{"id":156,"date":"2022-09-11T17:12:39","date_gmt":"2022-09-11T09:12:39","guid":{"rendered":"https:\/\/blog.taiyang.space\/?p=156"},"modified":"2024-01-07T21:39:38","modified_gmt":"2024-01-07T13:39:38","slug":"rsa%e7%ae%80%e8%bf%b0","status":"publish","type":"post","link":"https:\/\/blog.taiyang.space\/?p=156","title":{"rendered":"RSA\u7b80\u8ff0"},"content":{"rendered":"<h2>1. \u6982\u8ff0<\/h2>\n<p>RSA\u7b97\u6cd5\u5e38\u7528\u4e8e\u975e\u5bf9\u79f0\u52a0\u5bc6\uff0c\u975e\u5bf9\u79f0\u52a0\u5bc6\u6d41\u7a0b\u5982\u4e0b\uff1a<\/p>\n<ol start=\"\">\n<li>\u4e59\u65b9\u751f\u6210\u4e24\u628a\u5bc6\u94a5(\u516c\u94a5\u548c\u79c1\u94a5)\u3002\u516c\u94a5\u662f\u516c\u5f00\u7684\uff0c\u4efb\u4f55\u4eba\u90fd\u53ef\u4ee5\u83b7\u5f97\uff0c\u79c1\u94a5\u5219\u662f\u4fdd\u5bc6\u7684\u3002<\/li>\n<li>\u7532\u65b9\u83b7\u53d6\u4e59\u65b9\u7684\u516c\u94a5\uff0c\u7136\u540e\u7528\u5b83\u5bf9\u4fe1\u606f\u52a0\u5bc6\u3002<\/li>\n<li>\u4e59\u65b9\u5f97\u5230\u52a0\u5bc6\u540e\u7684\u4fe1\u606f\uff0c\u7528\u79c1\u94a5\u89e3\u5bc6\u3002<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p>\u7279\u70b9\uff1a<\/p>\n<ul>\n<li>\u7b97\u6cd5\u662f\u53ef\u9006\u7684\uff0c\u5373\u7528\u516c\u94a5\u52a0\u5bc6\u7684\u4fe1\u606f\u53ef\u4ee5\u7528\u79c1\u94a5\u89e3\u5bc6\uff1b\u7528\u79c1\u94a5\u52a0\u5bc6\u7684\u4fe1\u606f\u53ef\u4ee5\u88ab\u516c\u94a5\u89e3\u5bc6<\/li>\n<li>\u4e0d\u80fd\u7531\u516c\u94a5\u63a8\u7b97\u51fa\u79c1\u94a5\uff08\u524d\u63d0\u662f\u6240\u53d6\u7684\u8d28\u6570\u4e0d\u662f\u7279\u522b\u5c0f\uff09<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h2>2. \u6570\u8bba\u57fa\u7840<\/h2>\n<h3>2.1 \u7d20\u6570<\/h3>\n<p>\u7d20\u6570\u53c8\u79f0\u8d28\u6570\uff0c\u6307\u5728\u4e00\u4e2a\u5927\u4e8e1\u7684\u81ea\u7136\u6570\u4e2d\uff0c\u9664\u4e861\u548c\u6b64\u6574\u6570\u81ea\u8eab\u5916\uff0c\u4e0d\u80fd\u88ab\u5176\u4ed6\u81ea\u7136\u6570\u6574\u9664\u7684\u6570\u3002\u8fd9\u4e2a\u6982\u5ff5\uff0c\u6211\u4eec\u5728\u4e0a\u521d\u4e2d\uff0c\u751a\u81f3\u5c0f\u5b66\u7684\u65f6\u5019\u90fd\u5b66\u8fc7\u4e86\uff0c\u8fd9\u91cc\u5c31\u4e0d\u518d\u8fc7\u591a\u89e3\u91ca\u4e86\u3002<\/p>\n<p>&nbsp;<\/p>\n<h3>2.2 \u6a21\u8fd0\u7b97<\/h3>\n<p>\u6a21\u8fd0\u7b97\u5373\u6c42\u4f59\u8fd0\u7b97\u3002\u201c\u6a21\u201d\u662f\u201cMod\u201d\u7684\u97f3\u8bd1\u3002\u548c\u6a21\u8fd0\u7b97\u7d27\u5bc6\u76f8\u5173\u7684\u4e00\u4e2a\u6982\u5ff5\u662f\u201c\u540c\u4f59\u201d\u3002\u6570\u5b66\u4e0a\uff0c\u5f53\u4e24\u4e2a\u6574\u6570\u9664\u4ee5\u540c\u4e00\u4e2a\u6b63\u6574\u6570\uff0c\u82e5\u5f97\u76f8\u540c\u4f59\u6570\uff0c\u5219\u4e8c\u6574\u6570\u540c\u4f59\u3002<\/p>\n<p>\u4e24\u4e2a\u6574\u6570a\uff0cb\uff0c\u82e5\u5b83\u4eec\u9664\u4ee5\u6b63\u6574\u6570m\u6240\u5f97\u7684\u4f59\u6570\u76f8\u7b49\uff0c\u5219\u79f0a\uff0cb\u5bf9\u4e8e\u6a21m\u540c\u4f59\uff0c\u8bb0\u4f5c: a \u2261 b \\ (mod m)\uff1b\u8bfb\u4f5c\uff1aa\u540c\u4f59\u4e8eb\u6a21m\uff0c\u6216\u8005\uff0ca\u4e0eb\u5173\u4e8e\u6a21m\u540c\u4f59\u3002\u4f8b\u5982\uff1a26 \u2261 14 \\ (mod 12)\u3002<\/p>\n<p>&nbsp;<\/p>\n<h3>2.3 \u4e92\u8d28\u5173\u7cfb<\/h3>\n<p>\u5982\u679c\u4e24\u4e2a\u6b63\u6574\u6570\uff0c\u9664\u4e861\u4ee5\u5916\uff0c\u6ca1\u6709\u5176\u4ed6\u516c\u56e0\u5b50\uff0c\u6211\u4eec\u5c31\u79f0\u8fd9\u4e24\u4e2a\u6570\u662f\u4e92\u8d28\u5173\u7cfb\uff08coprime\uff09\u3002\u6bd4\u5982\uff0c15\u548c32\u6ca1\u6709\u516c\u56e0\u5b50\uff0c\u6240\u4ee5\u5b83\u4eec\u662f\u4e92\u8d28\u5173\u7cfb\u3002\u8fd9\u8bf4\u660e\uff0c\u4e0d\u662f\u8d28\u6570\u4e5f\u53ef\u4ee5\u6784\u6210\u4e92\u8d28\u5173\u7cfb\u3002<\/p>\n<p>\u5173\u4e8e\u4e92\u8d28\u5173\u7cfb\uff0c\u4e0d\u96be\u5f97\u5230\u4ee5\u4e0b\u7ed3\u8bba\uff1a<\/p>\n<ul>\n<li>\u4efb\u610f\u4e24\u4e2a\u8d28\u6570\u6784\u6210\u4e92\u8d28\u5173\u7cfb\uff0c\u6bd4\u598213\u548c61\u3002<\/li>\n<li>\u4e00\u4e2a\u6570\u662f\u8d28\u6570\uff0c\u53e6\u4e00\u4e2a\u6570\u53ea\u8981\u4e0d\u662f\u524d\u8005\u7684\u500d\u6570\uff0c\u4e24\u8005\u5c31\u6784\u6210\u4e92\u8d28\u5173\u7cfb\uff0c\u6bd4\u59823\u548c10\u3002<\/li>\n<li>\u5982\u679c\u4e24\u4e2a\u6570\u4e4b\u4e2d\uff0c\u8f83\u5927\u7684\u90a3\u4e2a\u6570\u662f\u8d28\u6570\uff0c\u5219\u4e24\u8005\u6784\u6210\u4e92\u8d28\u5173\u7cfb\uff0c\u6bd4\u598297\u548c57\u3002<\/li>\n<li>1\u548c\u4efb\u610f\u4e00\u4e2a\u81ea\u7136\u6570\u662f\u90fd\u662f\u4e92\u8d28\u5173\u7cfb\uff0c\u6bd4\u59821\u548c99\u3002<\/li>\n<li>p\u662f\u5927\u4e8e1\u7684\u6574\u6570\uff0c\u5219p\u548cp-1\u6784\u6210\u4e92\u8d28\u5173\u7cfb\uff0c\u6bd4\u598257\u548c56\u3002<\/li>\n<li>p\u662f\u5927\u4e8e1\u7684\u5947\u6570\uff0c\u5219p\u548cp-2\u6784\u6210\u4e92\u8d28\u5173\u7cfb\uff0c\u6bd4\u598217\u548c15\u3002<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h3>2.4 \u6b27\u62c9\u51fd\u6570<\/h3>\n<p>\u8bf7\u601d\u8003\u4ee5\u4e0b\u95ee\u9898\uff1a<\/p>\n<blockquote><p>\u4efb\u610f\u7ed9\u5b9a\u6b63\u6574\u6570n\uff0c\u8bf7\u95ee\u5728\u5c0f\u4e8e\u7b49\u4e8en\u7684\u6b63\u6574\u6570\u4e4b\u4e2d\uff0c\u6709\u591a\u5c11\u4e2a\u4e0en\u6784\u6210\u4e92\u8d28\u5173\u7cfb\uff1f\uff08\u6bd4\u5982\uff0c\u57281\u52308\u4e4b\u4e2d\uff0c\u6709\u591a\u5c11\u4e2a\u6570\u4e0e8\u6784\u6210\u4e92\u8d28\u5173\u7cfb\uff1f\uff09<\/p><\/blockquote>\n<p>\u8ba1\u7b97\u8fd9\u4e2a\u503c\u7684\u65b9\u6cd5\u5c31\u53eb\u505a\u6b27\u62c9\u51fd\u6570\uff0c\u4ee5\u03c6(n)\u8868\u793a\u3002\u57281\u52308\u4e4b\u4e2d\uff0c\u4e0e8\u5f62\u6210\u4e92\u8d28\u5173\u7cfb\u7684\u662f1\u30013\u30015\u30017\uff0c\u6240\u4ee5 \u03c6(8) = 4\u3002<\/p>\n<p>\u03c6(n)\u7684\u8ba1\u7b97\u65b9\u6cd5\u5e76\u4e0d\u590d\u6742\uff0c\u4f46\u662f\u4e3a\u4e86\u5f97\u5230\u6700\u540e\u90a3\u4e2a\u516c\u5f0f\uff0c\u9700\u8981\u4e00\u6b65\u6b65\u8ba8\u8bba\u3002<\/p>\n<p>&nbsp;<\/p>\n<h4>\u7b2c1\u79cd\u60c5\u51b5<\/h4>\n<p>\u5982\u679cn=1\uff0c\u5219 \u03c6(1) = 1 \u3002\u56e0\u4e3a1\u4e0e\u4efb\u4f55\u6570\uff08\u5305\u62ec\u81ea\u8eab\uff09\u90fd\u6784\u6210\u4e92\u8d28\u5173\u7cfb\u3002<\/p>\n<p>&nbsp;<\/p>\n<h4>\u7b2c2\u79cd\u60c5\u51b5<\/h4>\n<p>\u5982\u679cn\u662f\u8d28\u6570\uff0c\u5219 \u03c6(n)=n-1 \u3002\u56e0\u4e3a\u8d28\u6570\u4e0e\u5c0f\u4e8e\u5b83\u7684\u6bcf\u4e00\u4e2a\u6570\uff0c\u90fd\u6784\u6210\u4e92\u8d28\u5173\u7cfb\u3002\u6bd4\u59825\u4e0e1\u30012\u30013\u30014\u90fd\u6784\u6210\u4e92\u8d28\u5173\u7cfb\u3002<\/p>\n<p>&nbsp;<\/p>\n<h4>\u7b2c3\u79cd\u60c5\u51b5<\/h4>\n<p>\u5982\u679cn\u662f\u8d28\u6570\u7684\u67d0\u4e00\u4e2a\u6b21\u65b9\uff0c\u5373 n = p^k (p\u4e3a\u8d28\u6570\uff0ck\u4e3a\u5927\u4e8e\u7b49\u4e8e1\u7684\u6574\u6570)\uff0c\u5219<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n58\" cid=\"n58\" mdtype=\"math_block\">\n<div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\">\n<div class=\"MathJax_SVG_Display\" style=\"text-align: center;\"><span class=\"MathJax_SVG\" id=\"MathJax-Element-1-Frame\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"18.13ex\" height=\"2.928ex\" viewBox=\"0 -956.9 7805.9 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex; max-width: 100%;\"><defs><path stroke-width=\"0\" id=\"E1-MJMATHI-3C6\" d=\"M92 210Q92 176 106 149T142 108T185 85T220 72L235 70L237 71L250 112Q268 170 283 211T322 299T370 375T429 423T502 442Q547 442 582 410T618 302Q618 224 575 152T457 35T299 -10Q273 -10 273 -12L266 -48Q260 -83 252 -125T241 -179Q236 -203 215 -212Q204 -218 190 -218Q159 -215 159 -185Q159 -175 214 -2L209 0Q204 2 195 5T173 14T147 28T120 46T94 71T71 103T56 142T50 190Q50 238 76 311T149 431H162Q183 431 183 423Q183 417 175 409Q134 361 114 300T92 210ZM574 278Q574 320 550 344T486 369Q437 369 394 329T323 218Q309 184 295 109L286 64Q304 62 306 62Q423 62 498 131T574 278Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMATHI-6B\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMAIN-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMAIN-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMAIN-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path stroke-width=\"0\" id=\"E1-MJMAIN-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use xlink:href=\"#E1-MJMATHI-3C6\" x=\"0\" y=\"0\"><\/use><use xlink:href=\"#E1-MJMAIN-28\" x=\"654\" y=\"0\"><\/use><g transform=\"translate(1043,0)\"><use xlink:href=\"#E1-MJMATHI-70\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E1-MJMATHI-6B\" x=\"711\" y=\"583\"><\/use><\/g><use xlink:href=\"#E1-MJMAIN-29\" x=\"2014\" y=\"0\"><\/use><use xlink:href=\"#E1-MJMAIN-3D\" x=\"2681\" y=\"0\"><\/use><g transform=\"translate(3736,0)\"><use xlink:href=\"#E1-MJMATHI-70\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E1-MJMATHI-6B\" x=\"711\" y=\"583\"><\/use><\/g><use xlink:href=\"#E1-MJMAIN-2212\" x=\"4930\" y=\"0\"><\/use><g transform=\"translate(5930,0)\"><use xlink:href=\"#E1-MJMATHI-70\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(503,412)\"><use transform=\"scale(0.707)\" xlink:href=\"#E1-MJMATHI-6B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E1-MJMAIN-2212\" x=\"521\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E1-MJMAIN-31\" x=\"1299\" y=\"0\"><\/use><\/g><\/g><\/g><\/svg><\/span><\/div>\n<p><script type=\"math\/tex; mode=display\" id=\"MathJax-Element-1\">\u03c6(p^k)=p^k-p^{k-1}<\/script><\/p>\n<\/div>\n<\/div>\n<p>\u6bd4\u5982 \u03c6(8) = \u03c6(2^3) =2^3 &#8211; 2^2 = 8 -4 = 4\u3002<\/p>\n<p>&nbsp;<\/p>\n<p>\u8fd9\u662f\u56e0\u4e3a\u53ea\u6709\u5f53\u4e00\u4e2a\u6570\u4e0d\u5305\u542b\u8d28\u6570p\uff0c\u624d\u53ef\u80fd\u4e0en\u4e92\u8d28\u3002\u800c\u5305\u542b\u8d28\u6570p\u7684\u6570\u4e00\u5171\u6709p^(k-1)\u4e2a\uff0c\u53731\u00d7p\u30012\u00d7p\u30013\u00d7p\u3001\u2026\u3001p^(k-1)\u00d7p\uff0c\u628a\u5b83\u4eec\u53bb\u9664\uff0c\u5269\u4e0b\u7684\u5c31\u662f\u4e0en\u4e92\u8d28\u7684\u6570\u3002<\/p>\n<p>\u4e0a\u9762\u7684\u5f0f\u5b50\u8fd8\u53ef\u4ee5\u5199\u6210\u4e0b\u9762\u7684\u5f62\u5f0f\uff1a<\/p>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n63\" cid=\"n63\" mdtype=\"math_block\">\n<div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\">\n<div class=\"MathJax_SVG_Display\" style=\"text-align: center;\"><span class=\"MathJax_SVG\" id=\"MathJax-Element-2-Frame\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"31.295ex\" height=\"5.496ex\" viewBox=\"0 -1409.3 13474.3 2366.2\" role=\"img\" focusable=\"false\" style=\"vertical-align: -2.223ex; max-width: 100%;\"><defs><path stroke-width=\"0\" id=\"E2-MJMATHI-3C6\" d=\"M92 210Q92 176 106 149T142 108T185 85T220 72L235 70L237 71L250 112Q268 170 283 211T322 299T370 375T429 423T502 442Q547 442 582 410T618 302Q618 224 575 152T457 35T299 -10Q273 -10 273 -12L266 -48Q260 -83 252 -125T241 -179Q236 -203 215 -212Q204 -218 190 -218Q159 -215 159 -185Q159 -175 214 -2L209 0Q204 2 195 5T173 14T147 28T120 46T94 71T71 103T56 142T50 190Q50 238 76 311T149 431H162Q183 431 183 423Q183 417 175 409Q134 361 114 300T92 210ZM574 278Q574 320 550 344T486 369Q437 369 394 329T323 218Q309 184 295 109L286 64Q304 62 306 62Q423 62 498 131T574 278Z\"><\/path><path stroke-width=\"0\" id=\"E2-MJMAIN-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 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transform=\"translate(1043,0)\"><use xlink:href=\"#E2-MJMATHI-70\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E2-MJMATHI-6B\" x=\"711\" y=\"583\"><\/use><\/g><use xlink:href=\"#E2-MJMAIN-29\" x=\"2014\" y=\"0\"><\/use><use xlink:href=\"#E2-MJMAIN-3D\" x=\"2681\" y=\"0\"><\/use><g transform=\"translate(3736,0)\"><use xlink:href=\"#E2-MJMATHI-70\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E2-MJMATHI-6B\" x=\"711\" y=\"583\"><\/use><\/g><use xlink:href=\"#E2-MJMAIN-2212\" x=\"4930\" y=\"0\"><\/use><g transform=\"translate(5930,0)\"><use xlink:href=\"#E2-MJMATHI-70\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(503,412)\"><use transform=\"scale(0.707)\" xlink:href=\"#E2-MJMATHI-6B\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E2-MJMAIN-2212\" x=\"521\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E2-MJMAIN-31\" x=\"1299\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E2-MJMAIN-3D\" x=\"8083\" y=\"0\"><\/use><g transform=\"translate(9139,0)\"><use xlink:href=\"#E2-MJMATHI-70\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E2-MJMATHI-6B\" x=\"711\" y=\"583\"><\/use><\/g><use xlink:href=\"#E2-MJMAIN-28\" x=\"10110\" y=\"0\"><\/use><use xlink:href=\"#E2-MJMAIN-31\" x=\"10499\" y=\"0\"><\/use><use xlink:href=\"#E2-MJMAIN-2212\" x=\"11222\" y=\"0\"><\/use><g transform=\"translate(12000,0)\"><g transform=\"translate(342,0)\"><rect stroke=\"none\" width=\"623\" height=\"60\" x=\"0\" y=\"220\"><\/rect><use xlink:href=\"#E2-MJMAIN-31\" x=\"61\" y=\"676\"><\/use><use xlink:href=\"#E2-MJMATHI-70\" x=\"60\" y=\"-686\"><\/use><\/g><\/g><use xlink:href=\"#E2-MJMAIN-29\" x=\"13085\" y=\"0\"><\/use><\/g><\/svg><\/span><\/div>\n<p><script type=\"math\/tex; mode=display\" id=\"MathJax-Element-2\">\u03c6(p^k)=p^k-p^{k-1}=p^k(1-\\frac 1p)<\/script><\/p>\n<\/div>\n<\/div>\n<p>\u53ef\u4ee5\u770b\u51fa\uff0c\u4e0a\u9762\u7684\u7b2c2\u79cd\u60c5\u51b5\u662f k=1 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xlink:href=\"#E10-MJMAIN-28\" x=\"3155\" y=\"0\"><\/use><use xlink:href=\"#E10-MJMATHI-6D\" x=\"3544\" y=\"0\"><\/use><use xlink:href=\"#E10-MJMATHI-6F\" x=\"4422\" y=\"0\"><\/use><use xlink:href=\"#E10-MJMATHI-64\" x=\"4907\" y=\"0\"><\/use><g transform=\"translate(5597,0)\"><use xlink:href=\"#E10-MJMAIN-31\"><\/use><use xlink:href=\"#E10-MJMAIN-30\" x=\"500\" y=\"0\"><\/use><\/g><use xlink:href=\"#E10-MJMAIN-29\" x=\"6597\" y=\"0\"><\/use><\/g><\/svg><\/span><\/div>\n<p><script type=\"math\/tex; mode=display\" id=\"MathJax-Element-10\">7^{4k}=1(mod\\,10)<\/script><\/p>\n<\/div>\n<\/div>\n<p>\u56e0\u6b64\uff0c7\u7684\u4efb\u610f\u6b21\u65b9\u7684\u4e2a\u4f4d\u6570\uff08\u4f8b\u59827\u7684222\u6b21\u65b9\uff09\uff0c\u5fc3\u7b97\u5c31\u53ef\u4ee5\u7b97\u51fa\u6765\u3002<\/p>\n<p>&nbsp;<\/p>\n<p>\u6b27\u62c9\u5b9a\u7406\u6709\u4e00\u4e2a\u7279\u6b8a\u60c5\u51b5\u3002<\/p>\n<blockquote><p>\u5047\u8bbe\u6b63\u6574\u6570a\u4e0e\u8d28\u6570p\u4e92\u8d28\uff0c\u56e0\u4e3a\u8d28\u6570p\u7684\u03c6(p)\u7b49\u4e8ep-1\uff0c\u5219\u6b27\u62c9\u5b9a\u7406\u53ef\u4ee5\u5199\u6210<\/p><\/blockquote>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n102\" cid=\"n102\" mdtype=\"math_block\">\n<div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\">\n<div class=\"MathJax_SVG_Display\" style=\"text-align: center;\"><span class=\"MathJax_SVG\" id=\"MathJax-Element-11-Frame\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"16.387ex\" height=\"2.928ex\" viewBox=\"0 -956.9 7055.6 1260.5\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex; max-width: 100%;\"><defs><path stroke-width=\"0\" id=\"E11-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path stroke-width=\"0\" id=\"E11-MJMATHI-70\" d=\"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 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id=\"MathJax-Element-11\">a^{p-1}=1(mod\\,p)<\/script><\/p>\n<\/div>\n<\/div>\n<p>\u8fd9\u5c31\u662f\u8457\u540d\u7684<a href=\"https:\/\/baike.baidu.com\/item\/%E8%B4%B9%E9%A9%AC%E5%B0%8F%E5%AE%9A%E7%90%86\/4776158\">\u8d39\u9a6c\u5c0f\u5b9a\u7406<\/a>\u3002\u5b83\u662f\u6b27\u62c9\u5b9a\u7406\u7684\u7279\u4f8b\u3002<\/p>\n<p>\u6b27\u62c9\u5b9a\u7406\u662fRSA\u7b97\u6cd5\u7684\u6838\u5fc3\u3002\u7406\u89e3\u4e86\u8fd9\u4e2a\u5b9a\u7406\uff0c\u5c31\u53ef\u4ee5\u7406\u89e3RSA\u3002<\/p>\n<p>&nbsp;<\/p>\n<h3>2.6 \u6a21\u53cd\u5143\u7d20<\/h3>\n<p>\u8fd8\u5269\u4e0b\u6700\u540e\u4e00\u4e2a\u6982\u5ff5\uff1a<\/p>\n<blockquote><p>\u5982\u679c\u4e24\u4e2a\u6b63\u6574\u6570a\u548cn\u4e92\u8d28\uff0c\u90a3\u4e48\u4e00\u5b9a\u53ef\u4ee5\u627e\u5230\u6574\u6570b\uff0c\u4f7f\u5f97ab-1\u88abn\u6574\u9664\uff0c\u6216\u8005\u8bf4ab\u88abn\u9664\u7684\u4f59\u6570\u662f1\u3002<\/p><\/blockquote>\n<div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n110\" cid=\"n110\" mdtype=\"math_block\">\n<div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\">\n<div class=\"MathJax_SVG_Display\" style=\"text-align: center;\"><span class=\"MathJax_SVG\" id=\"MathJax-Element-12-Frame\" tabindex=\"-1\" style=\"font-size: 100%; display: inline-block;\"><svg xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"14.452ex\" height=\"2.577ex\" viewBox=\"0 -806.1 6222.2 1109.7\" role=\"img\" focusable=\"false\" style=\"vertical-align: -0.705ex; max-width: 100%;\"><defs><path stroke-width=\"0\" id=\"E12-MJMATHI-61\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 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x=\"3399\" y=\"0\"><\/use><use xlink:href=\"#E13-MJMAIN-D7\" x=\"4150\" y=\"0\"><\/use><g transform=\"translate(5150,0)\"><use xlink:href=\"#E13-MJMATHI-61\" x=\"0\" y=\"0\"><\/use><g transform=\"translate(529,412)\"><use transform=\"scale(0.707)\" xlink:href=\"#E13-MJMATHI-3C6\" x=\"0\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E13-MJMAIN-28\" x=\"654\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E13-MJMATHI-6E\" x=\"1043\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E13-MJMAIN-29\" x=\"1643\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E13-MJMAIN-2212\" x=\"2032\" y=\"0\"><\/use><use transform=\"scale(0.707)\" xlink:href=\"#E13-MJMAIN-31\" x=\"2810\" y=\"0\"><\/use><\/g><\/g><use xlink:href=\"#E13-MJMAIN-3D\" x=\"8398\" y=\"0\"><\/use><use xlink:href=\"#E13-MJMAIN-31\" x=\"9453\" y=\"0\"><\/use><use xlink:href=\"#E13-MJMAIN-28\" x=\"9953\" y=\"0\"><\/use><use xlink:href=\"#E13-MJMATHI-6D\" x=\"10342\" y=\"0\"><\/use><use xlink:href=\"#E13-MJMATHI-6F\" x=\"11220\" y=\"0\"><\/use><use xlink:href=\"#E13-MJMATHI-64\" x=\"11705\" y=\"0\"><\/use><use xlink:href=\"#E13-MJMATHI-6E\" x=\"12395\" y=\"0\"><\/use><use xlink:href=\"#E13-MJMAIN-29\" x=\"12995\" y=\"0\"><\/use><\/g><\/svg><\/span><\/div>\n<p><script type=\"math\/tex; mode=display\" id=\"MathJax-Element-13\">a^{\u03c6(n)}=a\\times a^{\u03c6(n)-1}=1(mod\\,n)<\/script><\/p>\n<\/div>\n<\/div>\n<p>\u53ef\u4ee5\u770b\u5230\uff0ca\u7684 \u03c6(n)-1 \u6b21\u65b9\uff0c\u5c31\u662fa\u7684\u6a21\u53cd\u5143\u7d20\u3002<\/p>\n<p>&nbsp;<\/p>\n<h2>3. RSA\u7b97\u6cd5<\/h2>\n<h4>3.1 \u968f\u673a\u9009\u62e9\u4e24\u4e2a\u4e0d\u76f8\u7b49\u7684\u8d28\u6570p\u548cq<\/h4>\n<p>\u8fd9\u8fb9\u6211\u4eec\u9009\u62e961\u548c53.\uff08\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u8fd9\u4e24\u4e2a\u8d28\u6570\u8d8a\u5927\uff0c\u5c31\u8d8a\u96be\u7834\u89e3\u3002\uff09<\/p>\n<p>&nbsp;<\/p>\n<h4>3.2 \u8ba1\u7b97p\u548cq\u7684\u4e58\u79efn\u3002<\/h4>\n<p>\u628a61\u548c53\u76f8\u4e58\u3002<\/p>\n<blockquote><p>n = 61\u00d753 = 3233<\/p><\/blockquote>\n<p>n\u7684\u957f\u5ea6\u5c31\u662f\u5bc6\u94a5\u957f\u5ea6\u30023233\u5199\u6210\u4e8c\u8fdb\u5236\u662f110010100001\uff0c\u4e00\u5171\u670912\u4f4d\uff0c\u6240\u4ee5\u8fd9\u4e2a\u5bc6\u94a5\u5c31\u662f12\u4f4d\u3002\u5b9e\u9645\u5e94\u7528\u4e2d\uff0cRSA\u5bc6\u94a5\u4e00\u822c\u662f1024\u4f4d\uff0c\u91cd\u8981\u573a\u5408\u5219\u4e3a2048\u4f4d\u3002<\/p>\n<p>&nbsp;<\/p>\n<h4>3.3 \u8ba1\u7b97n\u7684\u6b27\u62c9\u51fd\u6570\u03c6(n)\u3002<\/h4>\n<blockquote><p>n\u662f\u8d28\u6570\uff0c\u5219 \u03c6(n)=n-1<br \/>\nn = p1 \u00d7 p2<br \/>\n\u03c6(n) = \u03c6(p1p2) = \u03c6(p1)\u03c6(p2)<br \/>\n=&gt; \u03c6(n) = (p-1)(q-1)<\/p><\/blockquote>\n<p>\u7b97\u51fa\u03c6(3233)\u7b49\u4e8e60\u00d752\uff0c\u53733120\u3002<\/p>\n<p>&nbsp;<\/p>\n<h4>3.4 \u968f\u673a\u9009\u62e9\u4e00\u4e2a\u6574\u6570e\uff0c\u6761\u4ef6\u662f1&lt; e &lt; \u03c6(n)\uff0c\u4e14e\u4e0e\u03c6(n) \u4e92\u8d28\u3002<\/h4>\n<p>\u5c31\u57281\u52303120\u4e4b\u95f4\uff0c\u968f\u673a\u9009\u62e9\u4e8617\u3002\uff08\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u5e38\u5e38\u9009\u62e965537\u3002\uff09<\/p>\n<p>&nbsp;<\/p>\n<h4>3.5 \u8ba1\u7b97e\u5bf9\u4e8e\u03c6(n)\u7684\u6a21\u53cd\u5143\u7d20d\u3002<\/h4>\n<p>\u6240\u8c13\u201d\u6a21\u53cd\u5143\u7d20\u201d\u5c31\u662f\u6307\u6709\u4e00\u4e2a\u6574\u6570d\uff0c\u53ef\u4ee5\u4f7f\u5f97ed\u88ab\u03c6(n)\u9664\u7684\u4f59\u6570\u4e3a1\u3002<\/p>\n<blockquote><p>ed \u2261 1 (mod \u03c6(n))<\/p><\/blockquote>\n<p>\u8fd9\u4e2a\u5f0f\u5b50\u7b49\u4ef7\u4e8e<\/p>\n<blockquote><p>ed &#8211; 1 = k\u03c6(n)<\/p><\/blockquote>\n<p>\u4e8e\u662f\uff0c\u627e\u5230\u6a21\u53cd\u5143\u7d20d\uff0c\u5b9e\u8d28\u4e0a\u5c31\u662f\u5bf9\u4e0b\u9762\u8fd9\u4e2a\u4e8c\u5143\u4e00\u6b21\u65b9\u7a0b\u6c42\u89e3\u3002(-k = y)<\/p>\n<blockquote><p>ex + \u03c6(n)y = 1<\/p><\/blockquote>\n<p>\u5df2\u77e5 e=17, \u03c6(n)=3120\uff0c<\/p>\n<blockquote><p>17x + 3120y = 1<\/p><\/blockquote>\n<p>\u8fd9\u4e2a\u65b9\u7a0b\u53ef\u4ee5\u7528<a href=\"https:\/\/zhuanlan.zhihu.com\/p\/58241990\">\u201c\u6269\u5c55\u6b27\u51e0\u91cc\u5f97\u7b97\u6cd5\u201d(\u53c8\u53eb\u8f97\u8f6c\u76f8\u9664\u6cd5)<\/a>\u6c42\u89e3\uff0c\u6b64\u5904\u7701\u7565\u5177\u4f53\u8fc7\u7a0b\u3002\u603b\u4e4b\uff0c\u7b97\u51fa\u4e00\u7ec4\u6574\u6570\u89e3\u4e3a (x,y)=(2753,-15)\uff0c\u5373 d=2753\u3002<\/p>\n<p>\u81f3\u6b64\u6240\u6709\u8ba1\u7b97\u5b8c\u6210\u3002<\/p>\n<p>&nbsp;<\/p>\n<h4>3.6 \u5c06n\u548ce\u5c01\u88c5\u6210\u516c\u94a5\uff0cn\u548cd\u5c01\u88c5\u6210\u79c1\u94a5\u3002<\/h4>\n<p>\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0cn=3233\uff0ce=17\uff0cd=2753\uff0c\u6240\u4ee5\u516c\u94a5\u5c31\u662f (3233,17)\uff0c\u79c1\u94a5\u5c31\u662f\uff083233, 2753\uff09\u3002<\/p>\n<hr>\n<p>OK\uff0c\u4e0a\u9762\u4e00\u5927\u6bb5\u90fd\u662f\u6211\u4ece\u7f51\u4e0acv\u4e0b\u6765\u7684\u3002<a href=\"https:\/\/zhuanlan.zhihu.com\/p\/48249182\">\u539f\u6587\u94fe\u63a5\ud83c\udf39<\/a> \u5199\u7684\u7279\u522b\u597d\uff01\u4e0b\u9762\u7684Python\u7248RSA\u7b97\u6cd5\u5c31\u662f\u6839\u636e\u8fd9\u7bc7\u6587\u7ae0\u5199\u7684\uff08\u8fd9\u4e2a\u662f\u6211\u7eaf\u624b\u6495\u7684\ud83c\udf39\uff09<\/p>\n<p>&nbsp;<\/p>\n<h2>4. Python\u5b9e\u73b0<\/h2>\n<p>\u7531\u4e8e\u53ea\u4f9b\u5b66\u4e60\u7406\u89e3\u7528\uff0c\u8be5\u4ee3\u7801\u6709\u5f88\u591a\u4e0d\u8db3\uff0c\u4f46\u662f\u80fd\u8dd1\uff0c\u8fd8\u8bf7\u89c1\u8c05\ud83c\udf39<\/p>\n<blockquote><p>\u6211\u548c\u4ee3\u7801\u6709\u4e00\u4e2a\u80fd\u8dd1\u5c31\u884c\ud83c\udf39<\/p><\/blockquote>\n<p>&nbsp;<\/p>\n<p>\u67092\u79cd\u6a21\u5f0f\uff1a<\/p>\n<ul>\n<li>\u81ea\u52a8\u751f\u6210\u8d28\u6570p\u3001q<\/li>\n<li>\u624b\u52a8\u8f93\u5165\u8d28\u6570p\u3001q<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<h3>4.1 \u5b8c\u6574\u4ee3\u7801<\/h3>\n<pre><code class=\"language-python\" lang=\"python\"># Author\uff1ataiyang\n# welcome to https:\/\/taiyang.space\n \nimport random\nimport math\n \n'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\ndescription\uff1a\u5224\u65ad\u8f93\u5165\u7684p\u3001q\u662f\u4e0d\u662f\u8d28\u6570\nparam\uff1ap\u3001q\nreturn\uff1a\u5982\u679c\u662f\u8fd4\u56de True\n'''\n \n \ndef judgePandQ(p, q):\n    ls = []\n    ls.append(p)\n    ls.append(q)\n    for x in range(0, 2):\n        if int(ls[x]) &gt; 1:\n            # \u67e5\u770b\u56e0\u5b50\n            for i in range(2, int(ls[x])):\n                if (int(ls[x]) % i) == 0:\n                    print(int(ls[x]), \"\u4e0d\u662f\u8d28\u6570\")\n                    print(i, \"\u4e58\u4e8e\", int(ls[x]) \/ i, \"\u662f\", int(ls[x]))\n                    return False\n                else:\n                    return True\n        # \u5982\u679c\u8f93\u5165\u7684\u6570\u5b57\u5c0f\u4e8e\u6216\u7b49\u4e8e 1\uff0c\u4e0d\u662f\u8d28\u6570\n        else:\n            print(int(ls[x]), \"\u4e0d\u662f\u8d28\u6570\")\n \n \n'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\ndescription\uff1a\u8ba1\u7b97\u51fa N \u548c L\nparam\uff1ap\u3001q\nreturn\uff1aN\u3001L\n'''\n \n \ndef getNandL(p, q):\n    N = p * q\n    L = (p - 1) * (q - 1)\n    return N, L\n \n \n'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\ndescription\uff1a\u8ba1\u7b97\u51fa e\nparam\uff1ap\u3001q\nreturn\uff1a\u4e00\u4e2a\u5b58\u653e\u6240\u6709E\u7684\u5217\u8868\n'''\n \n \ndef getE(p, q):\n    N, L = getNandL(p, q)\n    listOfE = []\n    for i in range(2, L):\n        if math.gcd(i, L) == 1:\n            listOfE.append(i)\n    return listOfE\n \n \n'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\ndescription\uff1a\u8ba1\u7b97\u51fa d\nparam\uff1ap\u3001q\nreturn\uff1a\u4e00\u4e2a\u5b57\u5178\uff1b\u7ed3\u6784{e1:d1, e2:d2}   \u6bcf\u4e00\u4e2ae\u90fd\u6709\u5bf9\u5e94\u7684d\u5e76\u4e14\u53bb\u6389\u91cd\u590d\u7684\n'''\n \n \ndef getdict_EandD(p, q):\n    N, L = getNandL(p, q)\n    listOfE = getE(p, q)\n    dict_EandD = {}\n    for e in listOfE:\n        for d in range(2, L):\n            # \u6784\u5efa\u5b57\u5178\u5e76\u53bb\u91cd\n            if (e * d) % L == 1 and e != d:\n                dict_EandD[e] = d\n    return dict_EandD\n \n \n'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\ndescription\uff1a\u8ba1\u7b97\u516c\u94a5\u548c\u79c1\u94a5\nparam\uff1ap\u3001q\nreturn\uff1apublicKey, privateKey, N, L\uff0cE\uff0cD\n'''\n \n \ndef getKey(p, q):\n    if judgePandQ(p, q):\n        N, L = getNandL(p, q)\n        listOfE = getE(p, q)\n        dict_EandD = getdict_EandD(p, q)\n \n        # \u83b7\u53d6E\u7684\u503c\n        print(\"\u8fd9\u662f\u6240\u6709\u53ef\u7528\u7684E\\n\", listOfE, \"\\n\")\n        E = int(input(\"\u8bf7\u9009\u62e9\u4e00\u4e2a\u559c\u6b22\u7684\u503c\uff1a\"))\n        while (True):\n            if E not in listOfE:\n                print(\"\u4f60\u8f93\u5165\u7684E\u4e0d\u5728\u7ed9\u51fa\u7684\u5217\u8868\u4e2d\uff0c\u8bf7\u91cd\u65b0\u8f93\u5165\uff01\\n\")\n                E = int(input(\"\u8bf7\u9009\u62e9\u4e00\u4e2a\u559c\u6b22\u7684\u503c\uff1a\"))\n            else:\n                break\n \n        # \u6839\u636eE\u7684\u503c\u5728dict_EandD\u5b57\u5178\u4e2d\u627eD\u7684\u503c\n        D = dict_EandD.get(E)\n \n        # \u6784\u5efa\u516c\u94a5\u3001\u79c1\u94a5\n        # \u516c\u94a5\uff1a\uff08N\uff0cE\uff09\n        # \u79c1\u94a5\uff1a\uff08N\uff0cD\uff09\n        publicKey = (N, E)\n        privateKey = (N, D)\n        return publicKey, privateKey, N, L, E, D\n    else:\n        print(\"\u8f93\u5165\u7684p\uff0cq\u4e0d\u662f\u8d28\u6570\uff01\u8bf7\u91cd\u65b0\u8f93\u5165\")\n \n \n'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\ndescription\uff1a\u81ea\u52a8\u83b7\u5f97p\u3001q\nparam\uff1anull\nreturn\uff1ap\u3001q\n'''\n \n \ndef autoGetPandQ():\n    # \u8003\u8651\u5230\u6027\u80fd\u7684\u539f\u56e0\uff0c\u8fd9\u91cc\u53ea\u751f\u621010\u2014\u2014100\u7684\u7d20\u6570\n    primeList = []\n    for i in range(10, 100):\n        isPrime = True\n        for j in range(2, i):\n            if i % j == 0:\n                isPrime = False\n        if isPrime:\n            primeList.append(i)\n \n    p = random.choice(primeList)\n    q = random.choice(primeList)\n \n    # \u9632\u6b62\u51fa\u73b0\u4e00\u6837\u7684\u60c5\u51b5\n    while (True):\n        if p == q:\n            p = random.choice(primeList)\n            q = random.choice(primeList)\n        else:\n            break\n    return p, q\n \n \n'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\ndescription\uff1a\u6253\u5370p\u3001q\u3001N\u3001L\u3001publicKey\u3001privateKey\u3001E\nparam\uff1ap\u3001q\u3001N\u3001L\u3001publicKey\u3001privateKey\nreturn\uff1anull\n'''\n \n \ndef printParam(publicKey, privateKey, N, L, E, D, p, q):\n    print()\n    print(\"p\u4e3a\uff1a\", p, \" q\u4e3a\uff1a\", q)\n    print(\"\u8ba1\u7b97\u51fa\u7684N\u4e3a\uff1a\", N)\n    print(\"\u8ba1\u7b97\u51fa\u7684L\u4e3a\uff1a\", L)\n    print(\"\u4f60\u9009\u62e9\u7684E\u4e3a\uff1a\", E)\n    print(\"\u8be5E\u5bf9\u5e94\u7684D\u4e3a\uff1a\", D)\n    print(\"\u8ba1\u7b97\u51fa\u7684\u516c\u94a5\u4e3a\uff1a\", publicKey)\n    print(\"\u8ba1\u7b97\u51fa\u7684\u79c1\u94a5\u4e3a\uff1a\", privateKey)\n \n \n'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\ndescription\uff1a\u5bf9\u660e\u6587\u8fdb\u884c\u52a0\u5bc6\nparam\uff1aplainText\uff0cN\uff0cE\nreturn\uff1a\u52a0\u5bc6\u540e\u7684\u5bc6\u6587 cipherText\n'''\n \n \ndef rsaEncode(plainText, N, E):\n    cipherText = plainText % N\n    for i in range(1, E):\n        cipherText = (cipherText * (plainText % N)) % N\n    return cipherText\n \n \n'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\ndescription\uff1a\u5bf9\u5bc6\u6587\u8fdb\u884c\u89e3\u5bc6\nparam\uff1acipherText\uff0cN\uff0c\nreturn\uff1anull\n'''\n \n \ndef rsaDecode(cipherText, N):\n    plainText = cipherText % N\n    D = int(input(\"\u8bf7\u8f93\u5165\u79c1\u94a5\u4e2d\u7684D\uff1a\"))\n    for i in range(1, D):\n        plainText = (plainText * (cipherText % N)) % N\n    return plainText\n \n \n'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\ndescription\uff1aRSA\u7b97\u6cd5\uff0c\u8fdb\u884c\u52a0\u89e3\u5bc6\nparam\uff1ap\uff0cq\nreturn\uff1anull\n'''\n \n \ndef rsaMain(p, q):\n    publicKey, privateKey, N, L, E, D = getKey(p, q)\n    printParam(publicKey, privateKey, N, L, E, D, p, q)\n    plainText = int(input(\"\\n\u52a0\u5bc6\\n\u8bf7\u8f93\u5165\u4f60\u8981\u52a0\u5bc6\u7684\u660e\u6587\uff1a\"))\n    cipherText = rsaEncode(plainText, N, E)\n    print('\u4f60\u52a0\u5bc6\u540e\u7684\u5bc6\u6587\u662f\uff1a', cipherText)\n \n    print(\"\\n\u89e3\u5bc6\")\n    plainText = rsaDecode(cipherText, N)\n    print('\u4f60\u6240\u89e3\u5bc6\u7684\u660e\u6587\u662f\uff1a', plainText)\n \n \n''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\n \n \ndef main():\n    print(\"*************************************\")\n    print(\"*******        RSA\u7b97\u6cd5        ********\")\n    print(\"*******  \u81ea\u52a8\u751f\u6210\u8d28\u6570p\u3001q\uff1a\u8f93\u51651 ********\")\n    print(\"*******  \u624b\u52a8\u8f93\u5165\u8d28\u6570p\u3001q\uff1a\u8f93\u51652 ********\")\n    print(\"*************************************\")\n    option = int(input(\"\u8bf7\u8f93\u5165\u4f60\u8981\u9009\u62e9\u7684\u6a21\u5f0f\uff1a\"))\n \n    while (True):\n        if option == 1:\n            p, q = autoGetPandQ()\n            rsaMain(p, q)\n            break\n \n        elif option == 2:\n            p = int(input(\"\u8bf7\u8f93\u5165p\uff1a\"))\n            q = int(input(\"\u8bf7\u8f93\u5165q\uff1a\"))\n            rsaMain(p, q)\n            break\n \n        else:\n            print(\"\u8f93\u5165\u6709\u8bef\uff01\u8bf7\u91cd\u65b0\u8f93\u5165\uff01\")\n            option = int(input(\"\u8bf7\u8f93\u5165\u4f60\u8981\u9009\u62e9\u7684\u6a21\u5f0f\uff1a\"))\n \n \nif __name__ == '__main__':\n    try:\n        main()\n    except ValueError:\n        print(\"\u8f93\u5165\u6709\u8bef\uff01\")\n \n<\/code><\/pre>\n<p>&nbsp;<\/p>\n<h3>4.2 \u8fd0\u884c\u622a\u56fe<\/h3>\n<ol start=\"\">\n<li>\u81ea\u52a8\u751f\u6210\u8d28\u6570p\u3001q<\/li>\n<\/ol>\n<p><img decoding=\"async\" src=\"https:\/\/picture.chentaiyang.space\/PicGoPicGoimage-20220911164601938.png\" referrerpolicy=\"no-referrer\"><\/p>\n<p>&nbsp;<\/p>\n<ol start=\"2\">\n<li>\u624b\u52a8\u8f93\u5165\u8d28\u6570p\u3001q<\/li>\n<\/ol>\n<p><img decoding=\"async\" src=\"https:\/\/picture.chentaiyang.space\/PicGoPicGoimage-20220911164739087.png\" referrerpolicy=\"no-referrer\"><\/p>\n<p>&nbsp;<\/p>\n<h2>5. \u5199\u5728\u6700\u540e<\/h2>\n<p>\u8fd9\u91cc\u9762\u7684\u4ee3\u7801\u53ea\u80fd\u7528\u505a\u7406\u89e3RSA\u7b97\u6cd5<\/p>\n<p>\u901a\u7528\u7684RSA\u7b97\u6cd5Python\u7248\u653e\u5728\u8fd9\u91cc\u4e86\uff1a<a href=\"https:\/\/github.com\/kisstaiyang\/DataEncryption\/blob\/main\/RSA.py\">\u6233\u6211\ud83c\udf39<\/a><\/p>\n<p>\u8fd9\u4e2a\u4e5f\u662f\u6211\u7eaf\u624b\u6495\u7684\uff0c\u5927\u5bb6\u51d1\u5408\u7740\u7528\ud83c\udf39<\/p>\n<p>&nbsp;<\/p>\n<p>\u672c\u6587\u5b8c<\/p>\n<p>\u656c\u7231\u4e0e\u660e\u5929\ud83c\udf39<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u975e\u5bf9\u79f0\u52a0\u5bc6\u7b97\u6cd5\u4e4bRSA\u3001Python\u5b9e\u73b0<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[57,56],"class_list":["post-156","post","type-post","status-publish","format-standard","hentry","category-crypto","tag-python","tag-rsa"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v22.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>RSA\u7b80\u8ff0 - #\u762b\u7537\u65e5\u8bb0<\/title>\n<meta name=\"robots\" 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