{"id":301478,"date":"2024-05-20T18:09:16","date_gmt":"2024-05-20T10:09:16","guid":{"rendered":"https:\/\/docs.pingcode.com\/ask\/ask-ask\/301478.html"},"modified":"2024-05-20T18:09:43","modified_gmt":"2024-05-20T10:09:43","slug":"%e5%a6%82%e4%bd%95%e7%94%a8c-%e5%86%99%e4%b8%80%e4%b8%aa%e6%9c%89%e9%99%90%e5%9f%9f%e7%9a%84%e7%b1%bb%ef%bc%88f_pn%ef%bc%89","status":"publish","type":"post","link":"https:\/\/docs.pingcode.com\/ask\/301478.html","title":{"rendered":"\u5982\u4f55\u7528C  \u5199\u4e00\u4e2a\u6709\u9650\u57df\u7684\u7c7b\uff08F_p^n\uff09"},"content":{"rendered":"<p style=\"text-align:center\"><img decoding=\"async\" src=\"https:\/\/cdn-kb.worktile.com\/kb\/wp-content\/uploads\/2024\/04\/26213652\/fd857195-da35-4e72-99da-f163ac46ee88.webp\" alt=\"\u5982\u4f55\u7528C  \u5199\u4e00\u4e2a\u6709\u9650\u57df\u7684\u7c7b\uff08F_p^n\uff09\" \/><\/p>\n<p><p>\u5728C++\u4e2d\uff0c\u5199\u4e00\u4e2a\u6709\u9650\u57df\u7684\u7c7b\uff08F_p^n\uff09\uff0c<strong>\u9996\u5148\u9700\u7406\u89e3\u6709\u9650\u57df\u7684\u57fa\u672c\u6982\u5ff5\u3001C++\u7684\u7c7b\u5b9a\u4e49\u548c\u64cd\u4f5c\u91cd\u8f7d<\/strong>\u3002\u6709\u9650\u57df\uff08Finite Field\uff09\uff0c\u4e5f\u79f0\u4f3d\u7f57\u534e\u57df\uff08Galois Field\uff09\uff0c\u8868\u793a\u6709\u9650\u4e2a\u5143\u7d20\u7684\u6570\u5b66\u57df\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5143\u7d20\u90fd\u80fd\u8fdb\u884c\u52a0\u3001\u51cf\u3001\u4e58\u3001\u9664\u8fd0\u7b97\u800c\u4e14\u7ed3\u679c\u4ecd\u5728\u8be5\u57df\u5185\u3002\u5bf9\u4e8eF_p^n\uff0cp\u662f\u7d20\u6570\uff0cn\u4e3a\u6b63\u6574\u6570\uff0c\u8868\u793a\u5728\u6a21p\u7684\u610f\u4e49\u4e0b\u8fdb\u884c\u8fd0\u7b97\u3002\u6b64\u7c7b\u5c01\u88c5\u4e86\u6709\u9650\u57df\u7684\u5143\u7d20\u53ca\u5176\u8fd0\u7b97\u3002<strong>\u5c01\u88c5\u6709\u9650\u57df\u7684\u6838\u5fc3\u5728\u4e8e\u5b9e\u73b0\u5143\u7d20\u95f4\u7684\u52a0\u3001\u51cf\u3001\u4e58\u4ee5\u53ca\u6c42\u9006\u7b49\u57fa\u672c\u8fd0\u7b97<\/strong>\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u5c06\u8be6\u7ec6\u63cf\u8ff0\u5982\u4f55\u5b9e\u73b0\u52a0\u6cd5\u8fd0\u7b97\u3002<\/p>\n<\/p>\n<p><h3>\u4e00\u3001\u7406\u89e3\u6709\u9650\u57dfF_p^n<\/h3>\n<\/p>\n<p><p>\u5728\u6784\u5efa\u7c7b\u4e4b\u524d\uff0c\u5148\u8981\u7406\u89e3F_p^n\u7684\u6570\u5b66\u57fa\u7840\u3002\u6709\u9650\u57dfF_p^n\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u53ef\u4ee5\u89c6\u4e3a\u4e00\u4e2an\u9636\u591a\u9879\u5f0f\uff0c\u591a\u9879\u5f0f\u7684\u7cfb\u6570\u662f\u5c0f\u4e8ep\u7684\u975e\u8d1f\u6574\u6570\u3002\u56e0\u6b64\uff0c\u8be5\u57df\u4e2d\u6709p^n\u4e2a\u5143\u7d20\u3002\u5143\u7d20\u95f4\u7684\u52a0\u3001\u51cf\u3001\u4e58\u8fd0\u7b97\u662f\u5728\u6a21p\u7684\u6761\u4ef6\u4e0b\u8fdb\u884c\u7684\uff0c\u4fdd\u8bc1\u4e86\u8fd0\u7b97\u7ed3\u679c\u4ecd\u7136\u5c5e\u4e8e\u8be5\u57df\u3002<\/p>\n<\/p>\n<p><h3>\u4e8c\u3001\u5b9a\u4e49F_p^n\u7684\u7c7b\u6846\u67b6<\/h3>\n<\/p>\n<p><p>\u5b9a\u4e49F_p^n\u7c7b\u65f6\uff0c\u8981\u9996\u5148\u58f0\u660e\u5b58\u50a8\u5143\u7d20\u7684\u6570\u636e\u7ed3\u6784\uff0c\u4ee5\u53ca\u5b9e\u73b0\u57fa\u672c\u8fd0\u7b97\u7684\u6210\u5458\u51fd\u6570\u3002<\/p>\n<\/p>\n<p><pre><code class=\"language-cpp\">class FiniteFieldElement {<\/p>\n<p>private:<\/p>\n<p>    unsigned int p;   \/\/ \u7d20\u6570p<\/p>\n<p>    unsigned int n;   \/\/ \u6b63\u6574\u6570n<\/p>\n<p>    std::vector&lt;int&gt; coeffs; \/\/ \u591a\u9879\u5f0f\u7cfb\u6570<\/p>\n<p>public:<\/p>\n<p>    FiniteFieldElement(unsigned int p, unsigned int n, const std::vector&lt;int&gt;&amp; coeffs);<\/p>\n<p>    \/\/ \u5176\u4ed6\u6210\u5458\u51fd\u6570\u58f0\u660e<\/p>\n<p>};<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><h3>\u4e09\u3001\u5b9e\u73b0\u52a0\u6cd5\u8fd0\u7b97<\/h3>\n<\/p>\n<p><p>\u5728\u6709\u9650\u57df\u4e2d\uff0c\u5143\u7d20\u7684\u52a0\u6cd5\u8fd0\u7b97\u8f83\u4e3a\u76f4\u89c2\uff1a\u5bf9\u5e94\u4f4d\u7cfb\u6570\u76f8\u52a0\uff0c\u7ed3\u679c\u5bf9\u7d20\u6570p\u53d6\u6a21\u3002<\/p>\n<\/p>\n<p><pre><code class=\"language-cpp\">FiniteFieldElement FiniteFieldElement::operator+(const FiniteFieldElement&amp; other) const {<\/p>\n<p>    std::vector&lt;int&gt; resultCoeffs(n, 0); \/\/ \u521d\u59cb\u5316\u7ed3\u679c\u591a\u9879\u5f0f\u7cfb\u6570<\/p>\n<p>    for (size_t i = 0; i &lt; n; ++i) {<\/p>\n<p>        resultCoeffs[i] = (this-&gt;coeffs[i] + other.coeffs[i]) % p; \/\/ \u5bf9\u5e94\u4f4d\u76f8\u52a0\u5e76\u53d6\u6a21<\/p>\n<p>    }<\/p>\n<p>    return FiniteFieldElement(p, n, resultCoeffs);<\/p>\n<p>}<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><h3>\u56db\u3001\u5b9e\u73b0\u51cf\u6cd5\u3001\u4e58\u6cd5\u548c\u6c42\u9006<\/h3>\n<\/p>\n<p><p><strong>\u51cf\u6cd5\u8fd0\u7b97\u4e5f\u9075\u5faa\u76f8\u4f3c\u7684\u903b\u8f91<\/strong>\uff0c\u6bcf\u4f4d\u7cfb\u6570\u76f8\u51cf\u4e4b\u540e\u5bf9p\u53d6\u6a21\u3002\u4e58\u6cd5\u548c\u6c42\u9006\u5219\u66f4\u4e3a\u590d\u6742\u3002<\/p>\n<\/p>\n<p><h4>\u51cf\u6cd5\u8fd0\u7b97<\/h4>\n<\/p>\n<p><p>\u51cf\u6cd5\u64cd\u4f5c\u7c7b\u4f3c\u4e8e\u52a0\u6cd5\uff0c\u53ea\u662f\u6bcf\u4f4d\u4e0a\u7684\u8fd0\u7b97\u53d8\u4e3a\u76f8\u51cf\u3002<\/p>\n<\/p>\n<p><h4>\u4e58\u6cd5\u8fd0\u7b97<\/h4>\n<\/p>\n<p><p>\u4e58\u6cd5\u8fd0\u7b97\u6d89\u53ca\u5230\u591a\u9879\u5f0f\u7684\u4e58\u6cd5\uff0c\u6700\u540e\u7ed3\u679c\u4e5f\u9700\u8981\u901a\u8fc7\u67d0\u4e9b\u7b97\u6cd5\uff08\u4f8b\u5982\u957f\u9664\u6cd5\uff09\u6a21\u4e00\u4e2a\u751f\u6210\u591a\u9879\u5f0f\u6765\u4fdd\u8bc1\u7ed3\u679c\u4f9d\u65e7\u5728F_p^n\u4e2d\u3002<\/p>\n<\/p>\n<p><h4>\u6c42\u9006\u8fd0\u7b97<\/h4>\n<\/p>\n<p><p>\u6c42\u9006\u662fF_p^n\u4e2d\u8f83\u4e3a\u590d\u6742\u7684\u64cd\u4f5c\u4e4b\u4e00\uff0c\u901a\u5e38\u53ef\u4ee5\u901a\u8fc7\u6269\u5c55\u7684\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5\u5b9e\u73b0\u3002<\/p>\n<\/p>\n<p><h3>\u4e94\u3001\u6574\u5408\u6709\u9650\u57df\u8fd0\u7b97<\/h3>\n<\/p>\n<p><p>\u5b9a\u4e49\u597d\u57fa\u672c\u7684\u8fd0\u7b97\u540e\uff0c\u53ef\u4ee5\u8fdb\u4e00\u6b65\u5b9e\u73b0\u6bd4\u8f83\u8fd0\u7b97\u7b26\u3001\u6307\u6570\u8fd0\u7b97\u7b49\u9ad8\u7ea7\u529f\u80fd\uff0c\u4f7f\u5f97F_p^n\u7c7b\u66f4\u52a0\u5f3a\u5927\u3001\u6613\u7528\u3002<\/p>\n<\/p>\n<p><h4>\u6bd4\u8f83\u8fd0\u7b97\u7b26<\/h4>\n<\/p>\n<p><p>\u6bd4\u8f83\u4e24\u4e2a\u6709\u9650\u57df\u5143\u7d20\u662f\u5426\u76f8\u7b49\uff0c\u9700\u8981\u6bd4\u8f83\u5b83\u4eec\u7684\u5168\u90e8\u7cfb\u6570\u3002<\/p>\n<\/p>\n<p><h4>\u6307\u6570\u8fd0\u7b97<\/h4>\n<\/p>\n<p><p>\u6307\u6570\u8fd0\u7b97\u53ef\u4ee5\u5229\u7528\u8fde\u7eed\u7684\u4e58\u6cd5\u6216\u8005\u5feb\u901f\u5e42\u7b97\u6cd5\u5b9e\u73b0\uff0c\u662f\u6784\u5efa\u9ad8\u7ea7\u6570\u5b66\u7ed3\u6784\u7684\u57fa\u7840\u3002<\/p>\n<\/p>\n<p><h3>\u516d\u3001\u6d4b\u8bd5\u548c\u4f7f\u7528<\/h3>\n<\/p>\n<p><p>\u5b9a\u4e49\u597d\u7c7b\u4e4b\u540e\uff0c\u901a\u8fc7\u5177\u4f53\u7684\u5b9e\u4f8b\u6765\u6d4b\u8bd5\u6240\u6709\u529f\u80fd\u662f\u5426\u6309\u9884\u671f\u5de5\u4f5c\u3002<\/p>\n<\/p>\n<p><pre><code class=\"language-cpp\">#include &lt;iostream&gt;<\/p>\n<p>#include &lt;vector&gt;<\/p>\n<p>\/\/ \u76f8\u5173\u7c7b\u7684\u5b9a\u4e49<\/p>\n<p>int m<a href=\"https:\/\/docs.pingcode.com\/blog\/59162.html\" target=\"_blank\">AI<\/a>n() {<\/p>\n<p>    FiniteFieldElement a(7, 3, {1, 2, 4}); \/\/ \u521d\u59cb\u5316\u4e00\u4e2a\u6709\u9650\u57df\u5143\u7d20<\/p>\n<p>    FiniteFieldElement b(7, 3, {3, 5, 6});<\/p>\n<p>    FiniteFieldElement c = a + b; \/\/ \u52a0\u6cd5\u8fd0\u7b97<\/p>\n<p>    \/\/ \u6d4b\u8bd5\u5176\u4ed6\u8fd0\u7b97...<\/p>\n<p>    return 0;<\/p>\n<p>}<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u901a\u8fc7\u4e0a\u8ff0\u6b65\u9aa4\uff0c\u53ef\u4ee5\u5728C++\u4e2d\u5b9e\u73b0\u4e00\u4e2a\u6709\u9650\u57dfF_p^n\u7684\u7c7b\uff0c<strong>\u5173\u952e\u5728\u4e8e\u7cbe\u786e\u5b9a\u4e49\u548c\u5b9e\u73b0\u6709\u9650\u57df\u7684\u6570\u5b66\u8fd0\u7b97<\/strong>\u3002\u8fd9\u4e0d\u4ec5\u8981\u6c42\u5bf9\u6570\u5b66\u539f\u7406\u6709\u6df1\u5165\u7406\u89e3\uff0c\u8fd8\u9700\u8981\u638c\u63e1C++\u7c7b\u7684\u8bbe\u8ba1\u548c\u64cd\u4f5c\u91cd\u8f7d\u7b49\u7f16\u7a0b\u6280\u5de7\u3002<\/p>\n<\/p>\n<h2><strong>\u76f8\u5173\u95ee\u7b54FAQs\uff1a<\/strong><\/h2>\n<p><strong>Q\uff1a\u5982\u4f55\u4f7f\u7528C\u8bed\u8a00\u7f16\u5199\u4e00\u4e2a\u8868\u793a\u6709\u9650\u57dfF_p^n\u7684\u7c7b\uff1f<\/strong><\/p>\n<p>A\uff1a\u5728C\u8bed\u8a00\u4e2d\uff0c\u53ef\u4ee5\u4f7f\u7528\u7ed3\u6784\u4f53\u548c\u76f8\u5e94\u7684\u51fd\u6570\u6765\u6784\u5efa\u4e00\u4e2a\u6709\u9650\u57dfF_p^n\u7684\u7c7b\u3002\u9996\u5148\uff0c\u53ef\u4ee5\u5b9a\u4e49\u4e00\u4e2a\u7ed3\u6784\u4f53\u6765\u8868\u793a\u6709\u9650\u57df\u4e2d\u7684\u5143\u7d20\uff0c\u8be5\u7ed3\u6784\u4f53\u5e94\u5305\u542b\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\u6765\u5b58\u50a8\u5143\u7d20\u7684\u7cfb\u6570\u3002\u7136\u540e\uff0c\u53ef\u4ee5\u5b9a\u4e49\u4e00\u7cfb\u5217\u7684\u51fd\u6570\u6765\u5b9e\u73b0\u6709\u9650\u57df\u7c7b\u7684\u5404\u79cd\u64cd\u4f5c\uff0c\u6bd4\u5982\u52a0\u6cd5\u3001\u4e58\u6cd5\u3001\u6307\u6570\u8fd0\u7b97\u7b49\u3002\u8fd9\u4e9b\u51fd\u6570\u53ef\u4ee5\u63a5\u53d7\u4e24\u4e2a\u6709\u9650\u57df\u5143\u7d20\u4f5c\u4e3a\u53c2\u6570\uff0c\u5e76\u8fd4\u56de\u76f8\u5e94\u7684\u7ed3\u679c\u3002\u901a\u8fc7\u8fd9\u6837\u7684\u8bbe\u8ba1\uff0c\u53ef\u4ee5\u65b9\u4fbf\u5730\u4f7f\u7528C\u8bed\u8a00\u6765\u64cd\u4f5c\u548c\u8ba1\u7b97\u6709\u9650\u57df\u4e2d\u7684\u5143\u7d20\u3002<\/p>\n<p><strong>Q\uff1a\u5982\u4f55\u5b9e\u73b0\u6709\u9650\u57dfF_p^n\u4e2d\u7684\u5143\u7d20\u52a0\u6cd5\u548c\u4e58\u6cd5\u8fd0\u7b97\uff1f<\/strong><\/p>\n<p>A\uff1a\u8981\u5b9e\u73b0\u6709\u9650\u57dfF_p^n\u4e2d\u7684\u5143\u7d20\u52a0\u6cd5\u548c\u4e58\u6cd5\u8fd0\u7b97\uff0c\u53ef\u4ee5\u5148\u5b9a\u4e49\u4e00\u4e2a\u8868\u793a\u6709\u9650\u57df\u5143\u7d20\u7684\u7ed3\u6784\u4f53\u3002\u5728\u8be5\u7ed3\u6784\u4f53\u4e2d\uff0c\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u6574\u6570\u6570\u7ec4\u6765\u8868\u793a\u8be5\u5143\u7d20\u7684\u5404\u4e2a\u7cfb\u6570\u3002\u5bf9\u4e8e\u52a0\u6cd5\u8fd0\u7b97\uff0c\u53ef\u4ee5\u9010\u4f4d\u5bf9\u7cfb\u6570\u8fdb\u884c\u76f8\u52a0\uff0c\u5e76\u5904\u7406\u8fdb\u4f4d\u7684\u60c5\u51b5\u3002\u5bf9\u4e8e\u4e58\u6cd5\u8fd0\u7b97\uff0c\u53ef\u4ee5\u4f7f\u7528\u4e24\u4e2a\u6709\u9650\u57df\u5143\u7d20\u7684\u7cfb\u6570\u8fdb\u884c\u4e58\u6cd5\u8fd0\u7b97\uff0c\u5e76\u6839\u636e\u5143\u7d20\u7684\u5b9a\u4e49\u8fdb\u884c\u7cfb\u6570\u7684\u8c03\u6574\u548c\u8fdb\u4f4d\u7684\u5904\u7406\u3002\u901a\u8fc7\u8fd9\u6837\u7684\u65b9\u5f0f\uff0c\u53ef\u4ee5\u5728C\u8bed\u8a00\u4e2d\u5b9e\u73b0\u6709\u9650\u57dfF_p^n\u4e2d\u5143\u7d20\u7684\u57fa\u672c\u8fd0\u7b97\u3002<\/p>\n<p><strong>Q\uff1a\u9664\u4e86\u52a0\u6cd5\u548c\u4e58\u6cd5\u8fd0\u7b97\uff0c\u6709\u9650\u57dfF_p^n\u4e2d\u8fd8\u53ef\u4ee5\u8fdb\u884c\u54ea\u4e9b\u64cd\u4f5c\uff1f<\/strong><\/p>\n<p>A\uff1a\u9664\u4e86\u52a0\u6cd5\u548c\u4e58\u6cd5\u8fd0\u7b97\u5916\uff0c\u6709\u9650\u57dfF_p^n\u4e2d\u7684\u5143\u7d20\u8fd8\u53ef\u4ee5\u8fdb\u884c\u6307\u6570\u8fd0\u7b97\u3001\u9006\u8fd0\u7b97\u7b49\u64cd\u4f5c\u3002\u6307\u6570\u8fd0\u7b97\u53ef\u4ee5\u5bf9\u6709\u9650\u57df\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u5e42\u8fd0\u7b97\uff0c\u5373\u5c06\u5143\u7d20\u8fde\u7eed\u76f8\u4e58n\u6b21\u3002\u9006\u8fd0\u7b97\u53ef\u4ee5\u6c42\u89e3\u6709\u9650\u57df\u4e2d\u7684\u5143\u7d20\u7684\u4e58\u6cd5\u9006\u5143\u7d20\uff0c\u5373\u4f7f\u5f97\u5143\u7d20\u4e58\u4ee5\u5176\u9006\u5143\u7d20\u7b49\u4e8e1\u3002\u8fd9\u4e9b\u64cd\u4f5c\u53ef\u4ee5\u901a\u8fc7\u9010\u4f4d\u8fd0\u7b97\u4ee5\u53ca\u4e00\u4e9b\u6570\u5b66\u516c\u5f0f\u6765\u5b9e\u73b0\u3002\u5728\u7f16\u5199\u6709\u9650\u57df\u7c7b\u65f6\uff0c\u53ef\u4ee5\u4e3a\u6bcf\u4e2a\u64cd\u4f5c\u5b9a\u4e49\u76f8\u5e94\u7684\u51fd\u6570\uff0c\u5e76\u901a\u8fc7\u8c03\u7528\u8fd9\u4e9b\u51fd\u6570\u6765\u5b8c\u6210\u6240\u9700\u7684\u64cd\u4f5c\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"\u5728C++\u4e2d\uff0c\u5199\u4e00\u4e2a\u6709\u9650\u57df\u7684\u7c7b\uff08F_p^n\uff09\uff0c\u9996\u5148\u9700\u7406\u89e3\u6709\u9650\u57df\u7684\u57fa\u672c\u6982\u5ff5\u3001C++\u7684\u7c7b\u5b9a\u4e49\u548c\u64cd\u4f5c\u91cd\u8f7d\u3002\u6709\u9650\u57df\uff08Fi 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