{"id":380,"date":"2022-09-24T23:19:40","date_gmt":"2022-09-24T15:19:40","guid":{"rendered":"https:\/\/blog.cirno.fun\/?p=380"},"modified":"2024-09-12T09:33:39","modified_gmt":"2024-09-12T01:33:39","slug":"%e5%bf%ab%e9%80%9f%e5%b9%82","status":"publish","type":"post","link":"https:\/\/blog.sssn.tech\/?p=380","title":{"rendered":"\u5feb\u901f\u5e42"},"content":{"rendered":"<p>\u5feb\u901f\u5e42\u4e3b\u8981\u89e3\u51b3\u5f62\u5982<\/p>\n<pre><code class=\"language-katex\">a^k \\quad mod \\quad p<\/code><\/pre>\n<p>\u7684\u95ee\u9898<\/p>\n<p>\u4e00\u4e2a\u975e\u5e38\u7b80\u5355\u7684\u505a\u6cd5\u4fbf\u662f\u7528\u4e00\u4e2ak\u6b21\u7684\u5faa\u73af\uff0c\u6bcf\u4e00\u6b21\u5c06ans\u7d2f\u4e58\u4e00\u4e2aa\u518d\u5bf9p\u53d6\u6a21\uff0c\u8fd9\u6837\u7684\u6734\u7d20\u7b97\u6cd5\u65f6\u95f4\u590d\u6742\u5ea6\u662f<code>O(k)<\/code><\/p>\n<p>\u73b0\u5728\u8981\u4ecb\u7ecd\u7684\u5feb\u901f\u5e42\u7b97\u6cd5\u53ef\u4ee5\u5728<code>O(\\log_{}{k} )<\/code>\u7684\u590d\u6742\u5ea6\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898<\/p>\n<p>\u5feb\u901f\u5e42\u7684\u6838\u5fc3\u601d\u60f3\u662f\uff1a<\/p>\n<ol>\n<li>\n<p>\u5bf9\u4e8e\u95ee\u9898<code>a^k \\quad mod \\quad p<\/code>\uff0c\u9996\u5148\u5904\u7406\u51fa<\/p>\n<pre><code class=\"language-katex\">a^{2^i},i \\in [0,\\log_{}{k}]<\/code><\/pre>\n<\/li>\n<li>\n<p>\u5c06k\u5206\u89e3\u6210\u4e8c\u8fdb\u5236\u5f62\u5f0f<\/p>\n<pre><code class=\"language-katex\">(k_i)_2<\/code><\/pre>\n<\/li>\n<li>\n<p>\u6b64\u65f6<\/p>\n<pre><code class=\"language-katex\">ans=a^k \\quad mod \\quad p =\\prod_{j}^{} m_j \\quad mod \\quad p \\\\\nj\u662fk\u7684\u4e8c\u8fdb\u5236\u5199\u6cd5\u4e2d1\u7684\u4f4d\u7f6e<\/code><\/pre>\n<\/li>\n<\/ol>\n<p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u5c06<code>a^k<\/code>\u4e2d\u7684<code>k<\/code>\u5199\u6210\u4e86\u4e8c\u8fdb\u5236\u7684\u5f62\u5f0f\uff0c\u7528\u5341\u8fdb\u5236\u4e0e\u4e8c\u8fdb\u5236\u7684\u8f6c\u6362\u5173\u7cfb\u6765\u8868\u793a\u8fd9\u4e2a\u7b54\u6848\uff0c\u800c\u4f7f\u7528\u7684\u6570\u5b57<code>a^{2^i}<\/code>\u53ef\u4ee5\u901a\u8fc7\u81ea\u4e58\u83b7\u5f97\u4e4b<\/p>\n<p>\u4e0b\u9762\u7528\u4e00\u4e2a\u4f8b\u5b50\u6765\u8f85\u52a9\u7406\u89e3\u4e0a\u9762\u7684\u8fc7\u7a0b \u5bf9\u4e8e\u95ee\u9898<\/p>\n<pre><code class=\"language-katex\">4^3 \\quad mod \\quad 9<\/code><\/pre>\n<p>\u5206\u89e3\u4e8c\u8fdb\u5236\u7684\u7ed3\u679c\u662f<\/p>\n<pre><code class=\"language-katex\">3=(11)_2<\/code><\/pre>\n<p>\u4e5f\u5c31\u662f\u8bf4<\/p>\n<pre><code class=\"language-katex\">4^3 = 4^{2^0} \\times  4^{2^1}<\/code><\/pre>\n<p>\u800c<\/p>\n<pre><code class=\"language-katex\">4^{2^0}=4 \\\\\n4^{2^1}=4 \\times 4 =4^2\\\\\n(\u5982\u679c\u9700\u8981)4^{2^2}=4^{2^2}<\/code><\/pre>\n<p>\u6240\u4ee5<\/p>\n<pre><code class=\"language-katex\">4^3 \\quad mod \\quad 9 = 1<\/code><\/pre>\n<p>\u4e0a\u4ee3\u7801<\/p>\n<pre><code class=\"language-cpp\">ll qmi(ll a,ll k,ll p){\/\/a ^ k % p\n    ll res=1;\n    while(k){\/\/while(k)k&gt;&gt;1;\u662f\u5c06k\u5206\u89e3\u4e3a\u4e8c\u8fdb\u5236\u6570\u7684\u6807\u51c6\u505a\u6cd5\n        if(k &amp; 1)   res *= (ll)a % p,res %= p; \/\/\u5982\u679ck\u7684\u4e8c\u8fdb\u5236\u672b\u5c3e\u662f1\u5219\u7d2f\u4e58\n        a *= (ll)a % p,a %= p; \/\/a\u81ea\u7d2f\u4e58\n        k &gt;&gt;= 1;\n    }\n    return res % p;\n}\n<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"<p>\u5feb\u901f\u5e42\u4e3b\u8981\u89e3\u51b3\u5f62\u5982 a^k \\quad mod \\quad p \u7684\u95ee\u9898 \u4e00\u4e2a\u975e\u5e38 [&#8230;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13,16],"tags":[],"class_list":["post-380","post","type-post","status-publish","format-standard","hentry","category-13","category-16"],"_links":{"self":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/380","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=380"}],"version-history":[{"count":1,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/380\/revisions"}],"predecessor-version":[{"id":747,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/380\/revisions\/747"}],"wp:attachment":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=380"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=380"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=380"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}