{"id":813,"date":"2026-03-04T12:18:55","date_gmt":"2026-03-04T04:18:55","guid":{"rendered":"https:\/\/vite66.cn\/?p=813"},"modified":"2026-03-05T11:11:53","modified_gmt":"2026-03-05T03:11:53","slug":"%e5%bf%ab%e9%80%9f%e5%b9%82","status":"publish","type":"post","link":"https:\/\/vite66.cn\/?p=813","title":{"rendered":"\u5feb\u901f\u5e42"},"content":{"rendered":"

\u5feb\u901f\u5e42\u5168\u77e5\u8bc6\u70b9<\/h1>\n

\u4e00\u3001\u5feb\u901f\u5e42\u6838\u5fc3\u4f18\u5316\u903b\u8f91<\/h2>\n

\u6838\u5fc3\uff1a\u901a\u8fc7\u5e95\u6570\u81ea\u4e58\u590d\u7528\u4e2d\u95f4\u7ed3\u679c\uff0c\u66ff\u4ee3\u91cd\u590d\u4e58\u6cd5\u8fd0\u7b97\uff0c\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4eceO(n)\uff08\u66b4\u529b\u7b97\u6cd5\uff09\u964d\u81f3O(log\u2096n)\uff08\u5feb\u901f\u5e42\uff09\uff1b\u4e8c\u8fdb\u5236\u3001\u4e09\u8fdb\u5236\u3001\u5341\u8fdb\u5236\u5feb\u901f\u5e42\u4f18\u5316\u903b\u8f91\u4e00\u81f4\uff0c\u5dee\u5f02\u4ec5\u5728\u4e8e\u6307\u6570\u62c6\u5206\u57fa\u5e95\u4e0e\u5e95\u6570\u81ea\u4e58\u65b9\u5f0f\u3002<\/p>\n

1. \u66b4\u529b\u7b97\u6cd5<\/h3>\n

\u8ba1\u7b97 $a^{10}$ \u9700\u8fde\u7eed\u6267\u884c10\u6b21a\u7684\u4e58\u6cd5\u8fd0\u7b97\uff0c\u5373 $a \\times a \\times \\dots \\times a$ \uff08\u517110\u6b21\u4e58\u6cd5\uff09\u3002<\/p>\n

\u75db\u70b9\uff1a\u4e58\u6cd5\u6b21\u6570\u4e0e\u6307\u6570b\u76f8\u7b49\uff0c\u9ad8\u6307\u6570\u573a\u666f\u6548\u7387\u4f4e\u4e0b\uff1b\u5b58\u5728\u5197\u4f59\u8ba1\u7b97\uff08\u5982\u8ba1\u7b97 $a^4$ \u65f6\uff0c\u5df2\u8ba1\u7b97\u7684 $a^2$ \u672a\u88ab\u590d\u7528\uff09\u3002<\/p>\n

2. \u6240\u6709\u8fdb\u5236\u5feb\u901f\u5e42\u7684\u7edf\u4e00\u4f18\u5316\u903b\u8f91<\/h3>\n

\u6838\u5fc3\u6570\u5b66\u57fa\u7840\uff1a $a^{m+n} = a^m \\times a^n$ \u3002\u4f18\u5316\u903b\u8f91\u5206\u4e09\u6b65\uff0c\u7ed3\u5408\u4e09\u79cd\u8fdb\u5236\u5177\u4f53\u8bf4\u660e\uff1a<\/p>\n

    \n
  1. \u6307\u6570\u62c6\u5206\uff1a\u5c06\u6307\u6570b\u62c6\u5206\u4e3a\u5bf9\u5e94\u8fdb\u5236\u57fa\u5e95\u7684\u5e42\u6b21\u548c\uff0c\u975e\u96f6\u9879\u6570\u91cf\u51b3\u5b9a\u4e58\u6cd5\u6b21\u6570\uff0c\u975e\u96f6\u9879\u8d8a\u5c11\uff0c\u4e58\u6cd5\u6b21\u6570\u8d8a\u5c11\u3002<\/li>\n<\/ol>\n
      \n
    • \n

      \u4e8c\u8fdb\u5236\uff08\u57fa\u5e952\uff09\uff1a $10 = 2^3 + 2^1$ \uff0c $a^{10}=a^8 \\times a^2$ \uff082\u6b21\u4e58\u6cd5\uff09\uff1b<\/p>\n<\/li>\n

    • \n

      \u4e09\u8fdb\u5236\uff08\u57fa\u5e953\uff09\uff1a $10 = 3^2 + 3^0$ \uff0c $a^{10}=a^9 \\times a^1$ \uff082\u6b21\u4e58\u6cd5\uff09\uff1b<\/p>\n<\/li>\n

    • \n

      \u5341\u8fdb\u5236\uff08\u57fa\u5e9510\uff09\uff1a $10 = 10^1$ \uff0c $a^{10}=a^{10}$ \uff081\u6b21\u4e58\u6cd5\uff09\u3002<\/p>\n<\/li>\n<\/ul>\n

        \n
      1. \u5e95\u6570\u81ea\u4e58\uff1a\u62c6\u5206\u540e\u6240\u9700\u5c0f\u5e42\u6b21\uff0c\u901a\u8fc7\u5e95\u6570\u8fed\u4ee3\u81ea\u4e58\u83b7\u53d6\uff0c\u907f\u514d\u5197\u4f59\u8ba1\u7b97\u3002<\/li>\n<\/ol>\n
          \n
        • \n

          \u4e8c\u8fdb\u5236\uff1a\u81ea\u4e58\uff08\u5e73\u65b9\uff09\uff0c $a \\to a^2 \\to a^4 \\to a^8$ \uff083\u6b21\u81ea\u4e58\uff09\uff1b<\/p>\n<\/li>\n

        • \n

          \u4e09\u8fdb\u5236\uff1a\u81ea\u4e58\uff08\u7acb\u65b9\uff09\uff0c $a \\to a^3 \\to a^9$ \uff082\u6b21\u81ea\u4e58\uff09\uff1b<\/p>\n<\/li>\n

        • \n

          \u5341\u8fdb\u5236\uff1a\u81ea\u4e58\uff0810\u6b21\u65b9\uff09\uff0c $a \\to a^{10}$ \uff089\u6b21\u81ea\u4e58\uff09\u3002<\/p>\n<\/li>\n<\/ul>\n

            \n
          1. \u53d6\u6a21\uff1aa\u3001b\u8f83\u5927\u65f6\uff0c\u81ea\u4e58\u7ed3\u679c\u6613\u6ea2\u51fa\uff0c\u6bcf\u6b65\u81ea\u4e58\u3001\u4e58\u6cd5\u540e\u5747\u9700\u6267\u884c\u53d6\u6a21\u64cd\u4f5c\uff0c\u4e0d\u6539\u53d8\u6700\u7ec8\u7ed3\u679c\uff0c\u9002\u7528\u4e8e\u6240\u6709\u8fdb\u5236\u3002<\/li>\n<\/ol>\n

            3. \u4f18\u5316\u6548\u679c\u5bf9\u6bd4<\/h3>\n\n\n\n\n\n\n\n\n
            \u8fd0\u7b97\u65b9\u5f0f<\/th>\n\u8ba1\u7b97 $a^{10}$ \u6b65\u9aa4<\/th>\n\u603b\u8fd0\u7b97\u6b21\u6570<\/th>\n\u590d\u6742\u5ea6<\/th>\n<\/tr>\n<\/thead>\n
            \u66b4\u529b<\/td>\na\u00d7a\u00d7a\u00d7a\u00d7a\u00d7a\u00d7a\u00d7a\u00d7a\u00d7a\uff0810\u6b21\u4e58\uff09<\/td>\n10\u6b21<\/td>\nO(n)<\/td>\n<\/tr>\n
            \u4e8c\u8fdb\u5236\u5feb\u901f\u5e42<\/td>\n\u81ea\u4e583\u6b21\uff08a\u2192a\u00b2\u2192a\u2074\u2192a\u2078\uff09\uff1b\u4e58\u6cd51\u6b21\uff08a\u2078\u00d7a\u00b2\uff09<\/td>\n4\u6b21<\/td>\nO(log\u2082n)<\/td>\n<\/tr>\n
            \u4e09\u8fdb\u5236\u5feb\u901f\u5e42<\/td>\n\u81ea\u4e582\u6b21\uff08a\u2192a\u00b3\u2192a\u2079\uff09\uff1b\u4e58\u6cd51\u6b21\uff08a\u2079\u00d7a\uff09<\/td>\n3\u6b21<\/td>\nO(log\u2083n)<\/td>\n<\/tr>\n
            \u5341\u8fdb\u5236\u5feb\u901f\u5e42<\/td>\n\u81ea\u4e589\u6b21\uff08a\u2192a\u00b9\u2070\uff09\uff1b\u4e58\u6cd50\u6b21<\/td>\n9\u6b21<\/td>\nO(log\u2081\u2080n)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

            \u7ed3\u8bba\uff1a\u6240\u6709\u8fdb\u5236\u5feb\u901f\u5e42\u5747\u901a\u8fc7\u201c\u6307\u6570\u62c6\u5206+\u5e95\u6570\u81ea\u4e58\u201d\u5b9e\u73b0\u4f18\u5316\uff0c\u5dee\u5f02\u4ec5\u5728\u4e8e\u62c6\u5206\u57fa\u5e95\uff082\u30013\u300110\uff09\uff1b\u57fa\u5e95\u8d8a\u5c0f\uff0c\u5e95\u6570\u81ea\u4e58\u64cd\u4f5c\u8d8a\u7b80\u5355\uff0c\u603b\u8fd0\u7b97\u6b21\u6570\u8d8a\u5c11\uff0c\u4e8c\u8fdb\u5236\u5feb\u901f\u5e42\u4f18\u5316\u6548\u7387\u6700\u4f18\u3002<\/p>\n

            \u4e8c\u3001\u4e0d\u540c\u8fdb\u5236\u5feb\u901f\u5e42\u5e95\u5c42\u5dee\u5f02<\/h2>\n\n\n\n\n\n\n\n
            \u5bf9\u6bd4\u7ef4\u5ea6<\/th>\n\u4e8c\u8fdb\u5236<\/th>\n\u4e09\u8fdb\u5236<\/th>\n\u5341\u8fdb\u5236<\/th>\n<\/tr>\n<\/thead>\n
            \u62c6\u5206\u57fa\u5e95<\/td>\n2\u7684\u5e42<\/td>\n3\u7684\u5e42<\/td>\n10\u7684\u5e42<\/td>\n<\/tr>\n
            \u5e95\u6570\u81ea\u4e58\u65b9\u5f0f<\/td>\n\u5e73\u65b9\uff081\u6b21\u81ea\u4e58\uff09<\/td>\n\u7acb\u65b9\uff082\u6b21\u81ea\u4e58\uff09<\/td>\n10\u6b21\u65b9\uff089\u6b21\u81ea\u4e58\uff09<\/td>\n<\/tr>\n
            \u4f18\u5316\u6548\u7387<\/td>\n\u6700\u9ad8\uff0c\u81ea\u4e58\u7b80\u5355\u3001\u5197\u4f59\u6700\u5c11<\/td>\n\u4e2d\u7b49\uff0c\u81ea\u4e58\u590d\u6742\u3001\u5c11\u91cf\u5197\u4f59<\/td>\n\u6700\u4f4e\uff0c\u81ea\u4e58\u6700\u590d\u6742\u3001\u5197\u4f59\u591a<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

            \u4e09\u3001C++\u4ee3\u7801\u5b9e\u73b0<\/h2>\n
            #include <iostream>\nusing namespace std;\n\n\/\/ \u4e8c\u8fdb\u5236\u5feb\u901f\u5e42\nlong long binpow(long long a, long long b, long long mod) {\n    long long res = 1;\n    a = a % mod;\n    while (b > 0) {\n        if (b & 1) {\n            res = (res * a) % mod;\n        }\n        a = (a * a) % mod;\n        b >>= 1;\n    }\n    return res;\n}\n\n\/\/ \u4e09\u8fdb\u5236\u5feb\u901f\u5e42\nlong long pow_ternary(long long a, long long b, long long mod) {\n    long long res = 1;\n    long long base = a % mod;\n    while (b > 0) {\n        int digit = b % 3;\n        for (int i = 0; i < digit; ++i) {\n            res = (res * base) % mod;\n        }\n        long long new_base = 1;\n        for (int i = 0; i < 3; ++i) {\n            new_base = (new_base * base) % mod;\n        }\n        base = new_base;\n        b = b \/ 3;\n    }\n    return res;\n}\n\n\/\/ \u5341\u8fdb\u5236\u5feb\u901f\u5e42\nlong long pow_decimal(long long a, long long b, long long mod) {\n    long long res = 1;\n    long long base = a % mod;\n    while (b > 0) {\n        int digit = b % 10;\n        for (int i = 0; i < digit; ++i) {\n            res = (res * base) % mod;\n        }\n        long long new_base = 1;\n        for (int i = 0; i < 10; ++i) {\n            new_base = (new_base * base) % mod;\n        }\n        base = new_base;\n        b = b \/ 10;\n    }\n    return res;\n}\n\nint main() {\n    cout << binpow(2, 10, 1e18) << endl;\n    cout << pow_ternary(2, 10, 7) << endl;\n    cout << pow_decimal(2, 123, 1000) << endl;\n    return 0;\n}<\/code><\/pre>\n
            \u56db\u3001\u53d6\u6a21\u8fd0\u7b97\u6838\u5fc3<\/h2>\n
            \u5feb\u901f\u5e42\u6838\u5fc3\u5e94\u7528\u4e3a\u8ba1\u7b97 $a^b \\mod \\text{mod}$ \uff0c\u57fa\u4e8e\u4ee5\u4e0b\u53d6\u6a21\u516c\u5f0f\uff0c\u76f8\u5173\u8bc1\u660e\u5982\u4e0b\uff1a<\/p>\n
            1. \u6838\u5fc3\u516c\u5f0f<\/h3>\n
              \n
            1. \n
              \u52a0\u6cd5\u53d6\u6a21\uff1a(a + b) % p = (a % p + b % p) % p<\/p>\n<\/li>\n
            2. \n
              \u4e58\u6cd5\u53d6\u6a21\uff1a(a \u00d7 b) % p = [(a % p) \u00d7 (b % p)] % p<\/p>\n<\/li>\n
            3. \n
              \u5e42\u53d6\u6a21\uff1a(a^b) % p = [(a % p)^b] % p<\/p>\n<\/li>\n<\/ol>\n
              2. \u516c\u5f0f\u4e25\u8c28\u8bc1\u660e<\/h3>\n
              \u57fa\u4e8e\u5e26\u4f59\u9664\u6cd5\uff1a\u4efb\u610f\u6574\u6570a\u53ef\u8868\u793a\u4e3aa = q\u00d7p + r\uff08q\u4e3a\u5546\uff0c0\u2264r<p\uff0cr = a%p\uff09\u3002<\/p>\n
              \uff081\uff09\u52a0\u6cd5\u53d6\u6a21\u516c\u5f0f\u8bc1\u660e<\/h4>\n
                \n
              1. \n
                \u8bbea = q\u2081\u00d7p + r\u2081\uff08r\u2081=a%p\uff09\uff0cb = q\u2082\u00d7p + r\u2082\uff08r\u2082=b%p\uff09\uff1b<\/p>\n<\/li>\n
              2. \n
                a+b = (q\u2081+q\u2082)\u00d7p + (r\u2081+r\u2082)\uff0c(q\u2081+q\u2082)\u00d7p mod p = 0\uff1b<\/p>\n<\/li>\n
              3. \n
                \u7ed3\u8bba\uff1a(a+b)%p = (r\u2081+r\u2082)%p = (a%p + b%p)%p\u3002<\/p>\n<\/li>\n<\/ol>\n
                \uff082\uff09\u4e58\u6cd5\u53d6\u6a21\u516c\u5f0f\u8bc1\u660e<\/h4>\n
                  \n
                1. \n
                  \u8bbea = q\u2081\u00d7p + r\u2081\uff0cb = q\u2082\u00d7p + r\u2082\uff1b<\/p>\n<\/li>\n
                2. \n
                  a\u00d7b = q\u2081q\u2082p\u00b2 + q\u2081r\u2082p + q\u2082r\u2081p + r\u2081r\u2082\uff0c\u524d\u4e09\u9879mod p = 0\uff1b<\/p>\n<\/li>\n
                3. \n
                  \u7ed3\u8bba\uff1a(a\u00d7b)%p = (r\u2081\u00d7r\u2082)%p = [(a%p)\u00d7(b%p)]%p\u3002<\/p>\n<\/li>\n<\/ol>\n
                  \uff083\uff09\u5e42\u53d6\u6a21\u516c\u5f0f\u8bc1\u660e<\/h4>\n
                    \n
                  1. \n
                    \u8bbea = q\u00d7p + r\uff08r = a%p\uff09\uff0c\u4e8c\u9879\u5f0f\u5c55\u5f00\uff1a $(x+y)^n = \\sum_{k=0}^{n} C_n^k x^{n-k} y^k$ \uff1b<\/p>\n<\/li>\n
                  2. \n
                    \u4ee3\u5165\u5f97 $a^b = (qp + r)^b = \\sum_{k=0}^{b} C_b^k (qp)^{b-k} r^k$ \uff0c\u9664 $r^b$ \u5916\u5747\u542bp\u56e0\u5b50\uff0cmod p = 0\uff1b<\/p>\n<\/li>\n
                  3. \n
                    \u7ed3\u8bba\uff1a(a^b)%p = r^b %p = [(a%p)^b]%p\u3002<\/p>\n<\/li>\n<\/ol>\n
                    3. \u53d6\u6a21\u7ec6\u8282<\/h3>\n
                    \u5173\u952e\u8bf4\u660e1<\/h4>\n
                    \u521d\u59cb\u6267\u884c $a = a \\mod \\text{mod}$ \uff1a\u2460 \u4f9d\u636e\u5e42\u53d6\u6a21\u516c\u5f0f\uff0c\u63d0\u524d\u53d6\u6a21\u4e0d\u6539\u53d8\u6700\u7ec8\u7ed3\u679c\uff1b\u2461 \u5c06a\u538b\u7f29\u81f3[0, mod-1]\u533a\u95f4\uff0c\u907f\u514d\u5e95\u6570\u5e73\u65b9\u8fd0\u7b97\u6ea2\u51fa\u3002<\/p>\n
                    \u5173\u952e\u8bf4\u660e2<\/h4>\n
                    \u6bcf\u6b65\u4e58\u6cd5\/\u5e73\u65b9\u540e\u53d6\u6a21\uff1a\u591a\u6b21\u81ea\u4e58\u540e\u6570\u503c\u6613\u6ea2\u51fa\uff0c\u53d6\u6a21\u53ef\u5c06\u6570\u503c\u9650\u5236\u5728[0, mod-1]\u533a\u95f4\uff0c\u9632\u6ea2\u51fa\u4e14\u4e0d\u6539\u53d8\u7ed3\u679c\u3002<\/p>\n
                    \u5173\u952e\u8bf4\u660e3<\/h4>\n
                    \u4ec5\u6700\u540e\u53d6\u6a21\u4e0d\u53ef\u884c\uff1a\u5927\u6307\u6570\u4e0b $a^b$ \u4f1a\u6ea2\u51fa\u81f4\u6570\u503c\u5f02\u5e38\uff0c\u6700\u7ec8\u53d6\u6a21\u65e0\u610f\u4e49\u3002<\/p>\n
                    \u4e94\u3001\u590d\u4e60\u5c0f\u7ed3<\/h2>\n
                      \n
                    1. \n
                      \u6838\u5fc3\u4f18\u5316\uff1a\u57fa\u4e8e $a^{m+n}=a^m \\times a^n$ \uff0c\u901a\u8fc7\u6307\u6570\u62c6\u5206\u4e0e\u5e95\u6570\u81ea\u4e58\uff0c\u5c06\u65f6\u95f4\u590d\u6742\u5ea6\u4eceO(n)\u964d\u81f3O(log\u2096n)\uff1b<\/p>\n<\/li>\n
                    2. \n
                      \u8fdb\u5236\u5dee\u5f02\uff1a\u62c6\u5206\u57fa\u5e95\u8d8a\u5c0f\uff0c\u5e95\u6570\u81ea\u4e58\u8d8a\u7b80\u5355\uff0c\u4f18\u5316\u6548\u7387\u8d8a\u9ad8\uff0c\u4e8c\u8fdb\u5236\u6700\u4f18\uff1b<\/p>\n<\/li>\n
                    3. \n
                      \u53d6\u6a21\u5173\u952e\uff1a\u7262\u8bb0\u4e09\u6761\u53d6\u6a21\u516c\u5f0f\uff0c\u521d\u59cb\u3001\u4e58\u6cd5\u3001\u5e73\u65b9\u8fd0\u7b97\u540e\u5747\u9700\u6267\u884c\u53d6\u6a21\u64cd\u4f5c\uff1b<\/p>\n<\/li>\n
                    4. \n
                      \u6613\u9519\u70b9\uff1a\u9057\u6f0f\u521d\u59cb\u53d6\u6a21\u3001\u672a\u53ca\u65f6\u53d6\u6a21\u3001\u6df7\u6dc6\u4e0d\u540c\u8fdb\u5236\u7684\u5e95\u6570\u81ea\u4e58\u65b9\u5f0f\u3002<\/p>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"
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