf\u5bf9\u5e94\u4e8e f(x) \u7684\u4e00\u4e2a\u56e0\u5b50\u3002<\/p>\n<\/p>\n<p><h3>\u4e8c\u3001\u574e\u6258-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5<\/h3>\n<\/p>\n<p><p>\u574e\u6258-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5\u662f\u4e00\u4e2a\u6982\u7387\u7b97\u6cd5\uff0c\u7528\u4e8e\u5206\u89e3\u6709\u9650\u57df\u4e0a\u7684\u591a\u9879\u5f0f\u3002<strong>\u5173\u952e\u5728\u4e8e\u5b83\u4f7f\u7528\u968f\u673a\u65b9\u6cd5\u6765\u5bfb\u627e\u591a\u9879\u5f0f\u7684\u56e0\u5b50<\/strong>\uff0c\u8fd9\u4e0e\u4f2f\u83b1\u574e\u666e\u7b97\u6cd5\u4f7f\u7528\u7ebf\u6027\u4ee3\u6570\u65b9\u6cd5\u4e0d\u540c\u3002\u4ee5\u4e0b\u662f\u7b97\u6cd5\u7684\u4e3b\u8981\u6b65\u9aa4\uff1a<\/p>\n<\/p>\n<p><p><strong>\u751f\u6210\u968f\u673a\u591a\u9879\u5f0f\uff1a<\/strong><\/p>\n<\/p>\n<p><p>\u7b97\u6cd5\u4ece\u968f\u673a\u9009\u62e9\u4e00\u4e2a\u591a\u9879\u5f0f\u5f00\u59cb\uff0c\u8be5\u591a\u9879\u5f0f\u7684\u5ea6\u6570\u5c0f\u4e8e\u7ed9\u5b9a\u591a\u9879\u5f0f f(x)\u3002\u8fd9\u4e2a\u968f\u673a\u591a\u9879\u5f0f\u4e0d\u4fdd\u8bc1\u4e0e f(x) \u6709\u5171\u540c\u7684\u6839\uff0c\u4f46\u4e0e f(x) \u7684\u67d0\u4e9b\u4e0d\u53ef\u7ea6\u56e0\u5b50\u6709\u8f83\u9ad8\u7684\u6982\u7387\u4ea7\u751f\u5173\u8054\u3002<\/p>\n<\/p>\n<p><p><strong>\u5229\u7528\u8fd9\u4e2a\u968f\u673a\u591a\u9879\u5f0f\uff1a<\/strong><\/p>\n<\/p>\n<p><p>\u7b97\u6cd5\u8ba1\u7b97\u8fd9\u4e2a\u968f\u673a\u9009\u62e9\u7684\u591a\u9879\u5f0f\u4e0e f(x) \u5728\u6a21 f(x) \u610f\u4e49\u4e0b\u7684\u6700\u5927\u6b21\u6570\u5e42\u6b21\uff0c\u4e00\u4e2a\u5178\u578b\u7684\u9009\u62e9\u662f q^(n\/2)\uff0c\u8fd9\u91cc\u7684 q \u4e3a\u57df\u7684\u7279\u5f81\uff0cn \u4e3a f(x) \u7684\u5ea6\u6570\u3002\u7136\u540e\u8ba1\u7b97 gcd(f(x), \u968f\u673a\u591a\u9879\u5f0f\u7684 q^(n\/2) \u6b21\u5e42 &#8211; 1)\uff0c\u4ece\u800c\u5f97\u5230 f(x) \u7684\u4e00\u4e2a\u975e\u5e73\u51e1\u56e0\u5b50\u3002\u82e5\u679c\u6ca1\u6709\u627e\u5230\uff0c\u5219\u91cd\u65b0\u9009\u62e9\u968f\u673a\u591a\u9879\u5f0f\u3002<\/p>\n<\/p>\n<p><p>\u901a\u8fc7\u4e0d\u65ad\u8fed\u4ee3\u8fd9\u4e2a\u8fc7\u7a0b\uff0c\u7b97\u6cd5\u6700\u7ec8\u5c06\u627e\u5230 f(x) \u7684\u6240\u6709\u4e0d\u53ef\u7ea6\u56e0\u5b50\u3002\u6b63\u56e0\u4e3a\u8fd9\u79cd\u968f\u673a\u6027\uff0c\u574e\u6258-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5\u7279\u522b\u9002\u5408\u4e8e\u8ba1\u7b97\u5927\u591a\u9879\u5f0f\u7684\u56e0\u5f0f\u5206\u89e3\u3002<\/p>\n<\/p>\n<p><h3>\u4e09\u3001\u4f2f\u83b1\u574e\u666e-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5<\/h3>\n<\/p>\n<p><p>\u4f2f\u83b1\u574e\u666e-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5\u878d\u5408\u4e86\u4f2f\u83b1\u574e\u666e\u7b97\u6cd5\u548c\u574e\u6258-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5\u7684\u601d\u60f3\u3002<strong>\u8be5\u7b97\u6cd5\u9996\u5148\u4f7f\u7528\u4f2f\u83b1\u574e\u666e\u7b97\u6cd5\u83b7\u5f97\u4e00\u4e2a\u975e\u5e73\u51e1\u56e0\u5b50\uff0c\u7136\u540e\u91c7\u7528\u7c7b\u4f3c\u4e8e\u574e\u6258-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5\u7684\u968f\u673a\u7b56\u7565<\/strong>\u8fdb\u884c\u8fed\u4ee3\uff0c\u76f4\u5230\u627e\u5230\u6240\u6709\u7684\u4e0d\u53ef\u7ea6\u56e0\u5b50\u3002\u901a\u8fc7\u8fd9\u79cd\u65b9\u5f0f\uff0c\u7b97\u6cd5\u540c\u65f6\u4fdd\u8bc1\u4e86\u8fd0\u7b97\u7684\u786e\u5b9a\u6027\u548c\u6548\u7387\u3002<\/p>\n<\/p>\n<p><p>\u5728\u4f2f\u83b1\u574e\u666e-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5\u4e2d\uff0c\u4e3b\u8981\u7684\u6b65\u9aa4\u5982\u4e0b\uff1a<\/p>\n<\/p>\n<p><p><strong>\u8fd0\u7528\u7ebf\u6027\u4ee3\u6570\uff1a<\/strong><\/p>\n<\/p>\n<p><p>\u5229\u7528\u4f2f\u83b1\u574e\u666e\u7b97\u6cd5\u7684\u8fc7\u7a0b\u5206\u89e3\u591a\u9879\u5f0f\uff0c\u5e76\u786e\u5b9a f(x) \u5206\u89e3\u6210\u4e0d\u53ef\u7ea6\u56e0\u5b50\u7684\u53ef\u80fd\u6027\u3002<\/p>\n<\/p>\n<p><p><strong>\u5f15\u5165\u968f\u673a\u6027\uff1a<\/strong><\/p>\n<\/p>\n<p><p>\u5728\u83b7\u5f97\u53ef\u80fd\u7684\u975e\u5e73\u51e1\u56e0\u5b50\u540e\uff0c\u8fd0\u7528\u4e00\u79cd\u968f\u673a\u5316\u7b56\u7565\uff0c\u901a\u8fc7\u968f\u673a\u9009\u62e9\u591a\u9879\u5f0f\u4ee5\u53ca\u8ba1\u7b97\u8fd9\u4e9b\u591a\u9879\u5f0f\u7279\u5b9a\u6b21\u5e42\u7684\u6700\u5927\u516c\u7ea6\u6570\uff08gcd\uff09\u6765\u786e\u5b9a\u4e00\u4e2a\u786e\u5207\u7684\u975e\u5e73\u51e1\u56e0\u5b50\u3002<\/p>\n<\/p>\n<p><p>\u8fd9\u4e24\u79cd\u65b9\u6cd5\u7684\u5171\u540c\u70b9\u5728\u4e8e\u5b83\u4eec\u90fd\u5c06\u56e0\u5f0f\u5206\u89e3\u7684\u95ee\u9898\u8f6c\u6362\u6210\u6570\u5b66\u4e0a\u66f4\u6613\u4e8e\u5904\u7406\u7684\u95ee\u9898\uff0c\u5373\u5206\u522b\u662f\u7ebf\u6027\u4ee3\u6570\u95ee\u9898\u548c\u6709\u6982\u7387\u65b9\u6cd5\u53ef\u4ee5\u89e3\u51b3\u7684\u95ee\u9898\u3002\u901a\u8fc7\u7ed3\u5408\u8fd9\u4e24\u79cd\u65b9\u6cd5\uff0c\u4f2f\u83b1\u574e\u666e-\u624e\u68ee\u8c6a\u65af\u7b97\u6cd5\u53ef\u4ee5\u6709\u6548\u4e14\u9ad8\u6548\u5730\u5728\u6709\u9650\u57df\u4e0a\u5206\u89e3\u591a\u9879\u5f0f\u3002<\/p>\n<\/p>\n<h2><strong>\u76f8\u5173\u95ee\u7b54FAQs\uff1a<\/strong><\/h2>\n<p><strong>1. 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