{"id":477241,"date":"2023-08-09T09:09:43","date_gmt":"2023-08-09T09:09:43","guid":{"rendered":""},"modified":"2024-07-01T04:50:32","modified_gmt":"2024-07-01T04:50:32","slug":"finite-field","status":"publish","type":"wiki","link":"https:\/\/oneproxy.pro\/cn\/wiki\/finite-field\/","title":{"rendered":"\u6709\u9650\u57df"},"content":{"rendered":"

\u6709\u9650\u57df\uff0c\u53c8\u79f0\u4f3d\u7f57\u74e6\u57df\uff0c\u662f\u62bd\u8c61\u4ee3\u6570\u7684\u91cd\u8981\u7ec4\u6210\u90e8\u5206\uff0c\u5728\u8bb8\u591a\u6570\u5b66\u548c\u8ba1\u7b97\u9886\u57df\u4e2d\u53d1\u6325\u7740\u91cd\u8981\u4f5c\u7528\u3002\u6709\u9650\u57df\u662f\u4e00\u4e2a\u5143\u7d20\u6570\u91cf\u6709\u9650\u7684\u57df\uff0c\u5728\u5bc6\u7801\u5b66\u3001\u7f16\u7801\u7406\u8bba\u3001\u8ba1\u7b97\u673a\u79d1\u5b66\u548c\u8bb8\u591a\u5176\u4ed6\u9886\u57df\u90fd\u6709\u7740\u91cd\u8981\u7684\u5e94\u7528\u3002<\/p>\n

\u65f6\u5149\u56de\u6eaf\uff1a\u6709\u9650\u57df\u7684\u8d77\u6e90\u548c\u65e9\u671f\u63d0\u53ca<\/h2>\n

\u6709\u9650\u57df\u6700\u521d\u662f\u5728\u8bd5\u56fe\u89e3\u51b3\u591a\u9879\u5f0f\u65b9\u7a0b\u7684\u80cc\u666f\u4e0b\u63cf\u8ff0\u7684\uff0c\u8fd9\u4e00\u8ffd\u6c42\u53ef\u4ee5\u8ffd\u6eaf\u5230\u53e4\u4ee3\u3002\u7136\u800c\uff0c\u8fd9\u4e00\u6982\u5ff5\u76f4\u5230 19 \u4e16\u7eaa\u624d\u9996\u6b21\u5f62\u5f0f\u5316\u3002\u6cd5\u56fd\u6570\u5b66\u5bb6\u57c3\u74e6\u91cc\u65af\u7279\u00b7\u4f3d\u7f57\u74e6 (\u00c9variste Galois) \u4e3a\u6709\u9650\u57df\u7684\u53d1\u5c55\u505a\u51fa\u4e86\u91cd\u5927\u8d21\u732e\uff0c\u4e3a\u4e86\u7eaa\u5ff5\u4ed6\uff0c\u6709\u9650\u57df\u901a\u5e38\u88ab\u79f0\u4e3a\u201c\u4f3d\u7f57\u74e6\u57df\u201d\u3002<\/p>\n

\u4f3d\u7f57\u534e\u7684\u5de5\u4f5c\u4e3a\u73b0\u4ee3\u7fa4\u8bba\u548c\u6709\u9650\u57df\u7684\u4e00\u822c\u7406\u8bba\u5960\u5b9a\u4e86\u57fa\u7840\u3002\u6709\u9650\u57df\u7684\u7cfb\u7edf\u7814\u7a76\u5728 20 \u4e16\u7eaa\u5f97\u5230\u4e86\u8fdb\u4e00\u6b65\u53d1\u5c55\uff0c\u7406\u67e5\u5fb7\u00b7\u6234\u5fb7\u91d1\u548c\u57c3\u7c73\u00b7\u8bfa\u7279\u7b49\u6570\u5b66\u5bb6\u505a\u51fa\u4e86\u91cd\u5927\u8d21\u732e\u3002<\/p>\n

\u6df1\u5165\u6316\u6398\uff1a\u7406\u89e3\u6709\u9650\u57df<\/h2>\n

\u6709\u9650\u57df\u672c\u8d28\u4e0a\u662f\u4e00\u7ec4\u6570\u5b57\uff0c\u6240\u6709\u57fa\u672c\u8fd0\u7b97\uff08\u52a0\u3001\u51cf\u3001\u4e58\u3001\u9664\uff0c\u4f46\u9664\u4ee5\u96f6\u9664\u5916\uff09\u5747\u5728\u6b64\u6570\u5b57\u4e0a\u5b9a\u4e49\uff0c\u5e76\u4e14\u5177\u6709\u6709\u7406\u6570\u3001\u5b9e\u6570\u6216\u590d\u6570\u6240\u671f\u671b\u7684\u5c5e\u6027\u3002<\/p>\n

\u6709\u9650\u57df\u6709\u4e24\u4e2a\u91cd\u8981\u5c5e\u6027\uff1a\u5e8f\u548c\u7279\u5f81\u3002\u5e8f\u662f\u6307\u57df\u4e2d\u5143\u7d20\u7684\u603b\u6570\uff0c\u800c\u7279\u5f81\u662f\u51b3\u5b9a\u57df\u7684\u7b97\u672f\u8fd0\u7b97\u7684\u5c5e\u6027\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6709\u9650\u57df\u7684\u5e8f\u59cb\u7ec8\u662f\u7d20\u6570\u6216\u7d20\u6570\u7684\u5e42\u3002<\/p>\n

\u5e55\u540e\uff1a\u6709\u9650\u57df\u7684\u5185\u90e8\u7ed3\u6784<\/h2>\n

\u5728\u6709\u9650\u57df\u7684\u5185\u90e8\u7ed3\u6784\u4e2d\uff0c\u6bcf\u4e2a\u5143\u7d20\u90fd\u53ef\u4ee5\u4e0e\u53e6\u4e00\u4e2a\uff08\u975e\u96f6\uff09\u5143\u7d20\u76f8\u52a0\u3001\u76f8\u51cf\u3001\u76f8\u4e58\u6216\u76f8\u9664\uff0c\u4ece\u800c\u5f97\u5230\u57df\u4e2d\u7684\u7b2c\u4e09\u4e2a\u5143\u7d20\u3002\u6b64\u5c5e\u6027\u79f0\u4e3a\u201c\u95ed\u5305\u201d\uff0c\u5b83\u5bf9\u4e8e\u6709\u9650\u57df\u7684\u529f\u80fd\u81f3\u5173\u91cd\u8981\u3002<\/p>\n

\u6b64\u5916\uff0c\u6709\u9650\u57df\u8fd8\u5177\u6709\u7ed3\u5408\u5f8b\u3001\u4ea4\u6362\u5f8b\u3001\u5206\u914d\u5f8b\u3001\u5355\u4f4d\u5143\u7684\u5b58\u5728\u548c\u9006\u5143\u7684\u5b58\u5728\u7b49\u6027\u8d28\u3002\u4ece\u672c\u8d28\u4e0a\u8bb2\uff0c\u6709\u9650\u57df\u5728\u6570\u5b66\u4e0a\u8868\u73b0\u5f97\u201c\u5f88\u597d\u201d\uff0c\u8fd9\u4f7f\u5f97\u5b83\u4eec\u5728\u5404\u79cd\u5e94\u7528\u4e2d\u975e\u5e38\u6709\u7528\u3002<\/p>\n

\u6709\u9650\u57df\u7684\u4e3b\u8981\u7279\u5f81<\/h2>\n

\u6709\u9650\u57df\u7684\u4e00\u4e9b\u4e3b\u8981\u7279\u5f81\u5305\u62ec\uff1a<\/p>\n

    \n
  1. \u72ec\u7279\u6027<\/strong>\uff1a\u5bf9\u4e8e\u6bcf\u4e2a\u7d20\u6570\u5e42 q\uff0c\u672c\u8d28\u4e0a\u53ea\u5b58\u5728\u4e00\u4e2a q \u9636\u6709\u9650\u57df\u3002<\/li>\n
  2. \u52a0\u6cd5\u548c\u4e58\u6cd5\u7ed3\u6784<\/strong>\uff1aq \u9636\u6709\u9650\u57df\u7684\u52a0\u6cd5\u7fa4\u7ed3\u6784\uff0c\u5176\u4e2d q = p^n\uff0c\u540c\u6784\u4e8e p \u9636\u5faa\u73af\u7fa4\u7684 n \u4e2a\u526f\u672c\u7684\u76f4\u63a5\u548c\u3002\u975e\u96f6\u5143\u7d20\u7684\u4e58\u6cd5\u7fa4\u662f q-1 \u9636\u5faa\u73af\u7fa4\u3002<\/li>\n
  3. \u5b50\u5b57\u6bb5\u7684\u5b58\u5728<\/strong>\uff1a\u5177\u6709 q = p^n \u4e2a\u5143\u7d20\u7684\u6709\u9650\u57df\u5bf9\u4e8e n \u7684\u6bcf\u4e2a\u9664\u6570 d \u90fd\u6709\u4e00\u4e2a\u5b50\u57df\u3002\u8fd9\u4e9b\u5b50\u57df\u4e2d\u7684\u6bcf\u4e00\u4e2a\u90fd\u662f\u591a\u9879\u5f0f x^(p^d) \u2013 x = 0 \u7684\u6240\u6709\u89e3\u7684\u96c6\u5408\u3002<\/li>\n<\/ol>\n

    \u7edf\u4e00\u4e2d\u7684\u591a\u6837\u6027\uff1a\u6709\u9650\u57df\u7684\u7c7b\u578b<\/h2>\n

    \u6709\u9650\u57df\u6309\u7167\u5176\u9636\u8fdb\u884c\u5206\u7c7b\uff0c\u901a\u5e38\u5c06q\u9636\u6709\u9650\u57df\u8bb0\u4e3aGF(q)\u3002\u4f8b\u5982\uff0c\u6709\u4e24\u4e2a\u5143\u7d20\u7684\u6709\u9650\u57df\u8bb0\u4e3aGF(2)\uff0c\u6709\u4e09\u4e2a\u5143\u7d20\u7684\u6709\u9650\u57df\u8bb0\u4e3aGF(3)\uff0c\u7b49\u7b49\u3002<\/p>\n

    \u6709\u9650\u57df\u7684\u9636\u5fc5\u987b\u662f\u7d20\u6570\u7684\u5e42\uff0c\u56e0\u6b64\u6709\u9650\u57df\u7684\u7c7b\u578b\u6709GF(p)\uff0cGF(p^2)\uff0cGF(p^3)\uff0cGF(p^4)\u7b49\uff0c\u5176\u4e2dp\u4e3a\u7d20\u6570\u3002<\/p>\n\n\n\n\n\n\n\n\n\n\n
    \u573a\u7684\u987a\u5e8f<\/th>\n\u6709\u9650\u57df\uff08GF\uff09<\/th>\n<\/tr>\n<\/thead>\n
    2<\/td>\nGF(2)<\/td>\n<\/tr>\n
    3<\/td>\nGF(3)<\/td>\n<\/tr>\n
    4<\/td>\nGF(4)<\/td>\n<\/tr>\n
    5<\/td>\nGF(5)<\/td>\n<\/tr>\n
    p<\/td>\nGF(p)<\/td>\n<\/tr>\n
    \u666e^<\/td>\nGF(p^n)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    \u6709\u9650\u57df\u7684\u5e94\u7528\u4e0e\u95ee\u9898\u89e3\u51b3<\/h2>\n

    \u6709\u9650\u57df\u5728\u8ba1\u7b97\u673a\u79d1\u5b66\u4e0e\u5de5\u7a0b\u4e2d\u8d77\u7740\u81f3\u5173\u91cd\u8981\u7684\u4f5c\u7528\uff0c\u7279\u522b\u662f\u5728\u6570\u636e\u4f20\u8f93\u548c\u52a0\u5bc6\u534f\u8bae\u4e2d\u3002\u5b83\u4eec\u5728\u7f16\u7801\u7406\u8bba\u4e2d\u5fc5\u4e0d\u53ef\u5c11\uff0c\u6709\u52a9\u4e8e\u7ea0\u6b63\u6570\u636e\u4f20\u8f93\u4e2d\u7684\u9519\u8bef\uff0c\u5728\u5bc6\u7801\u5b66\u4e2d\u5219\u63d0\u4f9b\u5b89\u5168\u7684\u4e92\u8054\u7f51\u901a\u4fe1\u3002<\/p>\n

    \u4f7f\u7528\u6709\u9650\u57df\u7684\u5e38\u89c1\u6311\u6218\u4e4b\u4e00\u662f\u6267\u884c\u64cd\u4f5c\u6240\u6d89\u53ca\u7684\u8ba1\u7b97\u590d\u6742\u6027\u3002\u8fd9\u79cd\u590d\u6742\u6027\u5728\u8f83\u5927\u7684\u57df\u4e2d\u5c24\u5176\u660e\u663e\u3002\u7136\u800c\uff0c\u8fd9\u4e2a\u95ee\u9898\u901a\u5e38\u53ef\u4ee5\u901a\u8fc7\u4f7f\u7528\u67e5\u627e\u8868\u6216\u5feb\u901f\u7b97\u6cd5\uff08\u4f8b\u5982\u7528\u4e8e\u6709\u9650\u57df\u4e2d\u591a\u9879\u5f0f\u4e58\u6cd5\u7684\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362 (FFT)\uff09\u6765\u7f13\u89e3\u3002<\/p>\n

    \u76f8\u4f3c\u6982\u5ff5\u7684\u6bd4\u8f83\u5206\u6790<\/h2>\n

    \u5c06\u6709\u9650\u57df\u4e0e\u5176\u4ed6\u7c7b\u4f3c\u6982\u5ff5\u8fdb\u884c\u6bd4\u8f83\uff0c\u533a\u5206\u6709\u9650\u57df\u548c\u73af\u6216\u7fa4\u975e\u5e38\u91cd\u8981\uff0c\u5b83\u4eec\u662f\u66f4\u4e00\u822c\u7684\u4ee3\u6570\u7ed3\u6784\u3002<\/p>\n\n\n\n\n\n\n\n\n\n\n
    \u8303\u56f4<\/th>\n\u6709\u9650\u57df<\/th>\n\u6212\u6307<\/th>\n\u56e2\u4f53<\/th>\n<\/tr>\n<\/thead>\n
    \u5173\u95ed<\/td>\n\u662f\u7684<\/td>\n\u662f\u7684<\/td>\n\u662f\u7684<\/td>\n<\/tr>\n
    \u5173\u8054\u6027<\/td>\n\u662f\u7684<\/td>\n\u662f\u7684<\/td>\n\u662f\u7684<\/td>\n<\/tr>\n
    \u8eab\u4efd\u5143\u7d20<\/td>\n\u662f\u7684<\/td>\n\u662f\u7684<\/td>\n\u662f\u7684<\/td>\n<\/tr>\n
    \u9006<\/td>\n\u662f\u7684<\/td>\n\u662f\uff08\u6dfb\u52a0\u5242\uff09<\/td>\n\u662f\u7684<\/td>\n<\/tr>\n
    \u4ea4\u6362\u6027<\/td>\n\u662f\uff08\u4e24\u79cd\u64cd\u4f5c\uff09<\/td>\n\u662f\uff08\u6dfb\u52a0\uff09<\/td>\n\u662f\u7684<\/td>\n<\/tr>\n
    \u5206\u914d\u6027<\/td>\n\u662f\u7684<\/td>\n\u662f\u7684<\/td>\n\u4e0d<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    \u4e0e\u6709\u9650\u57df\u76f8\u5173\u7684\u672a\u6765\u5c55\u671b<\/h2>\n

    \u5728\u672a\u6765\u6280\u672f\u9886\u57df\uff0c\u6709\u9650\u57df\u6709\u671b\u53d1\u6325\u91cd\u8981\u4f5c\u7528\u3002\u4f8b\u5982\uff0c\u91cf\u5b50\u8ba1\u7b97\u662f\u6709\u9650\u57df\u539f\u7406\u53ef\u80fd\u81f3\u5173\u91cd\u8981\u7684\u4e00\u4e2a\u9886\u57df\uff0c\u7279\u522b\u662f\u5728\u91cf\u5b50\u7ea0\u9519\u548c\u52a0\u5bc6\u7cfb\u7edf\u4e2d\u3002<\/p>\n

    \u6b64\u5916\uff0c\u968f\u7740\u673a\u5668\u5b66\u4e60\u548c\u4eba\u5de5\u667a\u80fd\u7684\u5174\u8d77\uff0c\u6709\u9650\u57df\u53ef\u4ee5\u627e\u5230\u65b0\u7684\u5e94\u7528\uff0c\u7279\u522b\u662f\u5728\u9690\u79c1\u4fdd\u62a4\u6570\u636e\u5206\u6790\u4e2d\uff0c\u4f8b\u5982\u540c\u6001\u52a0\u5bc6\u548c\u5b89\u5168\u591a\u65b9\u8ba1\u7b97\u3002<\/p>\n

    \u6709\u9650\u57df\u548c\u4ee3\u7406\u670d\u52a1\u5668<\/h2>\n

    \u867d\u7136\u6709\u9650\u57df\u53ef\u80fd\u6ca1\u6709\u76f4\u63a5\u5e94\u7528\u4e8e\u4ee3\u7406\u670d\u52a1\u5668\uff0c\u4f46\u5b83\u4eec\u5728\u4ee3\u7406\u670d\u52a1\u5668\u6240\u4f9d\u8d56\u7684\u5b89\u5168\u901a\u4fe1\u7684\u5e95\u5c42\u6280\u672f\u4e2d\u53d1\u6325\u7740\u91cd\u8981\u4f5c\u7528\u3002<\/p>\n

    \u4f8b\u5982\uff0c\u8bb8\u591a\u7528\u4e8e\u4fdd\u62a4\u7f51\u7edc\u4e0a\u6570\u636e\u4f20\u8f93\u7684\u52a0\u5bc6\u534f\u8bae\uff08\u4ee3\u7406\u670d\u52a1\u5668\u7684\u4e00\u9879\u5173\u952e\u529f\u80fd\uff09\u90fd\u4f9d\u8d56\u4e8e\u6709\u9650\u57df\u7b97\u6cd5\u3002\u5e7f\u6cdb\u7528\u4e8e\u7f51\u7edc\u52a0\u5bc6\u7684\u5b89\u5168\u5957\u63a5\u5b57\u5c42 (SSL) \u548c\u4f20\u8f93\u5c42\u5b89\u5168\u6027 (TLS) \u4f9d\u8d56\u4e8e\u5176\u52a0\u5bc6\u7b97\u6cd5\u4e2d\u6709\u9650\u57df\u7684\u6570\u5b66\u5c5e\u6027\u3002<\/p>\n

    \u76f8\u5173\u94fe\u63a5<\/h2>\n
      \n
    1. \u6709\u9650\u57df\uff1a\u7406\u8bba\u4e0e\u8ba1\u7b97<\/a><\/li>\n
    2. \u6709\u9650\u57df\u5728\u73b0\u4ee3\u5bc6\u7801\u5b66\u4e2d\u7684\u4f5c\u7528<\/a><\/li>\n
    3. \u6709\u9650\u57df\u53ca\u5176\u5e94\u7528<\/a><\/li>\n
    4. \u6709\u9650\u57df\u7b97\u6cd5\u53ca\u5176\u5728\u5bc6\u7801\u5b66\u4e2d\u7684\u4f5c\u7528<\/a><\/li>\n<\/ol>\n

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