{"id":1451,"date":"2020-04-22T12:26:27","date_gmt":"2020-04-22T04:26:27","guid":{"rendered":"https:\/\/mnihyc.com\/blog\/?p=1451"},"modified":"2020-04-22T12:57:27","modified_gmt":"2020-04-22T04:57:27","slug":"%e7%a0%94%e5%ad%a6-%e8%b4%a8%e5%9b%a0%e6%95%b0%e5%88%86%e8%a7%a3%e5%8f%8a%e7%b4%a0%e6%80%a7%e5%88%a4%e5%ae%9a","status":"publish","type":"post","link":"https:\/\/mnihyc.com\/blog\/archives\/1451","title":{"rendered":"[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a"},"content":{"rendered":"<h1 style=\"text-align: center;\">[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a<\/h1>\n<p style=\"text-align: center;\">\u65f6\u95f4\uff1a2018.10 ~ 2019.3<\/p>\n<p style=\"text-align: center;\">\u53c2\u52a0\u6210\u5458\uff1a<span style=\"color: #ff0000;\"><strong><span style=\"color: #000000;\">M<\/span><\/strong>odem_\u00a0 <\/span><span style=\"color: #ff0000;\"><span style=\"color: #000000;\">L<\/span>agoon \u00a0 <\/span><strong>_<\/strong><strong>Q<\/strong><span style=\"color: #ff0000;\">ijia \u00a0 <span style=\"color: #00ff00;\">mnihyc<\/span><\/span><\/p>\n<blockquote>\n<p>\u611f\u89c9\u8fd9\u4e2a\u662f\u4eba\u751f\u5dc5\u5cf0\u4e86\uff0c Lagoon \u4e0a\u6e05\u534e\uff0c\u5269\u4e0b\u6211\u4eec\u4e09\u4e5f\u9000\u5f79\u4e86\uff0c\u6446\u5728\u8fd9\u7559\u4f5c\u7eaa\u5ff5ww\u00a0 <span style=\"color: #ffffff;\">\u62ff\u6fd1\u6237\u53e3\u7684\u8bdd\u6765\u8bf4\uff0c\u5c31\u662f\u840c\u65b0\u90a3\u4f1a\u624d\u662f\u6700\u8f89\u714c\u7684\u65f6\u671fwww<\/span><\/p>\n<\/blockquote>\n<ul>\n<li>\n<h3><strong>\u3010\u5e8f\u8a00\u3011<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u8d28\u6570\u7684\u7814\u7a76\u4e00\u76f4\u662f\u6570\u5b66\u4e0e\u4fe1\u606f\u5b66\u9886\u57df\u4e2d\u7684\u91cd\u8981\u8bfe\u9898\uff0c\u8d28\u6570\u7684\u5224\u5b9a\u4e0e\u8d28\u56e0\u6570\u5206\u89e3\u5728\u73b0\u4ee3\u901a\u8baf\u4fdd\u5bc6\u9886\u57df\u4e2d\u66f4\u662f\u53d1\u6325\u7740\u91cd\u8981\u7684\u4f5c\u7528\uff0c\u672c\u8bfe\u9898\u5c0f\u7ec4\u5c06\u901a\u8fc7\u6b64\u6b21\u7814\u5b66\u673a\u4f1a\u5bf9\u4e0d\u540c\u89c4\u6a21\u4e0b\u81ea\u7136\u6570\u7d20\u6027\u5224\u5b9a\u53ca\u8d28\u56e0\u6570\u5206\u89e3\u7684\u6709\u6548\u7b97\u6cd5\u8fdb\u884c\u63a2\u7a76\uff0c\u52a0\u6df1\u5bf9\u8be5\u9886\u57df\u7684\u4e86\u89e3\u548c\u7406\u89e3\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>\n<h3><strong>\u30101.0\u3011\u7d20\u6570\u7684\u5b9a\u4e49<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u7d20\u6570\uff0c\u53c8\u79f0\u8d28\u6570\u3002\u4e00\u4e2a\u6570 \\(n\\) \u662f\u7d20\u6570\u5f53\u4e14\u4ec5\u5f53\u5b83\u662f\u5927\u4e8e \\(1\\) \u7684\u81ea\u7136\u6570\u4e14\u5b83\u7684\u56e0\u6570\u6709\u4e14\u4ec5\u6709 \\(1\\) \u4e0e \\(n\\) \u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>\n<h3><strong>\u30101.1\u3011\u8bb0\u53f7\u4e0e\u89c4\u5b9a<\/strong><\/h3>\n<\/li>\n<\/ul>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li>\u8bb0 \\(\\displaystyle\\mathbb{R}\\) \u8868\u793a\u5b9e\u6570\u96c6\uff0c\\(\\displaystyle\\mathbb{N}\\) \u8868\u793a\u81ea\u7136\u6570\u96c6\uff0c\\(P\\) \u8868\u793a\u7d20\u6570\u96c6\u3002<\/li>\n<li>\u52d2\u8ba9\u5fb7\u7b26\u53f7 \\(\\displaystyle\\left( {\\frac{n}{p}} \\right)\\)\u3002\u8bbe \\(\\displaystyle p \\in P,n \\in \\mathbb{N}\\)\u3002<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<ul>\n<li>\n<h3><strong>\u30101.2\u3011\u7d20\u6570\u7684\u4e00\u4e9b\u6027\u8d28<\/strong><\/h3>\n<ul>\n<li>\\(P\\) \u662f\u65e0\u9650\u96c6\u3002<\/li>\n<li>\u5bf9\u4e8e\u4efb\u610f\u5927\u4e8e \\(1\\) \u7684\u81ea\u7136\u6570\uff0c\u5b83\u8981\u4e48\u662f\u4e2a\u7d20\u6570\uff0c\u8981\u4e48\u53ef\u4ee5\u5206\u89e3\u4e3a\u82e5\u5e72\u7d20\u6570\u4e4b\u79ef\uff0c\u5e76\u4e14\u5728\u5ffd\u7565\u987a\u5e8f\u7684\u60c5\u51b5\u4e0b\uff0c\u8fd9\u6837\u7684\u5206\u89e3\u662f\u552f\u4e00\u7684\u3002<\/li>\n<li>\u5c0f\u4e8e \\(n\\) \u7684\u8d28\u6570\u5927\u7ea6\u6709 \\(\\ln n\\) \u4e2a\u3002<\/li>\n<li>\u4e00\u4e2a\u5408\u6570 \\(n\\) \u6700\u5c0f\u7684\u7d20\u56e0\u6570\u56e0\u6570\u4e00\u5b9a\u5c0f\u4e8e \\(\\sqrt n \\)\u3002<\/li>\n<li>\u8d39\u9a6c\u5c0f\u5b9a\u7406\uff1a\u8bbe \\(p\\) \u662f\u5927\u4e8e \\(2\\) \u7684\u7d20\u6570\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f\u6b63\u6574\u6570 \\(n\\) \u5747\u6709 \\(\\displaystyle \\begin{array}{*{20}{c}}{{n^{p &#8211; 1}} \\equiv 1}&amp;{(\\bmod p)}\\end{array}\\)\u3002<\/li>\n<li><strong>Mertens&#8217; second theorem<\/strong><strong>\u3002<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li>\n<h3><strong>\u30102.0\u3011\u7d20\u6027\u5224\u5b9a<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u56e0\u4e3a\u7d20\u6570\u96c6\u4e3a\u65e0\u9650\u96c6\uff0c\u5e76\u4e14\u7d20\u6570\u7684\u5206\u5e03\u6ca1\u6709\u89c4\u5f8b\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u5b9e\u73b0\u4e00\u4e2a\u7b97\u6cd5\u6765\u5224\u5b9a\u4e00\u4e2a\u6570\u662f\u5426\u4e3a\u7d20\u6570\u3002\u800c\u7d20\u6027\u5224\u5b9a\u7b97\u6cd5\u6b63\u662f\u8fd9\u6837\u4e00\u7c7b\u7b97\u6cd5\uff1a\u8f93\u5165\u4e00\u4e2a\u6574\u6570\uff0c\u8fd4\u56de\u8fd9\u4e2a\u6570\u662f\u201c\u7d20\u6570\u201d\u8fd8\u662f\u201c\u5408\u6570\u201d\u3002<\/p>\n<p><!--more--><\/p>\n<ul>\n<li>\n<h3><strong>\u30102.1\u3011\u66b4\u529b\u7d20\u6027\u5224\u5b9a<\/strong><\/h3>\n<ul>\n<li>\n<h4><strong>\u30102.1.1\u3011\u7b97\u6cd5\u8fc7\u7a0b<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li>\u8f93\u5165\u6570\u9700\u8981\u5224\u65ad\u7684\u6570 \\(n\\)\u3002<\/li>\n<li>\u679a\u4e3e\u6240\u6709\u5927\u4e8e \\(2\\) \u5c0f\u4e8e \\(\\sqrt n \\) \u7684\u6574\u6570 \\(d\\)\uff0c\u5224\u65ad \\(d\\) \u662f\u5426\u4e3a \\(n\\) \u7684\u56e0\u6570\u3002<\/li>\n<li>\u82e5\u5b58\u5728 \\(d\\) \u662f \\(n\\) \u7684\u56e0\u6570\uff0c\u5219\u8fd4\u56de \\(0\\)\uff1b\u5426\u5219\uff0c\u8fd4\u56de \\(1\\)\u3002<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4>\u30102.1.2\u3011\u6b63\u786e\u6027\u8bc1\u660e<\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u7531\u5b9a\u4e49\u53ef\u77e5\uff0c\u6ca1\u6709\u9664\u4e86 \\(1\\) \u4e0e \\(n\\) \u4ee5\u5916\u56e0\u6570\u7684\u5927\u4e8e \\(1\\) \u7684\u6b63\u6574\u6570 \\(n\\) \u4e3a\u7d20\u6570\uff0c\u56e0\u6b64\u7d20\u6027\u5224\u5b9a\u6700\u6734\u7d20\u7684\u65b9\u6cd5\u4fbf\u662f\u679a\u4e3e\u56e0\u6570\u3002\u4e0a\u8ff0\u7b97\u6cd5\u7684\u6b63\u786e\u6027\u57fa\u4e8e\u4ee5\u4e0b\u7ed3\u8bba\uff1a\u4e00\u4e2a\u6570\u5408\u6570\u6700\u5c0f\u7684\u975e \\(1\\) \u56e0\u6570\u4e00\u5b9a\u5c0f\u4e8e \\(\\sqrt n \\)\u3002\u8fd0\u7528\u53cd\u8bc1\u6cd5\uff0c\u5047\u8bbe \\(n\\) \u7684\u6700\u5c0f\u56e0\u6570 \\(d &gt; \\sqrt n \\)\uff0c\u90a3 \\(\\displaystyle \\frac{n}{d}\\) \u4e3a\u6574\u6570\u4e14 \\(\\displaystyle \\frac{d}{n} &lt; \\sqrt n \\)\uff0c\u5e76\u4e14\u7531\u4e8e \\(\\displaystyle n \\div \\frac{d}{n} = n\\)\uff0c\u56e0\u6b64 \\(\\displaystyle \\frac{d}{n}\\) \u662f\u6bd4 \\(d\\) \u5c0f\u7684 \\(n\\) \u7684\u56e0\u6570\uff0c\u4e0e \\(d\\) \u7684\u6700\u5c0f\u6027\u77db\u76fe\u3002\u6545\u8be5\u7ed3\u8bba\u6210\u7acb\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4>\u30102.1.3\u3011\u590d\u6742\u5ea6\u5206\u6790<\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u663e\u7136\u65f6\u95f4 \\(O(\\sqrt n )\\)\uff0c\u7a7a\u95f4 \\(O(1)\\)\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30102.1.4\u3011\u4ee3\u7801\u5b9e\u73b0<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1452\" src=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55.png\" alt=\"\" width=\"358\" height=\"164\" srcset=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55.png 358w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55-300x137.png 300w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55-150x69.png 150w\" sizes=\"auto, (max-width: 358px) 100vw, 358px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>\u4e0a\u8ff0\u7b97\u6cd5\u80fd\u901a\u5e38\u80fd\u5728\u5408\u7406\u7684\u65f6\u95f4\u5185\u5b9e\u73b0\u5355\u4e2a\u6570\u7684\u7d20\u6027\u5224\u5b9a\uff0c\u4f46\u5f53\u9700\u8981\u5224\u5b9a\u591a\u4e2a\u6570\u7684\u7d20\u6027\u65f6\u5f80\u5f80\u9700\u8981\u201c<strong>\u7b5b\u6cd5\u201d<\/strong>\u7684\u5e2e\u52a9\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>\n<h3><strong>\u30102.2\u3011\u57c3\u5f0f\u7b5b\u6cd5<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u5728\u6570\u8bba\u7b97\u6cd5\u4e2d\uff0c\u679a\u4e3e\u56e0\u6570\u5f80\u5f80\u6bd4\u679a\u4e3e\u500d\u6570\u56f0\u96be\u8bb8\u591a\u3002\u6b63\u96be\u5219\u53cd\uff0c\u65e2\u7136\u7d20\u6027\u5224\u5b9a\u96be\u4ee5\u5feb\u901f\u5b9e\u73b0\uff0c\u90a3\u4e48\u4e0d\u5982\u5c06\u76ee\u5149\u6295\u5411\u5408\u6570\uff0c\u627e\u51fa\u6240\u6709\u7684\u5408\u6570\uff0c\u90a3\u4e48\u5269\u4e0b\u7684\u4e0d\u5c31\u662f\u8d28\u6570\u4e86\u5417\uff1f\u7b5b\u6cd5\u6b63\u57fa\u4e8e\u8fd9\u4e00\u601d\u60f3\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30102.2.1\u3011\u7b97\u6cd5\u8fc7\u7a0b<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li>\u8f93\u5165 \\(n\\)\uff0c\u8868\u793a\u8981\u627e\u51fa \\([2,n]\\) \u4e2d\u7684\u6240\u6709\u7d20\u6570\u3002<\/li>\n<li>\u4f9d\u6b21\u679a\u4e3e \\([2,\\sqrt n ]\\) \u4e2d\u7684\u6570\uff0c\u5982\u679c \\(x\\) \u6ca1\u6709\u88ab\u7b5b\u6389\uff0c\u679a\u4e3e \\([2,n]\\) \u8303\u56f4\u5185\u6240\u6709\u6570 \\(x\\) \u7684\u500d\u6570\uff0c\u8bb0\u5f55\u4e3a\u5408\u6570\u3002<\/li>\n<li>\\([2,n]\\) \u4e2d\u5269\u4e0b\u7684\u6240\u6709\u6570\u8bb0\u5f55\u4e3a\u8d28\u6570\u3002<\/li>\n<li>\u7b97\u6cd5\u7ed3\u675f\uff0c\u53ef\u5728\u6570\u8868\u4e2d\u67e5\u8be2 \\(n\\) \u7684\u7d20\u6027\u3002<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30102.2.2\u3011\u6b63\u786e\u6027\u8bc1\u660e<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u540c\u6837\uff0c\u672c\u7b97\u6cd5\u53ea\u7528\u679a\u4e3e \\([2,\\sqrt n ]\\) \u4e2d\u7684\u6570\u3002<\/p>\n<p style=\"padding-left: 40px;\">\u5982\u679c\u4e00\u4e2a\u6570 \\(n\\) \u662f\u5408\u6570\uff0c\u90a3\u4e48\u5b83\u4e00\u5b9a\u5b58\u5728\u4e00\u4e2a\u5c0f\u4e8e \\(\\sqrt n \\) \u7684\u8d28\u56e0\u6570 \\(p\\)\uff0c\\(p\\) \u4e00\u5b9a\u5728 \\(n\\) \u4e4b\u524d\u88ab\u679a\u4e3e\u5230\uff0c\u5e76\u4e14\u5c06 \\(n\\) \u8bb0\u5f55\u4e3a\u5408\u6570\uff0c\u56e0\u6b64\u5269\u4e0b\u7684\u6570\u4e00\u5b9a\u662f\u7d20\u6570\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30102.2.3\u3011\u590d\u6742\u5ea6\u5206\u6790<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n\\log \\log n)\\)\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u3002<\/p>\n<p style=\"padding-left: 40px;\">\u7a7a\u95f4\u590d\u6742\u5ea6\u663e\u7136\u3002\u57c3\u5f0f\u7b5b\u6cd5\u7684\u74f6\u9888\u590d\u6742\u5ea6\u5728\u679a\u4e3e\u6bcf\u4e2a\u8d28\u6570\u7684\u500d\u6570\u4e0a\uff0c\u4e5f\u5c31\u662f \u3002\u7531<strong>Mertens&#8217; second theorem<\/strong>\uff0c\u53ef\u77e5 \uff08\\(M\\) \u4e3a<strong>Meissel\u2013Mertens<\/strong> \u5e38\u6570\uff09\uff0c\u56e0\u6b64\u57c3\u5f0f\u7b5b\u6cd5\u590d\u6742\u5ea6\u4e3a \\(O(n\\log \\log n)\\)\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30102.2.4\u3011\u4ee3\u7801\u5b9e\u73b0<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1453\" src=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/F7N239TL6M5YVR1YJE.png\" alt=\"\" width=\"352\" height=\"212\" srcset=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/F7N239TL6M5YVR1YJE.png 352w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/F7N239TL6M5YVR1YJE-300x181.png 300w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/F7N239TL6M5YVR1YJE-150x90.png 150w\" sizes=\"auto, (max-width: 352px) 100vw, 352px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>\u5c3d\u7ba1 \\(O(n\\log \\log n)\\) \u5df2\u7ecf\u5341\u5206\u63a5\u8fd1 \\(O(n)\\)\uff0c\u4f46\u4ecd\u6709\u4f18\u5316\u7684\u7a7a\u95f4\u3002\u6b27\u62c9\u7b5b\u6cd5\uff0c\u53c8\u79f0\u7ebf\u6027\u7b5b\u6cd5\u53ef\u5728\u4e25\u683c \\(O(n)\\) \u7684\u65f6\u95f4\u5185\u5224\u5b9a \\([2,n]\\) \u4e2d\u6240\u6709\u6570\u7684\u7d20\u6027\u3002\u5e76\u4e14\u7ebf\u6027\u7b5b\u6cd5\u8fd8\u53ef\u4ee5\u7528\u4e8e\u4e00\u4e9b\u79ef\u6027\u6570\u8bba\u51fd\u6570\u7684\u6c42\u503c\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>\n<h3><strong>\u30102.3\u3011\u7ebf\u6027\u7b5b\u6cd5<\/strong><\/h3>\n<ul>\n<li>\n<h4><strong>\u30102.3.1\u3011\u7b97\u6cd5\u8fc7\u7a0b<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li>\u8f93\u5165 \\(n\\)\uff0c\u8868\u793a\u5b9e\u73b0 \\([2,n]\\) \u5185\u7684\u7d20\u6027\u5224\u5b9a\u3002<\/li>\n<li>\u679a\u4e3e\u4ece\u5c0f\u5230\u5927 \\([2,n]\\) \u7684\u6570\uff0c\u679a\u4e3e\u5230\u6570 \\(m\\) \u65f6\uff0c\u82e5\u5176\u8fd8\u672a\u88ab\u6807\u8bb0\uff0c\u5219\u5c06\u5176\u52a0\u5165\u7d20\u6570\u96c6\u3002\u5bf9\u4e8e\u6240\u6709\u679a\u4e3e\u5230\u7684 \\(m\\)\uff0c\u4ece\u5c0f\u5230\u5927\u679a\u4e3e\u7d20\u6570\u96c6\u4e2d\u6240\u6709\u6ee1\u8db3 \\(m \\times p \\le n\\) \u7684\u7d20\u6570 \\(p\\)\uff0c\u5c06 \\(m \\times p\\) \u6807\u8bb0\u3002\u4e00\u65e6\u5728\u679a\u4e3e\u8fc7\u7a0b\u4e2d\u51fa\u73b0 \\(p|m\\)\uff0c\u5219\u8df3\u51fa\u7d20\u6570\u7684\u679a\u4e3e\u3002<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li style=\"list-style-type: none;\">\n<ol start=\"3\">\n<li>\u7b97\u6cd5\u7ed3\u675f\uff0c\u8fd4\u56de\u7d20\u6570\u96c6\u4e0e\u6570\u8868\u3002<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30102.3.2\u3011\u6b63\u786e\u6027\u8bc1\u660e<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u7ebf\u6027\u7b5b\u6cd5\u4e0e\u57c3\u5f0f\u7b5b\u6cd5\u7684\u57fa\u672c\u8fc7\u7a0b\u76f8\u540c\uff0c\u5173\u952e\u5728\u4e0e\u201c\u4e00\u65e6\u5728\u679a\u4e3e\u8fc7\u7a0b\u4e2d\u51fa\u73b0 \\(p|m\\)\uff0c\u5219\u8df3\u51fa\u7d20\u6570\u7684\u679a\u4e3e\u201d\u8fd9\u4e00\u6b65\u3002\u8fd9\u6837\u505a\u7684\u6b63\u786e\u6027\u57fa\u4e8e\u4ee3\u6570\u57fa\u672c\u5b9a\u7406\u3002\u56e0\u4e3a\u6211\u4eec\u662f\u4ece\u5c0f\u5230\u5927\u679a\u4e3e\u7d20\u6570\u7684\uff0c\u8fd9\u79cd\u5f39\u51fa\u4f7f\u5f97\u6bcf\u4e2a\u6570\u4f1a\u4e14\u53ea\u4f1a\u88ab\u5b83\u7684\u6700\u5c0f\u7d20\u56e0\u6570\u7b5b\u6389\u3002\u6b63\u662f\u8fd9\u79cd\u7279\u6027\u8d4b\u4e88\u4e86\u7ebf\u7b5b\u9012\u63a8\u5904\u7406\u79ef\u6027\u51fd\u6570\u503c\u7684\u80fd\u529b\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30102.3.3\u3011\u590d\u6742\u5ea6\u5206\u6790<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u56e0\u4e3a\u6bcf\u4e2a\u6570\u4f1a\u4e14\u53ea\u4f1a\u88ab\u5b83\u7684\u6700\u5c0f\u7d20\u56e0\u5b50\u7b5b\u53bb\u4e00\u904d\uff0c\u6240\u4ee5\u65f6\u95f4\u590d\u6742\u5ea6\u663e\u7136\u5c31\u662f \\(O(n)\\)\u3002\u7a7a\u95f4\u4e5f\u662f \\(O(n)\\)\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30102.3.4\u3011\u4ee3\u7801\u5b9e\u73b0<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1454\" src=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/QF8UJQSN2YVBKKONT_O.png\" alt=\"\" width=\"461\" height=\"307\" srcset=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/QF8UJQSN2YVBKKONT_O.png 461w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/QF8UJQSN2YVBKKONT_O-300x200.png 300w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/QF8UJQSN2YVBKKONT_O-150x100.png 150w\" sizes=\"auto, (max-width: 461px) 100vw, 461px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>\u5982\u679c\u5224\u5b9a\u51fa\u8303\u56f4\u5185\u6240\u6709\u6570\u7684\u7d20\u6027\uff0c\u90a3\u4e48\u7ebf\u6027\u7b5b\u6cd5\u5df2\u7ecf\u505a\u5230\u4e86\u65f6\u95f4\u4e0e\u7a7a\u95f4\u7ebf\u6027\uff0c\u65e0\u6cd5\u518d\u4f18\u5316\u4e86\u3002\u4f46\u662f\u5f88\u591a\u65f6\u5019\u6211\u4eec\u53ea\u9700\u5224\u5b9a\u51fa\u51e0\u4e2a\u5927\u6574\u6570\u7684\u7d20\u6027\uff0c\u8fd9\u65f6\u7ebf\u6027\u7684\u590d\u6742\u5ea6\u5c31\u6210\u4e86\u6d6a\u8d39\u3002\u800c Miller-Rabin \u7d20\u6027\u6d4b\u8bd5\u5c31\u662f\u4e00\u79cd\u9488\u5bf9\u5355\u4e2a\u6570\u7d20\u6027\u5224\u5b9a\u7684\u4f18\u79c0\u7b97\u6cd5\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>\n<h3><strong>\u30102.4\u3011Miller-Rabin\u7d20\u6027\u6d4b\u8bd5<\/strong><\/h3>\n<ul>\n<li>\n<h4><strong>\u30102.4.0\u3011\u524d\u7f6e\u7406\u8bba\u4e0e\u65b9\u6cd5<\/strong><\/h4>\n<ul>\n<li>\n<h4><strong>\u30102.4.0.1\u3011\u8d39\u9a6c\u5c0f\u5b9a\u7406<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u82e5 \\(p\\) \u662f\u4e00\u4e2a\u8d28\u6570\uff0c\\(a\\) \u662f\u4e00\u4e2a\u4e0d\u4e0e \\(p\\) \u4e92\u8d28\u7684\u6b63\u6574\u6570\u4e14 \\(1 \\le a &lt; p\\)\uff0c\u90a3\u4e48 \\(\\displaystyle \\begin{array}{*{20}{c}}{{a^{p &#8211; 1}} \\equiv 1}&amp;{(\\bmod p)}\\end{array}\\) \u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4>\u30102.4.0.2\u3011\u4e8c\u6b21\u63a2\u6d4b<\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u5982\u679c\u5bf9\u4e8e\u8d28\u6570 \\(p\\) \u6709\u6b63\u6574\u6570 \\(x &lt; p\\)\uff0c\u82e5 \\(\\displaystyle \\begin{array}{*{20}{c}}{{x^2} \\equiv 1}&amp;{(\\bmod p)}\\end{array}\\)\uff0c\u90a3\u4e48 \\(\\displaystyle \\begin{array}{*{20}{c}}{x =\u00a0 \\pm 1}&amp;{(\\bmod p)}\\end{array}\\) \u4e3a\u6b64\u65b9\u7a0b\u552f\u4e00\u7684\u4e24\u4e2a\u975e\u5e73\u51e1\u6839\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4>\u30102.4.1\u3011\u7b97\u6cd5\u7684\u8fc7\u7a0b\u4e0e\u6b63\u786e\u6027<\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u5bf9\u4e8e\u4e00\u4e2a\u5947\u8d28\u6570 \\(n\\)\uff0c\u8bb0 \\(\\displaystyle n &#8211; 1 = {2^k}d\\) \u4e14 \\(2\\not|d\\) \u3002<\/p>\n<p style=\"padding-left: 40px;\">\u82e5 \\(a\\not|d\\)\uff0c\u5219\u6613\u8bc1\u6709 \\(\\displaystyle \\begin{array}{*{20}{c}}{{a^d} \\equiv 1}&amp;{(\\bmod n)}\\end{array}\\) \u6216 \\(\\displaystyle \\begin{array}{*{20}{c}}{{a^d}\/{a^{2d}}\/{a^{{2^2}d}}\/ \\cdots \/{a^{{2^{k &#8211; 1}}d}} \\equiv\u00a0 &#8211; 1}&amp;{(\\bmod n)}\\end{array}\\)\u3002<\/p>\n<p style=\"padding-left: 40px;\">\u5bf9\u4e8e\u4e00\u4e2a\u8d28\u6570 \\(n\\)\uff0c\u7531 <em>Lemma.1<\/em> \u53ef\u5f97\uff0c\\(\\displaystyle \\begin{array}{*{20}{c}}{{a^{{2^k}d}} \\equiv {a^{n &#8211; 1}} \\equiv 1}&amp;{(\\bmod n)}\\end{array}\\)\u3002<\/p>\n<p style=\"padding-left: 40px;\">\u53c8\u56e0\u4e3a \\(\\displaystyle {a^{{2^r}d}} = {({a^{{2^{r &#8211; 1}}d}})^2}\\)\uff0c\u540c\u65f6\u7531 <em>Lemma.2<\/em> \u53ef\u77e5\uff0c\\(\\displaystyle \\begin{array}{*{20}{c}}{{a^{{2^{r &#8211; 1}}d}} \\equiv\u00a0 \\pm 1}&amp;{(\\bmod n)}\\end{array}\\)\uff1a<\/p>\n<p style=\"padding-left: 80px;\">\u82e5 \\(\\displaystyle \\begin{array}{*{20}{c}}{{a^{{2^{r &#8211; 1}}d}} \\equiv\u00a0 &#8211; 1}&amp;{(\\bmod n)}\\end{array}\\)\uff0c\u90a3\u4e48\u7b2c\u4e8c\u4e2a\u6761\u4ef6\u6210\u7acb\uff1b<\/p>\n<p style=\"padding-left: 80px;\">\u82e5 \\(\\displaystyle \\begin{array}{*{20}{c}}{{a^{{2^{r &#8211; 1}}d}} \\equiv 1}&amp;{(\\bmod n)}\\end{array}\\)\uff0c\u90a3\u4e48\u7ee7\u7eed\u5f80\u4e0b\u63a2\u6d4b\uff0c\u76f4\u5230\u5f97\u5230 \\(\\displaystyle \\begin{array}{*{20}{c}}{{a^d} \\equiv 1}&amp;{(\\bmod n)}\\end{array}\\)\uff0c\u90a3\u4e48\u5c31\u4e0e\u7b2c\u4e00\u4e2a\u6761\u4ef6\u77db\u76fe\u3002<\/p>\n<p style=\"padding-left: 40px;\">Miller-Rabin \u7684\u63a2\u6d4b\u65b9\u6cd5\uff1a\u82e5 \\(\\begin{array}{*{20}{c}}{{a^d}1}&amp;{(\\bmod n)}\\end{array}\\) \u4e14 \\(\\begin{array}{*{20}{c}}{{a^{{2^r}d}} &#8211; 1}&amp;{(\\bmod n)}\\end{array}\\)\uff08\\(0 \\le r &lt; k\\) \uff09\uff0c\u90a3\u4e48 \\(n\\) \u5c31\u4e00\u5b9a\u4e0d\u662f\u8d28\u6570\u3002<\/p>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 40px;\">\u4e00\u822c\u6765\u8bf4\uff0c\u9009\u62e9\u591a\u4e2a\u8d28\u6570\u7684\u5e95 \\(a\\) \u53ef\u4ee5\u6781\u5927\u7684\u964d\u4f4e\u9519\u8bef\u7387\uff0c\u82e5\u4f7f \\(k\\) \u8868\u793a\u63a2\u6d4b\u7684\u6b21\u6570\uff0c\u90a3\u4e48\u9519\u8bef\u7387\u53ef\u4ee5\u964d\u4f4e\u81f3\u5927\u7ea6 \\(\\displaystyle {4^{ &#8211; k}}\\) \u3002<\/p>\n<p style=\"padding-left: 40px;\">\u7814\u7a76\u8868\u660e\uff0c\\(n &lt; 3,215,031,751\\) \u65f6\uff0c\u6700\u5c11\u53ef\u4ee5\u9009\u62e9 \\(2,3,5,7\\)\uff1b<\/p>\n<p style=\"padding-left: 40px;\">\\(n &lt; 3,825,123,056,546,413,051\\) \u65f6\uff0c\u6700\u5c11\u53ef\u4ee5\u9009\u62e9 \\(2,3,5,7,11,13,17,19,23\\) \uff1b<\/p>\n<p style=\"padding-left: 40px;\">\\(n &lt; 18,446,744,073,709,551,616\\) \u65f6\uff0c\u6700\u5c11\u53ef\u4ee5\u9009\u62e9 \\(2,3,5,7,11,13,17,19,23,29,31,37\\) \u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4>\u30102.4.2\u3011\u7b97\u6cd5\u7684\u590d\u6742\u5ea6<\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u4f7f\u7528\u5feb\u901f\u5e42\u4e58\u6cd5\uff0c\\(n\\) \u8868\u793a\u9700\u8981\u63a2\u6d4b\u7684\u6570\uff0c\\(k\\) \u8868\u793a\u63a2\u6d4b\u7684\u6b21\u6570\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(k{\\log ^2}n)\\) \u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30102.4.3\u3011\u7b97\u6cd5\u5b9e\u73b0\u4ee3\u7801<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1455\" src=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/1_6DBJS4FV_86CHQX7I.png\" alt=\"\" width=\"578\" height=\"677\" srcset=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/1_6DBJS4FV_86CHQX7I.png 578w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/1_6DBJS4FV_86CHQX7I-256x300.png 256w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/1_6DBJS4FV_86CHQX7I-128x150.png 128w\" sizes=\"auto, (max-width: 578px) 100vw, 578px\" \/><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>\n<h3><strong>\u30103.0\u3011\u8d28\u56e0\u6570\u5206\u89e3\u7b97\u6cd5<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u5728\u6570\u8bba\u4e2d\uff0c\u5408\u6570\u901a\u5e38\u6ca1\u6709\u5f88\u597d\u7684\u6027\u8d28\u3002\u4f46\u7531\u4e8e\u4ee3\u6570\u57fa\u672c\u5b9a\u7406\u6210\u7acb\uff0c\u5e38\u5e38\u628a\u5408\u6570\u5206\u89e3\u4e3a\u5b83\u7684\u7d20\u56e0\u6570\u3002\u4f46\u7d20\u6570\u5206\u89e3\u5e76\u4e0d\u5bb9\u6613\u5b9e\u73b0\uff0c\u8457\u540d\u7684RSA\u516c\u94a5\u52a0\u5bc6\u7b97\u6cd5\u5c31\u662f\u5229\u7528\u4e24\u4e2a\u5927\u7d20\u6570\u76f8\u4e58\u5bb9\u6613\uff0c\u4f46\u5bf9\u5176\u4e58\u79ef\u8fdb\u884c\u5206\u89e3\u5374\u6781\u5176\u56f0\u96be\u7684\u7279\u70b9\u6765\u5b9e\u73b0\u52a0\u5bc6\u7684\u3002\u8d28\u56e0\u6570\u5206\u89e3\u540c\u6837\u6709\u7740\u8bb8\u591a\u4f18\u79c0\u7b97\u6cd5\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>\n<h3><strong>\u30103.1\u3011\u57fa\u4e8e\u7b5b\u6cd5\u7684\u8d28\u56e0\u6570\u5206\u89e3<\/strong><\/h3>\n<ul>\n<li>\n<h4>\u30103.1.1\u3011\u7b97\u6cd5\u6d41\u7a0b<\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li>\u7528\u7b5b\u6cd5\u6c42\u51fa \\([2,\\sqrt n ]\\) \u5185\u7684\u7d20\u6570\u3002<\/li>\n<li>\u679a\u4e3e\u6bcf\u4e2a\u5c0f\u4e8e\u5f53\u524d\u7684 \\(\\sqrt n \\) \u7684\u7d20\u6570\uff0c\u82e5 \\(p\\) \u662f \\(n\\) \u7684\u56e0\u6570\uff0c\u4e0d\u65ad\u7528 \\(\\displaystyle \\frac{p}{n}\\) \u53bb\u66f4\u65b0 \\(n\\)\uff0c\u76f4\u5230 \\(p\\) \u4e0d\u518d\u662f \\(n\\) \u7684\u56e0\u6570\uff0c\u8bbe\u8fd9\u6837\u4e00\u5171\u505a\u4e86 \\(q\\) \u6b21\uff0c\u5219\u5c06 \\({p^q}\\) \u8bb0\u5165 \\(n\\) \u7684\u7d20\u56e0\u6570\u5206\u89e3\u5f0f\u3002<\/li>\n<li>\u82e5\u679a\u4e3e\u7ed3\u675f\u4e4b\u540e\u5269\u4f59\u7684 \\(n\\) \u5927\u4e8e \\(1\\)\uff0c\u5c06\u5176\u52a0\u5165\u7d20\u56e0\u6570\u5206\u89e3\u5f0f\u3002<\/li>\n<li>\u7b97\u6cd5\u7ed3\u675f\uff0c\u8fd4\u56de\u7d20\u56e0\u6570\u5206\u89e3\u5f0f\u3002<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30103.1.2\u3011\u6b63\u786e\u6027\u8bc1\u660e<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u56e0\u4e3a\u679a\u4e3e\u7684\u90fd\u662f\u8d28\u6570 \\(p\\)\uff0c\u800c\u4e14\u4e0d\u65ad\u7528 \\(\\displaystyle \\frac{p}{n}\\) \u66f4\u65b0 \\(n\\) \u540e\uff0c\\(n\\) \u4e2d\u5df2\u4e0d\u518d\u5269\u4e0b\u8be5\u7d20\u56e0\u6570\u3002\u6700\u540e\u5269\u4e0b\u7684 \\(n\\) \u82e5\u4e0d\u4e3a \\(1\\)\uff0c\u5b83\u4e5f\u4e0d\u542b\u5c0f\u4e8e \\(\\sqrt n \\) \u7684\u7d20\u56e0\u6570\uff0c\u7531\u7ed3\u8bba\u30101.2\u3011\uff0c\u5b83\u4e5f\u662f\u4e00\u4e2a\u7d20\u6570\u3002\u6240\u4ee5 \\(n\\) \u4e2d\u7684\u6bcf\u4e00\u4e2a\u7d20\u56e0\u6570\u90fd\u88ab\u5b8c\u5168\u5206\u89e3\u51fa\u6765\uff0c\u7b97\u6cd5\u6709\u6b63\u786e\u6027\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30103.1.3\u3011\u590d\u6742\u5ea6\u5206\u6790<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u7b5b\u6cd5\u65f6\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\uff0c\u5355\u6b21\u8d28\u56e0\u6570\u5206\u89e3\u65f6\u95f4\u590d\u6742\u5ea6 \\(\\displaystyle O({n^{\\frac{1}{4}}})\\)\uff1b\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30103.1.4\u3011\u4ee3\u7801\u5b9e\u73b0<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1456\" src=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/UCAM_3BA4UMSIG@CRE.png\" alt=\"\" width=\"553\" height=\"324\" srcset=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/UCAM_3BA4UMSIG@CRE.png 553w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/UCAM_3BA4UMSIG@CRE-300x176.png 300w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/UCAM_3BA4UMSIG@CRE-150x88.png 150w\" sizes=\"auto, (max-width: 553px) 100vw, 553px\" \/><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>\n<h3><strong>\u30103.2\u3011Pollard-Rho\u7b97\u6cd5<\/strong><\/h3>\n<ul>\n<li>\n<h4><strong>\u30103.2.1\u3011\u7b97\u6cd5\u601d\u8def\u4e0e\u6b63\u786e\u6027\u8bc1\u660e<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u8be5\u7b97\u6cd5\u7528\u4e8e\u627e\u51fa\u4e00\u4e2a\u6570\u7684\u8d28\u56e0\u6570\u3002\u8be5\u7b97\u6cd5\u57fa\u4e8e\u4e00\u4e2a\u5f88\u666e\u901a\u7684\u60f3\u6cd5\uff1a\u968f\u673a\u4e00\u4e2a\u6570 \\(a\\)\uff0c\u9a8c\u8bc1\u5176\u662f\u5426\u662f \\(n\\) \u7684\u8d28\u56e0\u6570\u3002\u8fd9\u4e2a\u590d\u6742\u5ea6\u663e\u7136\u662f\u9519\u7684\u3002\u8fd9\u65f6\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u751f\u65e5\u6096\u8bba\u6765\u63d0\u9ad8\u968f\u673a\u5230\u56e0\u6570\u7684\u6982\u7387\u3002\u751f\u65e5\u6096\u8bba\u53ef\u62bd\u8c61\u4e3a\uff1a\u53d6 \\(k\\) \u4e2a \\([1,365]\\) \u95f4\u7684\u6574\u6570\uff0c\u5f53 \\(\\displaystyle k \\approx \\sqrt {365} \\) \u65f6\uff0c\\(k\\) \u4e2a\u6570\u4e2d\u6709\u4e24\u4e2a\u6570\u76f8\u540c\u7684\u6982\u7387\u5df2\u5341\u5206\u63a5\u8fd1 \\(\\displaystyle \\frac{1}{2}\\)\u3002\u6240\u4ee5\uff0c\u6211\u4eec\u53ef\u4ee5\u53d6 \\(\\sqrt n \\) \u4e2a \\([1,n]\\) \u95f4\u7684\u6570\uff0c\u8fd9\u6837 \\(k\\) \u4e2a\u6570\u4e2d\u6709\u4e24\u4e2a\u6570\u76f8\u51cf\u4e3a \\(n\\) \u7684\u56e0\u6570\u7684\u6982\u7387\u5df2\u5341\u5206\u63a5\u8fd1 \\(\\displaystyle \\frac{1}{2}\\)\uff0c\u63d0\u9ad8\u4e86\u968f\u673a\u5316\u7684\u6548\u7387\u3002<\/p>\n<p style=\"padding-left: 40px;\">\u7531\u4e8e\u7a7a\u95f4\u7684\u9650\u5236\uff0c\u6211\u4eec\u53ef\u4ee5\u6bcf\u6b21\u968f\u4e24\u4e2a\u6570 \\(p,q\\)\uff0c\u5224\u65ad\u5b83\u4eec\u7684\u5dee\u662f\u5426\u662f \\(n\\) \u7684\u56e0\u6570\u3002\u663e\u7136\u8fd9\u4e2a\u64cd\u4f5c\u4e0e\u4e0a\u9762\u4e0d\u7b49\u4ef7\uff0c\u4f46\u6211\u4eec\u6709\u4e00\u4e2a\u5f88\u597d\u7684\u968f\u673a\u51fd\u6570\uff1a<\/p>\n<p style=\"padding-left: 40px;\">\uff08\\(\\displaystyle \\begin{array}{*{20}{c}}{f(x) = ({x^2} + c)}&amp;{(\\bmod n)}\\end{array}\\)\uff0c\\(c\\) \u4e3a\u4e00\u4e2a\u968f\u673a\u6570\uff09\u3002<\/p>\n<p style=\"padding-left: 40px;\">\u4f46\u8fd9\u6837\u53ef\u80fd\u8fdb\u5165\u5faa\u73af\u3002\u4e00\u4e2a\u7b80\u6613\u7684\u5224\u73af\u65b9\u6cd5\u53eb\u505aFloyd\u5224\u73af\u7b97\u6cd5\uff0c\u5927\u610f\u4e3a\u8ba9\u53e6\u4e00\u4e2a\u6570 \\(q\\) \u6bcf\u6b21\u90fd\u6267\u884c\u4e24\u904d \\(f(q)\\)\uff0c\u5f53 \\(q\\) \u201c\u8ffd\u4e0a\u201d \\(p\\) \u65f6\u53ca\u51fa\u73b0\u73af\u3002\u540c\u65f6\u8fd8\u6709\u4e00\u4e2a\u66f4\u4e3a\u9ad8\u6548\u7684\u5224\u73af\u7b97\u6cd5\u2014\u2014Brent\u7684\u5224\u73af\u7b97\u6cd5\u3002\u5373\u53ea\u6709\u5728\u5faa\u73af\u4e86 \\(2\\) \u7684\u6b21\u5e42\u65f6\u624d\u79fb\u52a8 \\(p\\)\uff0c\u540c\u65f6\u628a \\(q\\) \u8bbe\u4e3a\u79fb\u52a8\u524d\u7684 \\(p\\) \uff0c\u5224\u65ad\u65f6\u76f4\u63a5\u5224\u65ad \\(p,q\\) \u662f\u5426\u76f8\u7b49\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30103.2.2\u3011\u590d\u6742\u5ea6\u5206\u6790<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u8fd9\u6837\u53ef\u80fd\u4e00\u6b21\u627e\u4e0d\u51fa\u8d28\u56e0\u6570\uff0c\u4f46\u6211\u4eec\u53ef\u4ee5\u591a\u6b21\u6539\u53d8 \\(c\\)\uff0c\u76f4\u5230\u627e\u5230\u4e3a\u6b62\u3002\u5728\u8fd9\u91cc\u8981\u8bb0\u5f97\u524d\u9762\u8bb2\u5f97\u751f\u65e5\u6096\u8bba\uff0c\u56e0\u6b64\u8fd9\u4e2a\u8fc7\u7a0b\u671f\u671b\u4e0d\u4f1a\u91cd\u590d\u5f88\u591a\u6b21\u3002<\/p>\n<p style=\"padding-left: 40px;\">\u4e8e\u662f\u671f\u671b\u590d\u6742\u5ea6\u4e3a \\(\\displaystyle O({n^{\\frac{1}{4}}})\\)\uff08\u5305\u542b Miller-Rabin \u5224\u65ad\u7d20\u6570\u53ca\u8f97\u8f6c\u76f8\u9664\u6c42 \\(\\gcd\\)\uff09\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30103.2.3\u3011\u6838\u5fc3\u4ee3\u7801<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1457\" src=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/B3@3JFCUGXLDRX3N26MO.png\" alt=\"\" width=\"566\" height=\"739\" srcset=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/B3@3JFCUGXLDRX3N26MO.png 566w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/B3@3JFCUGXLDRX3N26MO-230x300.png 230w, https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/B3@3JFCUGXLDRX3N26MO-115x150.png 115w\" sizes=\"auto, (max-width: 566px) 100vw, 566px\" \/><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>\n<h3><strong>\u30103.3\u3011\u4e8c\u6b21\u7b5b\u6cd5<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u4e8c\u6b21\u7b5b\u6cd5\u662f\u4e00\u79cd\u57fa\u4e8e\u8d39\u9a6c\u5206\u89e3\u7684\u3001\u8fd0\u7528\u6784\u9020\u6280\u5de7\u7684\u6781\u4e3a\u5f3a\u5927\u7684\u8d28\u56e0\u6570\u5206\u89e3\u7b97\u6cd5\uff0c\u5728\u5176\u8fc7\u7a0b\u4e2d\u9700\u8981\u5b9e\u73b0\u9ad8\u7cbe\u5ea6\u6570\u7684\u8fd0\u7b97\u3001Miller_Rabin\u300101\u9ad8\u65af\u6d88\u5143\u7b49\u5b50\u7b97\u6cd5\uff0c\u53ef\u4ee5\u79f0\u5f97\u4e0a\u662f\u8d28\u56e0\u6570\u5206\u89e3\u7b97\u6cd5\u7684\u96c6\u5927\u6210\u8005\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30103.3.1\u3011\u7b97\u6cd5\u8fc7\u7a0b<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u7531\u4e8e\u7b97\u6cd5\u8fc7\u7a0b\u7684\u590d\u6742\u6027\uff0c\u8fd9\u91cc\u5c06\u4e00\u4e9b\u5b50\u7b97\u6cd5\u7684\u8fc7\u7a0b\u76f4\u63a5\u5c01\u88c5\u3002<\/p>\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li style=\"list-style-type: none;\">\n<ol>\n<li>\u6784\u9020\u51fa\u4e00\u7ec4\u7d20\u6570\u57fa\u5e95 \\(S\\)\uff0c\u5176\u4e2d\u6bcf\u4e00\u4e2a\u7d20\u6570 \\(p\\) \u90fd\u8981\u6ee1\u8db3 \\(\\displaystyle \\left( {\\frac{{n\\bmod p}}{p}} \\right) = 1\\)\uff08\\(n\\) \u5bf9 \\(p\\) \u7684\u52d2\u8ba9\u5fb7\u7b26\u53f7\uff09\u3002<\/li>\n<li>\u9009\u62e9\u82e5\u5e72\u4e2a \\({x_i}\\)\uff0c\u8ba1\u7b97\u51fa\u4e00\u7cfb\u5217 \\(\\displaystyle Q({x_i}) = {({x_i} + \\left\\lfloor {\\sqrt n } \\right\\rfloor )^2} &#8211; n\\) \u7684\u503c\u7ec4\u6210\u96c6\u5408 \\(T\\)\u3002<\/li>\n<li>\u8fd0\u7528\u57fa\u4e8e\u7b5b\u6cd5\u7684\u7d20\u56e0\u6570\u5206\u89e3\u65b9\u6cd5\uff0c\u627e\u51fa \\(T\\) \u4e2d\u6240\u6709\u7d20\u56e0\u6570\u96c6\u5408 \\(s\\) \u662f \\(S\\) \u5b50\u96c6\u7684 \\(\\displaystyle Q({x_i})\\) \u7ec4\u6210\u96c6\u5408 \\(B\\)\u3002<\/li>\n<li>\u4ee5 \\(B\\) \u4e3a\u57fa\u5e95\u4f7f\u752801\u9ad8\u65af\u6d88\u5143\u6784\u9020\u51fa\u4e00\u7ec4 \\(j\\) \u4f7f\u5f97 \\(\\displaystyle \\Pi Q({x_j})\\) \u4e3a\u5e73\u65b9\u6570\u3002<\/li>\n<li>\u4e8e\u662f\u6211\u4eec\u6709 \\(\\displaystyle \\Pi Q({x_j}) \\equiv \\Pi {({x_j} + \\sqrt n )^2}(\\bmod n)\\)\uff0c\u7528\u8d39\u9a6c\u5206\u89e3\u6cd5\u627e\u51fa \\(n\\) \u7684\u4e00\u4e2a\u56e0\u6570\uff0c\u5982\u679c\u662f\u975e\u5e73\u51e1\u56e0\u5b50\uff0c\u9012\u5f52\u6c42\u89e3\uff0c\u5426\u5219\u6269\u5927 \\(\\displaystyle {{\\rm{x}}_i}\\) \u8303\u56f4\u91cd\u65b0\u8ba1\u7b97\u3002<\/li>\n<li>\u82e5\u5f53\u524d \\(n\\) \u4e3a\u8d28\u6570\uff0c\u76f4\u63a5\u8fd4\u56de\uff1b<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30103.3.2\u3011\u6b63\u786e\u6027\u8bc1\u660e<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u9996\u5148\u662f\u8d39\u9a6c\u5206\u89e3\u6cd5\uff1a<\/p>\n<p style=\"padding-left: 40px;\">\u8fd9\u79cd\u56e0\u6570\u5206\u89e3\u6cd5\u662f\u7531\u8d39\u9a6c\u63d0\u51fa\u5e76\u8bc1\u660e\u7684\u3002\u5c1d\u8bd5\u627e\u5230 \\(x,y\\) \u6ee1\u8db3 \\(\\displaystyle \\begin{array}{*{20}{c}}{{x^2} \\equiv {y^2}}&amp;{(\\bmod n)}\\end{array}\\) \uff0c\u8fd9\u6837\u5c31\u6709 \\(\\displaystyle \\begin{array}{*{20}{c}}{(x + y)(x &#8211; y) \\equiv 0}&amp;{(\\bmod n)}\\end{array}\\)\uff0c\u6545 \\(\\displaystyle \\gcd (x + y,x &#8211; y)\\) \u4e00\u5b9a\u662f \\(n\\) \u7684\u56e0\u6570\uff0c\u4e14\u5927\u6982\u7387\u662f\u975e\u5e73\u51e1\u56e0\u5b50\u3002<\/p>\n<p style=\"padding-left: 40px;\">\u7136\u540e\u662f \\(Q(x)\\) \u7684\u6784\u9020\uff1a<\/p>\n<p style=\"padding-left: 40px;\">\u57fa\u4e8e\u8d39\u9a6c\u5206\u89e3\u6cd5\uff0c\u6211\u4eec\u6784\u9020 \\(\\displaystyle Q(x) = {(x + \\left\\lfloor {\\sqrt n } \\right\\rfloor )^2} &#8211; n\\)\uff0c\u8fd9\u6837\u53ea\u8981\u627e\u5230 \\(x\\) \u4f7f \\(Q(x)\\) \u4e3a\u5b8c\u5168\u5e73\u65b9\u6570\uff0c\u5c31\u6709 \\(\\displaystyle \\begin{array}{*{20}{c}}{Q(x) = {{(x + \\left\\lfloor {\\sqrt n } \\right\\rfloor )}^2}}&amp;{(\\bmod n)}\\end{array}\\) \u800c\u5b8c\u5168\u5e73\u65b9\u6570\u8bf4\u767d\u4e86\u5c31\u662f\u6240\u6709\u7d20\u56e0\u6570\u4e2a\u6570\u90fd\u662f\u5076\u6570\u7684\u6570\u3002\u4f46\u76f4\u63a5\u6784\u9020\u51fa\u8fd9\u6837\u7684\u6570\u5e76\u4e0d\u5bb9\u6613\uff0c\u53ef\u4ee5\u5148\u9009\u62e9\u4e00\u4e2a\u7d20\u6570\u96c6 \\(S\\)\uff0c\u627e\u51fa\u82e5\u5e72\u53ef\u4ee5\u88ab \\(S\\) \u5185\u6570\u5b8c\u5168\u5206\u89e3\u7684 \\(Q({x_i})\\)\u3002\u867d\u7136\u5355\u4e2a \\(Q({x_i})\\) \u7684\u6bcf\u4e2a\u7d20\u56e0\u6570\u7684\u4e2a\u6570\u53ef\u80fd\u4e0d\u662f\u5076\u6570\uff0c\u4f46\u901a\u8fc7\u5c06 \\(Q({x_i})\\) \u76f8\u4e58\u6700\u7ec8\u5c31\u6709\u80fd\u529b\u5c06\u5176\u8865\u6210\u5076\u6570\u3002\u800c\u8fd9\u65f6\u5c31\u53ef\u4ee5\u5217\u4e00\u4e2a\u6a21 \\(2\\) \u610f\u4e49\u4e0b\u768401\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u6c42\u89e3\u51fa\u6bcf\u4e2a \\(Q({x_i})\\) \u5bf9\u5e94\u7684\u7cfb\u6570\u5373\u53ef\u3002<\/p>\n<p style=\"padding-left: 40px;\">\u8fd9\u6837\u5c31\u517c\u987e\u4e86\u6b63\u786e\u7387\u4e0e\u590d\u6742\u5ea6\u3002<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>\n<h4><strong>\u30103.3.3\u3011\u65f6\u95f4\u590d\u6742\u5ea6<\/strong><\/h4>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p style=\"padding-left: 40px;\">\u65f6\u95f4\u590d\u6742\u5ea6 \\(\\displaystyle {\\rm{O}}({e^{\\sqrt {\\ln n\\ln \\ln n} }})\\)\uff0c\u7a7a\u95f4\u590d\u6742\u5ea6 \\(O(n)\\)<\/p>\n<p style=\"padding-left: 40px;\">\u8bc1\u660e\u8f83\u4e3a\u7e41\u7410\uff0c\u6b64\u5904\u7565\u53bb<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>\n<h3><strong>\u3010\u5c3e\u58f0\u3011<\/strong><\/h3>\n<\/li>\n<\/ul>\n<p>\u4ee5\u4e0a\u7684\u79cd\u79cd\u7b97\u6cd5\u90fd\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u5b9e\u73b0\u7d20\u6570\u5224\u5b9a\u6216\u5b8c\u6210\u8d28\u56e0\u6570\u5206\u89e3\uff0c\u4ed6\u4eec\u5bf9\u5e94\u7740\u4e0d\u540c\u7684\u65f6\u95f4\u4e0e\u7a7a\u95f4\u89c4\u6a21\uff0c\u56e0\u6b64\u5728\u771f\u6b63\u4f7f\u7528\u65f6\u8981\u6ce8\u610f\u7b97\u6cd5\u7684\u9009\u62e9\u3002<\/p>\n<p>\u6709\u4eba\u8bf4\uff0c\u6570\u8bba\u662f\u6570\u5b66\u738b\u51a0\u4e0a\u7684\u7480\u74a8\u660e\u73e0\uff0c\u800c\u7d20\u6570\u6709\u662f\u6570\u8bba\u4e2d\u4e0d\u53ef\u6216\u7f3a\u7684\u4e00\u4e2a\u90e8\u5206\u3002\u4eba\u7c7b\u5173\u4e8e\u7d20\u6570\u7684\u7814\u7a76\u6c38\u8fdc\u4e0d\u4f1a\u505c\u6b47\uff0c\u7d20\u6027\u5224\u5b9a\u4e0e\u8d28\u56e0\u6570\u5206\u89e3\u7684\u7b97\u6cd5\u4e5f\u662f\u8fd9\u6837\u3002\u4e0a\u6587\u4e2d\u63d0\u5230\u7684\u4e8c\u6b21\u7b5b\u6cd5\u5df2\u7ecf\u662f\u8d28\u56e0\u6570\u5206\u89e3\u4e2d\u7684\u4f18\u79c0\u7b97\u6cd5\uff0c\u800c\u57282002\u5e74\uff0cAKS\u7d20\u6027\u6d4b\u8bd5\u4e5f\u6210\u4e3a\u4e86\u9996\u4e2a\u786e\u5b9a\u6027\u7684\u9ad8\u6548\u7d20\u6027\u5224\u5b9a\u7b97\u6cd5\u3002\u800c1994\u5e74\u79c0\u5c14\u7b97\u6cd5\u7684\u51fa\u73b0\uff0c\u4e5f\u7ed9\u4e86\u6211\u4eec\u4e00\u4e2a\u7f8e\u597d\u7684\u5c55\u671b\u2014\u2014\u91cf\u5b50\u8ba1\u7b97\u673a\u80fd\u5728\u5bf9\u6570\u65f6\u95f4\u5185\u771f\u6b63\u9ad8\u6548\u5730\u89e3\u51b3\u8d28\u56e0\u6570\u5206\u89e3\u95ee\u9898\u3002<\/p>\n<p>\u76f8\u4fe1\u5728\u672a\u6765\uff0c\u968f\u7740\u786c\u4ef6\u4e0e\u8f6f\u4ef6\u7684\u8fdb\u6b65\uff0c\u4e5f\u5c06\u4f1a\u6709\u66f4\u591a\u4f18\u79c0\u7684\u7b97\u6cd5\u6d8c\u73b0\uff0c\u8ba9\u7d20\u6570\u5728\u4fe1\u606f\u65f6\u4ee3\u7efd\u653e\u66f4\u52a0\u707f\u70c2\u7684\u5149\u5f69\u3002<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a \u65f6\u95f4\uff1a2018.10 ~ 2019.3 \u53c2\u52a0\u6210\u5458\uff1aModem_\u00a0 Lagoon &hellip; <a href=\"https:\/\/mnihyc.com\/blog\/archives\/1451\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[67,26,32],"tags":[],"class_list":["post-1451","post","type-post","status-publish","format-standard","hentry","category-topic","category-number-theory","category-combmath"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a - mnihyc&#039;s Blog<\/title>\n<meta name=\"description\" content=\"\u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a \u65f6\u95f4\uff1a2018.10 ~ 2019.3 \u53c2\u52a0\u6210\u5458\uff1aModem_\u00a0 Lagoon \u00a0 _Qijia \u00a0 mnihyc \u611f\u89c9\u8fd9\u4e2a\u662f\u4eba\u751f\u5dc5\u5cf0\u4e86\uff0c Lagoon \u4e0a\u6e05\u534e\uff0c\u5269\u4e0b\u6211\u4eec\u4e09\u4e5f\u9000\u5f79\u4e86\uff0c\u6446\u5728\u8fd9\u7559\u4f5c\u7eaa\u5ff5ww\u00a0 \u62ff\u6fd1\u6237\u53e3\u7684\u8bdd\u6765\u8bf4\uff0c\u5c31\u662f\u840c\u65b0\u90a3\u4f1a\u624d\u662f\u6700\u8f89\u714c\u7684\u65f6\u671fwww \u3010\u5e8f\u8a00\u3011\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a - mnihyc&#039;s Blog\" \/>\n<meta property=\"og:description\" content=\"\u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a \u65f6\u95f4\uff1a2018.10 ~ 2019.3 \u53c2\u52a0\u6210\u5458\uff1aModem_\u00a0 Lagoon \u00a0 _Qijia \u00a0 mnihyc \u611f\u89c9\u8fd9\u4e2a\u662f\u4eba\u751f\u5dc5\u5cf0\u4e86\uff0c Lagoon \u4e0a\u6e05\u534e\uff0c\u5269\u4e0b\u6211\u4eec\u4e09\u4e5f\u9000\u5f79\u4e86\uff0c\u6446\u5728\u8fd9\u7559\u4f5c\u7eaa\u5ff5ww\u00a0 \u62ff\u6fd1\u6237\u53e3\u7684\u8bdd\u6765\u8bf4\uff0c\u5c31\u662f\u840c\u65b0\u90a3\u4f1a\u624d\u662f\u6700\u8f89\u714c\u7684\u65f6\u671fwww \u3010\u5e8f\u8a00\u3011\" \/>\n<meta property=\"og:url\" content=\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451\" \/>\n<meta property=\"og:site_name\" content=\"mnihyc&#039;s Blog\" \/>\n<meta property=\"article:published_time\" content=\"2020-04-22T04:26:27+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2020-04-22T04:57:27+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55.png\" \/>\n<meta name=\"author\" content=\"mnihyc\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@mnihyc\" \/>\n<meta name=\"twitter:site\" content=\"@mnihyc\" \/>\n<meta name=\"twitter:label1\" content=\"\u4f5c\u8005\" \/>\n\t<meta name=\"twitter:data1\" content=\"mnihyc\" \/>\n\t<meta name=\"twitter:label2\" content=\"\u9884\u8ba1\u9605\u8bfb\u65f6\u95f4\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 \u5206\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#article\",\"isPartOf\":{\"@id\":\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451\"},\"author\":{\"name\":\"mnihyc\",\"@id\":\"https:\/\/mnihyc.com\/blog\/#\/schema\/person\/61e167d6d591fdd20dcfee2cf848a751\"},\"headline\":\"[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a\",\"datePublished\":\"2020-04-22T04:26:27+00:00\",\"dateModified\":\"2020-04-22T04:57:27+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451\"},\"wordCount\":690,\"commentCount\":1,\"publisher\":{\"@id\":\"https:\/\/mnihyc.com\/blog\/#\/schema\/person\/61e167d6d591fdd20dcfee2cf848a751\"},\"image\":{\"@id\":\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#primaryimage\"},\"thumbnailUrl\":\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55.png\",\"articleSection\":[\"\u4e13\u9898\",\"\u6570\u8bba\",\"\u79bb\u6563\u6570\u5b66\"],\"inLanguage\":\"zh-Hans\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451\",\"url\":\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451\",\"name\":\"[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a - mnihyc&#039;s Blog\",\"isPartOf\":{\"@id\":\"https:\/\/mnihyc.com\/blog\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#primaryimage\"},\"image\":{\"@id\":\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#primaryimage\"},\"thumbnailUrl\":\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55.png\",\"datePublished\":\"2020-04-22T04:26:27+00:00\",\"dateModified\":\"2020-04-22T04:57:27+00:00\",\"description\":\"\u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a \u65f6\u95f4\uff1a2018.10 ~ 2019.3 \u53c2\u52a0\u6210\u5458\uff1aModem_\u00a0 Lagoon \u00a0 _Qijia \u00a0 mnihyc \u611f\u89c9\u8fd9\u4e2a\u662f\u4eba\u751f\u5dc5\u5cf0\u4e86\uff0c Lagoon \u4e0a\u6e05\u534e\uff0c\u5269\u4e0b\u6211\u4eec\u4e09\u4e5f\u9000\u5f79\u4e86\uff0c\u6446\u5728\u8fd9\u7559\u4f5c\u7eaa\u5ff5ww\u00a0 \u62ff\u6fd1\u6237\u53e3\u7684\u8bdd\u6765\u8bf4\uff0c\u5c31\u662f\u840c\u65b0\u90a3\u4f1a\u624d\u662f\u6700\u8f89\u714c\u7684\u65f6\u671fwww \u3010\u5e8f\u8a00\u3011\",\"breadcrumb\":{\"@id\":\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#breadcrumb\"},\"inLanguage\":\"zh-Hans\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"zh-Hans\",\"@id\":\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#primaryimage\",\"url\":\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55.png\",\"contentUrl\":\"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55.png\",\"width\":358,\"height\":164},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"\u9996\u9875\",\"item\":\"https:\/\/mnihyc.com\/blog\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mnihyc.com\/blog\/#website\",\"url\":\"https:\/\/mnihyc.com\/blog\/\",\"name\":\"mnihyc&#039;s Blog\",\"description\":\"Welcome!\",\"publisher\":{\"@id\":\"https:\/\/mnihyc.com\/blog\/#\/schema\/person\/61e167d6d591fdd20dcfee2cf848a751\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mnihyc.com\/blog\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"zh-Hans\"},{\"@type\":[\"Person\",\"Organization\"],\"@id\":\"https:\/\/mnihyc.com\/blog\/#\/schema\/person\/61e167d6d591fdd20dcfee2cf848a751\",\"name\":\"mnihyc\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"zh-Hans\",\"@id\":\"https:\/\/mnihyc.com\/blog\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8d111f863afc3f98816bc96220f97077d470a96f41088de9f19530fc480f8e72?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8d111f863afc3f98816bc96220f97077d470a96f41088de9f19530fc480f8e72?s=96&d=mm&r=g\",\"caption\":\"mnihyc\"},\"logo\":{\"@id\":\"https:\/\/mnihyc.com\/blog\/#\/schema\/person\/image\/\"}}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a - mnihyc&#039;s Blog","description":"\u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a \u65f6\u95f4\uff1a2018.10 ~ 2019.3 \u53c2\u52a0\u6210\u5458\uff1aModem_\u00a0 Lagoon \u00a0 _Qijia \u00a0 mnihyc \u611f\u89c9\u8fd9\u4e2a\u662f\u4eba\u751f\u5dc5\u5cf0\u4e86\uff0c Lagoon \u4e0a\u6e05\u534e\uff0c\u5269\u4e0b\u6211\u4eec\u4e09\u4e5f\u9000\u5f79\u4e86\uff0c\u6446\u5728\u8fd9\u7559\u4f5c\u7eaa\u5ff5ww\u00a0 \u62ff\u6fd1\u6237\u53e3\u7684\u8bdd\u6765\u8bf4\uff0c\u5c31\u662f\u840c\u65b0\u90a3\u4f1a\u624d\u662f\u6700\u8f89\u714c\u7684\u65f6\u671fwww \u3010\u5e8f\u8a00\u3011","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451","og_locale":"zh_CN","og_type":"article","og_title":"[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a - mnihyc&#039;s Blog","og_description":"\u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a \u65f6\u95f4\uff1a2018.10 ~ 2019.3 \u53c2\u52a0\u6210\u5458\uff1aModem_\u00a0 Lagoon \u00a0 _Qijia \u00a0 mnihyc \u611f\u89c9\u8fd9\u4e2a\u662f\u4eba\u751f\u5dc5\u5cf0\u4e86\uff0c Lagoon \u4e0a\u6e05\u534e\uff0c\u5269\u4e0b\u6211\u4eec\u4e09\u4e5f\u9000\u5f79\u4e86\uff0c\u6446\u5728\u8fd9\u7559\u4f5c\u7eaa\u5ff5ww\u00a0 \u62ff\u6fd1\u6237\u53e3\u7684\u8bdd\u6765\u8bf4\uff0c\u5c31\u662f\u840c\u65b0\u90a3\u4f1a\u624d\u662f\u6700\u8f89\u714c\u7684\u65f6\u671fwww \u3010\u5e8f\u8a00\u3011","og_url":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451","og_site_name":"mnihyc&#039;s Blog","article_published_time":"2020-04-22T04:26:27+00:00","article_modified_time":"2020-04-22T04:57:27+00:00","og_image":[{"url":"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55.png","type":"","width":"","height":""}],"author":"mnihyc","twitter_card":"summary_large_image","twitter_creator":"@mnihyc","twitter_site":"@mnihyc","twitter_misc":{"\u4f5c\u8005":"mnihyc","\u9884\u8ba1\u9605\u8bfb\u65f6\u95f4":"3 \u5206"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#article","isPartOf":{"@id":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451"},"author":{"name":"mnihyc","@id":"https:\/\/mnihyc.com\/blog\/#\/schema\/person\/61e167d6d591fdd20dcfee2cf848a751"},"headline":"[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a","datePublished":"2020-04-22T04:26:27+00:00","dateModified":"2020-04-22T04:57:27+00:00","mainEntityOfPage":{"@id":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451"},"wordCount":690,"commentCount":1,"publisher":{"@id":"https:\/\/mnihyc.com\/blog\/#\/schema\/person\/61e167d6d591fdd20dcfee2cf848a751"},"image":{"@id":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#primaryimage"},"thumbnailUrl":"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55.png","articleSection":["\u4e13\u9898","\u6570\u8bba","\u79bb\u6563\u6570\u5b66"],"inLanguage":"zh-Hans","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/cf.mnihyc.com\/blog\/archives\/1451#respond"]}]},{"@type":"WebPage","@id":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451","url":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451","name":"[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a - mnihyc&#039;s Blog","isPartOf":{"@id":"https:\/\/mnihyc.com\/blog\/#website"},"primaryImageOfPage":{"@id":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#primaryimage"},"image":{"@id":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#primaryimage"},"thumbnailUrl":"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55.png","datePublished":"2020-04-22T04:26:27+00:00","dateModified":"2020-04-22T04:57:27+00:00","description":"\u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a \u65f6\u95f4\uff1a2018.10 ~ 2019.3 \u53c2\u52a0\u6210\u5458\uff1aModem_\u00a0 Lagoon \u00a0 _Qijia \u00a0 mnihyc \u611f\u89c9\u8fd9\u4e2a\u662f\u4eba\u751f\u5dc5\u5cf0\u4e86\uff0c Lagoon \u4e0a\u6e05\u534e\uff0c\u5269\u4e0b\u6211\u4eec\u4e09\u4e5f\u9000\u5f79\u4e86\uff0c\u6446\u5728\u8fd9\u7559\u4f5c\u7eaa\u5ff5ww\u00a0 \u62ff\u6fd1\u6237\u53e3\u7684\u8bdd\u6765\u8bf4\uff0c\u5c31\u662f\u840c\u65b0\u90a3\u4f1a\u624d\u662f\u6700\u8f89\u714c\u7684\u65f6\u671fwww \u3010\u5e8f\u8a00\u3011","breadcrumb":{"@id":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#breadcrumb"},"inLanguage":"zh-Hans","potentialAction":[{"@type":"ReadAction","target":["https:\/\/cf.mnihyc.com\/blog\/archives\/1451"]}]},{"@type":"ImageObject","inLanguage":"zh-Hans","@id":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#primaryimage","url":"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55.png","contentUrl":"https:\/\/mnihyc.com\/blog\/wp-content\/uploads\/2020\/04\/3XN79LE@AZYHGKAE55.png","width":358,"height":164},{"@type":"BreadcrumbList","@id":"https:\/\/cf.mnihyc.com\/blog\/archives\/1451#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"\u9996\u9875","item":"https:\/\/mnihyc.com\/blog"},{"@type":"ListItem","position":2,"name":"[\u7814\u5b66] \u8d28\u56e0\u6570\u5206\u89e3\u53ca\u7d20\u6027\u5224\u5b9a"}]},{"@type":"WebSite","@id":"https:\/\/mnihyc.com\/blog\/#website","url":"https:\/\/mnihyc.com\/blog\/","name":"mnihyc&#039;s Blog","description":"Welcome!","publisher":{"@id":"https:\/\/mnihyc.com\/blog\/#\/schema\/person\/61e167d6d591fdd20dcfee2cf848a751"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mnihyc.com\/blog\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"zh-Hans"},{"@type":["Person","Organization"],"@id":"https:\/\/mnihyc.com\/blog\/#\/schema\/person\/61e167d6d591fdd20dcfee2cf848a751","name":"mnihyc","image":{"@type":"ImageObject","inLanguage":"zh-Hans","@id":"https:\/\/mnihyc.com\/blog\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8d111f863afc3f98816bc96220f97077d470a96f41088de9f19530fc480f8e72?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8d111f863afc3f98816bc96220f97077d470a96f41088de9f19530fc480f8e72?s=96&d=mm&r=g","caption":"mnihyc"},"logo":{"@id":"https:\/\/mnihyc.com\/blog\/#\/schema\/person\/image\/"}}]}},"_links":{"self":[{"href":"https:\/\/mnihyc.com\/blog\/wp-json\/wp\/v2\/posts\/1451","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mnihyc.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mnihyc.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mnihyc.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mnihyc.com\/blog\/wp-json\/wp\/v2\/comments?post=1451"}],"version-history":[{"count":0,"href":"https:\/\/mnihyc.com\/blog\/wp-json\/wp\/v2\/posts\/1451\/revisions"}],"wp:attachment":[{"href":"https:\/\/mnihyc.com\/blog\/wp-json\/wp\/v2\/media?parent=1451"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mnihyc.com\/blog\/wp-json\/wp\/v2\/categories?post=1451"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mnihyc.com\/blog\/wp-json\/wp\/v2\/tags?post=1451"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}