{"id":33031,"date":"2025-04-26T10:27:09","date_gmt":"2025-04-26T02:27:09","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/33031.html"},"modified":"2025-04-26T10:27:09","modified_gmt":"2025-04-26T02:27:09","slug":"%e6%af%8f%e6%97%a5c-c%e9%a2%98-%e5%a4%87%e6%88%98%e8%93%9d%e6%a1%a5%e6%9d%af-%ef%bc%88%e6%b4%9b%e8%b0%b7-p1226-%e5%bf%ab%e9%80%9f%e5%b9%82%e6%b1%82%e6%a8%a1%e9%a2%98%e8%a7%a3%ef%bc%89","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/33031.html","title":{"rendered":"\u6bcf\u65e5c\/c++\u9898 \u5907\u6218\u84dd\u6865\u676f \uff08[\u6d1b\u8c37 P1226] \u5feb\u901f\u5e42\u6c42\u6a21\u9898\u89e3\uff09"},"content":{"rendered":"<h2>[\u6d1b\u8c37 P1226] \u5feb\u901f\u5e42\u6c42\u6a21\u9898\u89e3<\/h2>\n<h3>&#x1f4cc; \u9898\u76ee\u94fe\u63a5<\/h3>\n<p>https:\/\/www.luogu.com.cn\/problem\/P1226<\/p>\n<h3>&#x1f4dd; \u9898\u76ee\u63cf\u8ff0<\/h3>\n<p>\u7ed9\u5b9a\u6b63\u6574\u6570 a\u3001b \u548c\u8d28\u6570 p&#xff0c;\u8981\u6c42\u8ba1\u7b97&#xff1a;<\/p>\n<p>a^b % p<\/p>\n<p>\u5176\u4e2d&#xff1a;<\/p>\n<ul>\n<li>1 \u2264 a \u2264 10^9<\/li>\n<li>0 \u2264 b \u2264 10^9<\/li>\n<li>2 \u2264 p \u2264 10^9<\/li>\n<\/ul>\n<h3>&#x1f4a1; \u89e3\u9898\u601d\u8def<\/h3>\n<h4>1. \u76f4\u63a5\u8ba1\u7b97\u4e0d\u53ef\u884c<\/h4>\n<p>\u7531\u4e8e b \u7684\u8303\u56f4\u53ef\u8fbe 10^9&#xff0c;\u76f4\u63a5\u8ba1\u7b97 a^b \u4f1a\u5bfc\u81f4\u7ed3\u679c\u8fc7\u5927&#xff0c;\u65e0\u6cd5\u5728\u5e38\u89c4\u6570\u636e\u7c7b\u578b\u4e2d\u5b58\u50a8\u3002\u56e0\u6b64&#xff0c;\u9700\u8981\u4e00\u79cd\u9ad8\u6548\u7684\u7b97\u6cd5\u6765\u8ba1\u7b97 a^b % p\u3002<\/p>\n<h4>2. \u5feb\u901f\u5e42\u7b97\u6cd5<\/h4>\n<p>\u5feb\u901f\u5e42\u7b97\u6cd5&#xff08;Exponentiation by Squaring&#xff09;\u662f\u4e00\u79cd\u9ad8\u6548\u8ba1\u7b97\u5927\u6574\u6570\u5e42\u6a21\u7684\u65b9\u6cd5\u3002\u5176\u57fa\u672c\u601d\u60f3\u662f\u5229\u7528\u4e8c\u8fdb\u5236\u5206\u89e3&#xff0c;\u5c06\u6307\u6570 b \u8f6c\u6362\u4e3a\u4e8c\u8fdb\u5236\u5f62\u5f0f&#xff0c;\u7136\u540e\u901a\u8fc7\u5e73\u65b9\u548c\u4e58\u6cd5\u7684\u65b9\u5f0f\u5feb\u901f\u8ba1\u7b97\u5e42\u503c\u3002<\/p>\n<h5>\u5feb\u901f\u5e42\u7684\u9012\u63a8\u5173\u7cfb&#xff1a;<\/h5>\n<ul>\n<li>\u5f53 b \u4e3a\u5076\u6570\u65f6&#xff1a;a^b &#061; (a^(b\/2))^2<\/li>\n<li>\u5f53 b \u4e3a\u5947\u6570\u65f6&#xff1a;a^b &#061; a * (a^(b-1))<\/li>\n<\/ul>\n<p>\u901a\u8fc7\u4e0d\u65ad\u5c06 b \u53f3\u79fb\u4e00\u4f4d&#xff08;\u5373\u9664\u4ee5 2&#xff09;&#xff0c;\u5e76\u6839\u636e b \u7684\u5947\u5076\u6027\u51b3\u5b9a\u662f\u5426\u4e58\u4e0a\u5f53\u524d\u7684 a&#xff0c;\u53ef\u4ee5\u5728\u5bf9\u6570\u65f6\u95f4\u5185\u8ba1\u7b97\u51fa\u7ed3\u679c\u3002<\/p>\n<h4>3. \u6a21\u8fd0\u7b97\u4f18\u5316<\/h4>\n<p>\u5728\u8ba1\u7b97\u8fc7\u7a0b\u4e2d&#xff0c;\u6bcf\u4e00\u6b65\u90fd\u5bf9\u7ed3\u679c\u53d6\u6a21 p&#xff0c;\u4ee5\u907f\u514d\u4e2d\u95f4\u7ed3\u679c\u6ea2\u51fa&#xff0c;\u5e76\u786e\u4fdd\u6700\u7ec8\u7ed3\u679c\u6ee1\u8db3\u9898\u76ee\u8981\u6c42\u3002<\/p>\n<h3>&#x1f4bb; \u4ee3\u7801\u5b9e\u73b0<\/h3>\n<p><span class=\"token macro property\"><span class=\"token directive-hash\">#<\/span><span class=\"token directive keyword\">include<\/span> <span class=\"token string\">&lt;bits\/stdc&#043;&#043;.h&gt;<\/span><\/span><br \/>\n<span class=\"token keyword\">using<\/span> <span class=\"token keyword\">namespace<\/span> std<span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token keyword\">long<\/span> <span class=\"token keyword\">long<\/span> a<span class=\"token punctuation\">,<\/span> b<span class=\"token punctuation\">,<\/span> p<span class=\"token punctuation\">;<\/span><\/p>\n<p><span class=\"token keyword\">int<\/span> <span class=\"token function\">main<\/span><span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n    ios<span class=\"token operator\">::<\/span><span class=\"token function\">sync_with_stdio<\/span><span class=\"token punctuation\">(<\/span><span class=\"token boolean\">false<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    cin<span class=\"token punctuation\">.<\/span><span class=\"token function\">tie<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    cout<span class=\"token punctuation\">.<\/span><span class=\"token function\">tie<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    cin <span class=\"token operator\">&gt;&gt;<\/span> a <span class=\"token operator\">&gt;&gt;<\/span> b <span class=\"token operator\">&gt;&gt;<\/span> p<span class=\"token punctuation\">;<\/span><br \/>\n    cout <span class=\"token operator\">&lt;&lt;<\/span> a <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;^&#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> b <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034; mod &#034;<\/span> <span class=\"token operator\">&lt;&lt;<\/span> p <span class=\"token operator\">&lt;&lt;<\/span> <span class=\"token string\">&#034;&#061;&#034;<\/span><span class=\"token punctuation\">;<\/span><\/p>\n<p>    <span class=\"token keyword\">long<\/span> <span class=\"token keyword\">long<\/span> ans <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span><br \/>\n    a <span class=\"token operator\">&#061;<\/span> a <span class=\"token operator\">%<\/span> p<span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ \u9632\u6b62 a \u5927\u4e8e p<\/span><\/p>\n<p>    <span class=\"token keyword\">while<\/span> <span class=\"token punctuation\">(<\/span>b<span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token punctuation\">(<\/span>b <span class=\"token operator\">&amp;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token punctuation\">{<\/span> <span class=\"token comment\">\/\/ \u5982\u679c b \u662f\u5947\u6570<\/span><br \/>\n            ans <span class=\"token operator\">&#061;<\/span> ans <span class=\"token operator\">*<\/span> a <span class=\"token operator\">%<\/span> p<span class=\"token punctuation\">;<\/span><br \/>\n        <span class=\"token punctuation\">}<\/span><br \/>\n        a <span class=\"token operator\">&#061;<\/span> a <span class=\"token operator\">*<\/span> a <span class=\"token operator\">%<\/span> p<span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ a^2 mod p<\/span><br \/>\n        b <span class=\"token operator\">&gt;&gt;&#061;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">;<\/span> <span class=\"token comment\">\/\/ b \u53f3\u79fb\u4e00\u4f4d&#xff0c;\u76f8\u5f53\u4e8e b \/&#061; 2<\/span><br \/>\n    <span class=\"token punctuation\">}<\/span><\/p>\n<p>    cout <span class=\"token operator\">&lt;&lt;<\/span> ans<span class=\"token punctuation\">;<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> <span class=\"token number\">0<\/span><span class=\"token punctuation\">;<\/span><br \/>\n<span class=\"token punctuation\">}<\/span><\/p>\n<h3>&#x1f9e9; \u4ee3\u7801\u89e3\u6790<\/h3>\n<li>\n<p>\u8f93\u5165\u5904\u7406&#xff1a;\u4f7f\u7528 cin \u8bfb\u53d6\u8f93\u5165&#xff0c;a\u3001b \u548c p \u5206\u522b\u8868\u793a\u5e95\u6570\u3001\u6307\u6570\u548c\u6a21\u6570\u3002<\/p>\n<\/li>\n<li>\n<p>\u521d\u59cb\u5316&#xff1a;\u5c06 ans \u521d\u59cb\u5316\u4e3a 1&#xff0c;\u7528\u4e8e\u5b58\u50a8\u6700\u7ec8\u7ed3\u679c\u3002\u4e3a\u4e86\u9632\u6b62\u4e2d\u95f4\u7ed3\u679c\u8fc7\u5927&#xff0c;\u5148\u5bf9 a \u53d6\u6a21 p\u3002<\/p>\n<\/li>\n<li>\n<p>\u5feb\u901f\u5e42\u5faa\u73af&#xff1a;<\/p>\n<ul>\n<li>\u5f53 b \u4e0d\u4e3a 0 \u65f6&#xff0c;\u8fdb\u5165\u5faa\u73af\u3002<\/li>\n<li>\u5982\u679c b \u662f\u5947\u6570&#xff08;\u5373 b &amp; 1 \u4e3a\u771f&#xff09;&#xff0c;\u5219\u5c06\u5f53\u524d\u7684 a \u4e58\u5230 ans \u4e0a&#xff0c;\u5e76\u5bf9 p \u53d6\u6a21\u3002<\/li>\n<li>\u5c06 a \u81ea\u8eab\u5e73\u65b9&#xff0c;\u5e76\u5bf9 p \u53d6\u6a21\u3002<\/li>\n<li>\u5c06 b \u53f3\u79fb\u4e00\u4f4d&#xff0c;\u76f8\u5f53\u4e8e\u5c06 b \u9664\u4ee5 2\u3002<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u8f93\u51fa\u7ed3\u679c&#xff1a;\u8f93\u51fa\u8ba1\u7b97\u7ed3\u679c\u3002<\/p>\n<\/li>\n<h3>\u23f1\ufe0f \u65f6\u95f4\u590d\u6742\u5ea6\u5206\u6790<\/h3>\n<p>\u5feb\u901f\u5e42\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(log b)&#xff0c;\u5176\u4e2d b \u662f\u6307\u6570\u3002\u7531\u4e8e\u6bcf\u6b21\u5faa\u73af\u5c06 b \u51cf\u534a&#xff0c;\u56e0\u6b64\u5faa\u73af\u6b21\u6570\u4e3a log b\u3002\u5728\u6bcf\u6b21\u5faa\u73af\u4e2d&#xff0c;\u8fdb\u884c\u5e38\u6570\u6b21\u7684\u4e58\u6cd5\u548c\u53d6\u6a21\u64cd\u4f5c&#xff0c;\u56e0\u6b64\u603b\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(log b)\u3002<\/p>\n<h3>&#x1f51a; \u603b\u7ed3<\/h3>\n<p>\u901a\u8fc7\u4f7f\u7528\u5feb\u901f\u5e42\u7b97\u6cd5&#xff0c;\u53ef\u4ee5\u5728\u5bf9\u6570\u65f6\u95f4\u5185\u8ba1\u7b97\u5927\u6574\u6570\u7684\u5e42\u6a21&#xff0c;\u907f\u514d\u4e86\u76f4\u63a5\u8ba1\u7b97\u53ef\u80fd\u5bfc\u81f4\u7684\u6ea2\u51fa\u95ee\u9898\u3002\u8be5\u7b97\u6cd5\u5728\u5904\u7406\u5927\u89c4\u6a21\u6570\u636e\u65f6\u975e\u5e38\u9ad8\u6548&#xff0c;\u662f\u89e3\u51b3\u6b64\u7c7b\u95ee\u9898\u7684\u5e38\u7528\u6280\u5de7\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6587\u7ae0\u6d4f\u89c8\u9605\u8bfb290\u6b21\uff0c\u70b9\u8d5e5\u6b21\uff0c\u6536\u85cf4\u6b21\u3002\u901a\u8fc7\u4f7f\u7528\u5feb\u901f\u5e42\u7b97\u6cd5\uff0c\u53ef\u4ee5\u5728\u5bf9\u6570\u65f6\u95f4\u5185\u8ba1\u7b97\u51fa\u5927\u6574\u6570\u7684\u5e42\u6a21\uff0c\u907f\u514d\u4e86\u76f4\u63a5\u8ba1\u7b97\u53ef\u80fd\u5bfc\u81f4\u7684\u6ea2\u51fa\u95ee\u9898\u3002\u8be5\u7b97\u6cd5\u5728\u5904\u7406\u5927\u89c4\u6a21\u6570\u636e\u65f6\u975e\u5e38\u9ad8\u6548\uff0c\u662f\u89e3\u51b3\u6b64\u7c7b\u95ee\u9898\u7684\u5e38\u7528\u6280\u5de7\u3002<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[55,2694,427,729],"topic":[],"class_list":["post-33031","post","type-post","status-publish","format-standard","hentry","category-server","tag-c","tag-c-c","tag-427","tag-729"],"yoast_head":"<!-- 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