{"id":2763,"date":"2024-03-31T08:42:20","date_gmt":"2024-03-31T00:42:20","guid":{"rendered":"https:\/\/www.aqwu.net\/wp\/?p=2763"},"modified":"2024-03-31T11:38:49","modified_gmt":"2024-03-31T03:38:49","slug":"%e4%b8%80%e9%ab%98%e4%b8%ad%e7%9a%84%e9%ab%98%e8%80%83%e6%a8%a1%e6%8b%9f%e5%a4%a7%e9%a2%98","status":"publish","type":"post","link":"https:\/\/www.aqwu.net\/wp\/?p=2763","title":{"rendered":"\u4e00\u9ad8\u4e2d\u7684\u9ad8\u8003\u6a21\u62df\u5927\u9898"},"content":{"rendered":"\n

\u8fd9\u636e\u8bf4\u662f\u6cb3\u5357\u4e00\u9ad8\u4e2d\u6a21\u62df\u8003\u8bd5\u7684\u6700\u540e\u5927\u9898\uff1a<\/p>\n\n\n\n

\u5728\u5bc6\u7801\u5b66\u9886\u57df,\u6b27\u62c9\u51fd\u6570\u662f\u975e\u5e38\u91cd\u8981\u7684,\u5176\u4e2d\u6700\u8457\u540d\u7684\u5e94\u7528\u5c31\u662f\u5728RSA<\/strong><\/em>\u52a0\u5bc6\u7b97\u6cd5\u4e2d\u7684\u5e94\u7528.\u8bbe p,q<\/em> \u662f\u4e24\u4e2a\u6b63\u6574\u6570,\u82e5 p,q<\/em> \u7684\u6700\u5927\u516c\u7ea6\u6570\u662f1,\u5219\u79f0 p,q<\/em> \u4e92\u7d20,\u5bf9\u4e8e\u4efb\u610f\u6b63\u6574\u6570 n<\/em> ,\u6b27\u62c9\u51fd\u6570\u662f\u4e0d\u8d85\u8fc7 n<\/em> \u4e14\u4e0e n<\/em> \u4e92\u7d20\u7684\u6b63\u6574\u6570\u7684\u4e2a\u6570,\u8bb0\u4e3a \u03c6(n).<\/p>\n\n\n\n

(1)\u8bd5\u6c42 \u03c6(3),\u03c6(9),\u03c6(7),\u03c6(21)\u7684\u503c;<\/p>\n\n\n\n

(2)\u8bbe n<\/em> \u662f\u4e00\u4e2a\u6b63\u6574\u6570 p,q<\/em> \u662f\u4e24\u4e2a\u4e0d\u540c\u7684\u7d20\u6570,\u8bd5\u6c42\u03c6(3n<\/em><\/sup>), \u03c6(pq)\u4e0e \u03c6(p)\u548c\u03c6(q)\u7684\u5173\u7cfb<\/p>\n\n\n\n

(3)RSA\u7b97\u6cd5\u662f\u4e00\u79cd\u975e\u5bf9\u79f0\u52a0\u5bc6\u7b97\u6cd5,\u5b83\u4f7f\u7528\u4e86\u4e24\u4e2a\u4e0d\u540c\u7684\u5bc6\u94a5:\u516c\u94a5\u548c\u79c1\u94a5,\u5177\u4f53\u800c\u8a00:
\u2460\u51c6\u5907\u4e24\u4e2a\u4e0d\u540c\u7684\u3001\u8db3\u591f\u5927\u7684\u7d20\u6570p,q<\/em>;
\u2461\u8ba1\u7b97n=pq<\/em>,\u6b27\u62c9\u51fd\u6570 \u03c6(n);
\u2462\u6c42\u6b63\u6574\u6570k,\u4f7f\u5f97kq\u9664\u4ee5\u03c6(n)\u7684\u4f59\u6570\u662f1;
\u2463\u5176\u4e2d(n,q)<\/em>\u79f0\u4e3a\u516c\u94a5,(n,k)<\/em>\u79f0\u4e3a\u79c1\u94a5.
\u5df2\u77e5\u8ba1\u7b97\u673a\u5de5\u7a0b\u5e08\u5728\u67d0 RSA<\/em><\/strong> \u52a0\u5bc6\u7b97\u6cd5\u4e2d\u516c\u5e03\u7684\u516c\u94a5\u662f(187,17).\u82e5\u6ee1\u8db3\u9898\u610f\u7684\u6b63\u6574\u6570k\u4ece\u5c0f\u5230\u5927\u6392\u5217\u5f97\u5230\u4e00\u5217\u6570\u8bb0\u4e3a\u6570\u5217{bn<\/sub><\/em>)\uff0c\u6570\u5217{cn<\/sub><\/em>)\u6ee1\u8db380cn<\/sub><\/em>=bn<\/sub><\/em>\u534147,\u6c42\u6570\u5217{tancn<\/sub>\u00b7tancn+1<\/sub>}\u7684\u524d\u9879\u548cTn<\/sub><\/p>\n\n\n\n

\u7b54\u6848\u5982\u4e0b<\/strong>\uff1a<\/p>\n\n\n\n

(1)<\/strong> \u5957\u516c\u5f0f\uff1a\u03c6(pk<\/sup>)<\/em> = pk<\/sup><\/em> – pk<\/sup><\/em>-1<\/sup> (p\u4e3a\u8d28\u6570\uff0ck\u4e3a\u5927\u4e8e\u7b49\u4e8e1\u7684\u6574\u6570)<\/p>\n\n\n\n

\u2460 \u03c6(3)= 31<\/sup> – 30<\/sup> = 3 – 1 = 2 (1,2)<\/strong><\/p>\n\n\n\n

\u2461 \u03c6(9) = \u03c6(32<\/sup>) = 32<\/sup> -32-1<\/sup> = 9-3 = 6 (1,2,4,5,7,8)<\/strong><\/p>\n\n\n\n

\u2462 \u03c6(7) = 71<\/sup> – 70<\/sup> = 7 – 1 = 6 (1,2,3,4,5,6)<\/strong><\/p>\n\n\n\n

\u2463 \u03c6(21) = \u03c6(3*7) = \u03c6(3) * \u03c6(7) = 2 * 6 = 12 (1,2,4,5,8,10,11,13,16,17,19,20)<\/strong><\/p>\n\n\n\n

(2)<\/strong><\/p>\n\n\n\n

\u2460 \u03c6(3n<\/em><\/sup>) = 3n<\/sup> – 3n-1<\/sup> = 3n<\/sup>(1-1\/3)= 2 * 3n-1<\/sup><\/p>\n\n\n\n

\u9a8c\u8bc1\uff0c\u5982\u679cn = 1, \u03c6(31<\/em><\/sup>) = 2 * 30<\/sup> = 2<\/p>\n\n\n\n

\u5982\u679c n = 2, \u03c6(32<\/em><\/sup>) = 2 * 32-1<\/sup> = 2 * 31<\/sup> = 6<\/p>\n\n\n\n

\u5982\u679c n = 3, \u03c6(33<\/em><\/sup>) = 2 * 33-1<\/sup> = 2 * 32<\/sup> = 18 (1,2,4,5,7,8,10,11,13,14,16,17,19,20,22,23,25,26)<\/strong><\/p>\n\n\n\n

\u2461 \u03c6(pq) = \u03c6(p)* \u03c6(q)<\/p>\n\n\n\n

(3) <\/strong>\u56e0\u4e3a n = 187, q = 17 \u5f97 p = 11<\/p>\n\n\n\n

\u03c6(187) = \u03c6(17) * \u03c6(11) = 16 * 10 = 160<\/p>\n\n\n\n

\u6c42 bn<\/sub><\/strong><\/p>\n\n\n\n

17k % 160 \u2261<\/em> 1<\/p>\n\n\n\n

17k + 160y = 1 <\/p>\n\n\n\n

\u6c42\u4e24\u4e2a\u6570\u6700\u5927\u516c\u7ea6\u6570\u7684\u8f97\u8f6c\u76f8\u9664\u6cd5<\/p>\n\n\n\n

160 \/ 17 = 9 … 7, \u6240\u4ee5 7 = 160 – 17 * 9<\/p>\n\n\n\n

17 \/ 7 = 2 … 3, \u6240\u4ee5 3 = 17 – 7 * 2<\/p>\n\n\n\n

7 \/ 3 = 2 … 1\uff0c \u6240\u4ee5 1 = 7 – 3 * 2<\/p>\n\n\n\n

\u6240\u4ee5 1 = 7 – 3 * 2 = 7 – (17 – 7 * 2) * 2 <\/p>\n\n\n\n

= 7 + 4 * 7 – 17 * 2 = 5 * 7 -17 * 2 <\/p>\n\n\n\n

= 5 * (160 – 17 * 9) -17 * 2 + <\/p>\n\n\n\n

= 5 * 160 -17 * (2 + 45)<\/p>\n\n\n\n

= 5 * 160 -17 * 47 <\/p>\n\n\n\n

\u5f97\uff0c5 * 160 – 17 * 47 = 1<\/p>\n\n\n\n

\u5f97 y = 5, k = -47<\/p>\n\n\n\n

\u8981\u6c42k \u662f\u6b63\u6574\u6570\uff0c\u53ef\u4ee5<\/p>\n\n\n\n

-17 * 47 = 1 + (- 5 * 160)<\/p>\n\n\n\n

\u4e24\u8fb9\u52a0\u4e0a 17 * 160<\/p>\n\n\n\n

17 * (160 – 47) = 1 + (17 – 5) * 160<\/p>\n\n\n\n

17 * 113 = 1 + 12 * 160 <\/p>\n\n\n\n

\u7b2c\u4e00\u4e2a k\u503c \u662f 113<\/p>\n\n\n\n

\u6c42\u7b2c\u4e8c\u4e2ak\u503c\uff0c\u7b2c\u4e00\u4e2ak\u503c + \u03c6(187) = 113 + 160 = 273<\/p>\n\n\n\n

\u6c42\u7b2c\u4e09\u4e2ak\u503c\uff0c\u662f\u7b2c\u4e8c\u4e2ak\u503c + \u03c6(187) = 273 + 160 = 433<\/p>\n\n\n\n

\u6c42\u7b2c\u56db\u4e2ak\u503c\uff0c\u662f\u7b2c\u4e09\u4e2ak\u503c + \u03c6(187) = 433+ 160 = 593<\/p>\n\n\n\n

bn<\/sub> = bn-1<\/sub> + 160<\/p>\n\n\n\n

bn<\/sub> = 113 + (n -1) * 160 = 160n – 47<\/p>\n\n\n\n

\u6c42 cn<\/sub><\/strong><\/p>\n\n\n\n

cn<\/sub> = (bn<\/sub> + 47 = 160n – 47 + 47)\/80 = 160n \/ 80 = 2n<\/p>\n\n\n\n

\u6c42 Tn<\/sub><\/em><\/strong><\/p>\n\n\n\n

tancn<\/sub>\u00b7tancn+1<\/sub> = tan2n * tan2(n+1)<\/p>\n\n\n\n

tan(2)= tan((2n+2)-2n)<\/p>\n\n\n\n

tan(n) = tan[2(n<\/em>+1)]\u2212tan(2n<\/em>)\u200b\/(1+tan[2(n<\/em>+1)]tan(2n<\/em>))<\/p>\n\n\n\n

Tn<\/sub><\/em>\u200b= tan[2(n<\/em>+1)]\u2212tan(2)\u200b\/(tan(2)\u22121)<\/p>\n","protected":false},"excerpt":{"rendered":"

\u8fd9\u636e\u8bf4\u662f\u6cb3\u5357\u4e00\u9ad8\u4e2d\u6a21\u62df\u8003\u8bd5\u7684\u6700\u540e\u5927\u9898\uff1a \u5728\u5bc6\u7801\u5b66\u9886\u57df,\u6b27\u62c9\u51fd\u6570\u662f\u975e\u5e38\u91cd\u8981\u7684,\u5176\u4e2d\u6700\u8457\u540d\u7684\u5e94\u7528\u5c31\u662f\u5728RSA\u52a0\u5bc6\u7b97 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[43],"tags":[390,402,401],"class_list":["post-2763","post","type-post","status-publish","format-standard","hentry","category-infoarticle","tag-rsa","tag-402","tag-401"],"views":1713,"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/www.aqwu.net\/wp\/index.php?rest_route=\/wp\/v2\/posts\/2763","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.aqwu.net\/wp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.aqwu.net\/wp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.aqwu.net\/wp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.aqwu.net\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2763"}],"version-history":[{"count":17,"href":"https:\/\/www.aqwu.net\/wp\/index.php?rest_route=\/wp\/v2\/posts\/2763\/revisions"}],"predecessor-version":[{"id":2783,"href":"https:\/\/www.aqwu.net\/wp\/index.php?rest_route=\/wp\/v2\/posts\/2763\/revisions\/2783"}],"wp:attachment":[{"href":"https:\/\/www.aqwu.net\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2763"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.aqwu.net\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2763"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.aqwu.net\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2763"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}