Merge pull request #18 from invisal/decrypt-using-crt-algorithm

Using Chinese Remainder Theorem

.vscode/settings.json

@@ -2,4 +2,8 @@
   "editor.tabSize": 2,
   "deno.enable": true,
   "files.eol": "\n",
+  "editor.formatOnSave": true,
+  "[typescript]": {
+    "editor.defaultFormatter": "denoland.vscode-deno"
+  }
 }

benchmark/rsa_decryption.ts

@@ -0,0 +1,61 @@
+import { RSAKey } from "../src/rsa/rsa_key.ts";
+import { RSA } from "./../mod.ts";
+
+async function benchmark(
+  callback: () => Promise<void>,
+  time: number,
+): Promise<string> {
+  const begin = Date.now();
+  for (let i = 0; i < time; i++) await callback();
+  const duration = Date.now() - begin;
+  return (
+    "Average time per operation: " +
+    (duration / time).toLocaleString(undefined, {
+      maximumFractionDigits: 3,
+      minimumFractionDigits: 3,
+    }) +
+    "ms"
+  );
+}
+
+const RSAWithoutCRT = new RSA(
+  new RSAKey({
+    n: 18860626341786571281488823024986320858450594409825894653370543036132043241142573896042059032222842653925236802058387508300831766517210547599910735316414515960981281788934868325895562621846910598716519620622800765932541992515497122274302135166053720433521886017494719475895327062855822141735315456613329660043977602681873027347568355824008701177948209784908095092287571982104446342906805480285143028837509076748632094733769028381835899214426100089454654381383340292544225149266603150745375542089871264998183586873913491071542891188124185517417658034185267204900711933432911892469528812455677450346478816563776340692357n,
+    e: 65537n,
+    d: 675889233296421535959584534071325460876697008626327098961835826056666939623318583538247287416696388446697983317835135300723690022117970081796395234018582764098060974026823771341271141202058228435371968592820841644512299636165137501280011132877728106486853881053264668544941537620562475421678333757690909726581512579603702598150422421865490996288244127528474781858718770957987603681469839575489191876915053252047160475432784849252000744700939988583801783684487872676506769116621572018367654125765915215495461243190782913460732307866572233187464540482209385816659449697777403383908159076257721991916366849321145825505n,
+    length: 256,
+  }),
+);
+
+const RSAWithCRT = new RSA(
+  new RSAKey({
+    n: 18860626341786571281488823024986320858450594409825894653370543036132043241142573896042059032222842653925236802058387508300831766517210547599910735316414515960981281788934868325895562621846910598716519620622800765932541992515497122274302135166053720433521886017494719475895327062855822141735315456613329660043977602681873027347568355824008701177948209784908095092287571982104446342906805480285143028837509076748632094733769028381835899214426100089454654381383340292544225149266603150745375542089871264998183586873913491071542891188124185517417658034185267204900711933432911892469528812455677450346478816563776340692357n,
+    e: 65537n,
+    d: 675889233296421535959584534071325460876697008626327098961835826056666939623318583538247287416696388446697983317835135300723690022117970081796395234018582764098060974026823771341271141202058228435371968592820841644512299636165137501280011132877728106486853881053264668544941537620562475421678333757690909726581512579603702598150422421865490996288244127528474781858718770957987603681469839575489191876915053252047160475432784849252000744700939988583801783684487872676506769116621572018367654125765915215495461243190782913460732307866572233187464540482209385816659449697777403383908159076257721991916366849321145825505n,
+    p: 144012741627856144888308881334281149724189531338704225677876939563376606824918324031374859287124236226502971065990666108124651521772921507380585859955952240309460362784798071545709251190489395954672667468167440559860853869293126718717957973299492588051948353555478537074738121854154013245465025431728566135973n,
+    q: 130964983574331118238688281723343099404807798188351894743739498160286888632365582428908430826736465978859171109808627043780421532713284022169182243601628532694521039146449170161837408903716062502861857136295244082497345869098293182658740508141543051039696970285876309414793236229439985580214897811694398138209n,
+    dp:
+      101745242578283949839304176626026302615765501625718038014206601700771528284898122271399648057012472126577972233967069321732572665229256933255045035440751035763135842618089328298662278539176953652032498422434484837613519933106647440836151945919299110608989008748123460390840662802554879248234151815614020617785n,
+    dq:
+      35678301062546466652937128368533312430740473760697540759490440516864704054843143700287335764233224950585953598646920476061700200574682590472688401624478932858201910873563078628399913004363131970125205568021351112332081315087369371243559409682455859030208122243679686715774576798456162206832122091795348953409n,
+    qi:
+      290194643186456996857651131997752205887237030546449364256620415456526041634563025067464278896891532150660234750257681796597575941847755684751705687345792123358974007946764700824215929859573834912100325612311985272735079519163425874861676228607347917173236103665132111683625679969874392886475792638645717652n,
+    length: 256,
+  }),
+);
+
+const ciper = await RSAWithCRT.encrypt("Hello World");
+
+console.log(
+  "With CRT",
+  await benchmark(async () => {
+    RSAWithCRT.decrypt(ciper);
+  }, 1000),
+);
+
+console.log(
+  "Without CRT",
+  await benchmark(async () => {
+    RSAWithoutCRT.decrypt(ciper);
+  }, 1000),
+);

src/rsa/rsa_internal.ts

@@ -4,6 +4,7 @@ import { i2osp, os2ip } from "./primitives.ts";
 import { concat, random_bytes } from "./../helper.ts";
 import { ber_decode, ber_simple } from "./basic_encoding_rule.ts";
 import { RawBinary } from "../binary.ts";
+import { RSAKey } from "./rsa_key.ts";
 
 /**
  * @param n public key modulus
@@ -19,8 +20,25 @@ export function rsaep(n: bigint, e: bigint, m: bigint): bigint {
  * @param d private key exponent
  * @param c ciphertext representative
  */
-export function rsadp(n: bigint, d: bigint, c: bigint): bigint {
-  return power_mod(c, d, n);
+export function rsadp(key: RSAKey, c: bigint): bigint {
+  if (!key.d) throw "Invalid RSA key";
+
+  if (key.dp && key.dq && key.qi && key.q && key.p) {
+    // Using the Chinese remainder algorithm
+    const m1 = power_mod(c % key.p, key.dp, key.p);
+    const m2 = power_mod(c % key.q, key.dq, key.q);
+
+    let h = 0n;
+    if (m1 >= m2) {
+      h = (key.qi * (m1 - m2)) % key.p;
+    } else {
+      h = (key.qi * (m1 - m2 + key.p * (key.p / key.q))) % key.p;
+    }
+
+    return (m2 + h * key.q) % (key.q * key.p);
+  } else {
+    return power_mod(c, key.d, key.n);
+  }
 }
 
 export function rsa_oaep_encrypt(
@@ -37,17 +55,15 @@ export function rsa_oaep_encrypt(
 }
 
 export function rsa_oaep_decrypt(
-  bytes: number,
-  n: bigint,
-  d: bigint,
+  key: RSAKey,
   c: Uint8Array,
   algorithm: "sha1" | "sha256",
 ) {
-  const em = rsadp(n, d, os2ip(c));
+  const em = rsadp(key, os2ip(c));
   const m = eme_oaep_decode(
     new Uint8Array(0),
-    i2osp(em, bytes),
-    bytes,
+    i2osp(em, key.length),
+    key.length,
     algorithm,
   );
   return m;
@@ -65,13 +81,8 @@ export function rsa_pkcs1_encrypt(
   return i2osp(c, bytes);
 }
 
-export function rsa_pkcs1_decrypt(
-  bytes: number,
-  n: bigint,
-  d: bigint,
-  c: Uint8Array,
-) {
-  const em = i2osp(rsadp(n, d, os2ip(c)), bytes);
+export function rsa_pkcs1_decrypt(key: RSAKey, c: Uint8Array) {
+  const em = i2osp(rsadp(key, os2ip(c)), key.length);
 
   if (em[0] !== 0) throw "Decryption error";
   if (em[1] !== 0x02) throw "Decryption error";
@@ -87,13 +98,13 @@ export function rsa_pkcs1_decrypt(
 }
 
 export function rsa_pkcs1_verify(
-  bytes: number,
-  n: bigint,
-  d: bigint,
+  key: RSAKey,
   s: Uint8Array,
   m: Uint8Array,
 ): boolean {
-  let em = i2osp(rsadp(n, d, os2ip(s)), bytes);
+  if (!key.e) throw "Invalid RSA key";
+
+  let em = i2osp(rsaep(key.n, key.e, os2ip(s)), key.length);
 
   if (em[0] !== 0) throw "Decryption error";
   if (em[1] !== 0x01) throw "Decryption error";
@@ -128,7 +139,23 @@ export function rsa_pkcs1_sign(
   message: Uint8Array,
 ): RawBinary {
   // deno-fmt-ignore
-  const oid = [0x30, 0x0d, 0x06, 0x09, 0x60, 0x86, 0x48, 0x01, 0x65, 0x03, 0x04, 0x02, 0x01, 0x05, 0x00]
+  const oid = [
+    0x30,
+    0x0d,
+    0x06,
+    0x09,
+    0x60,
+    0x86,
+    0x48,
+    0x01,
+    0x65,
+    0x03,
+    0x04,
+    0x02,
+    0x01,
+    0x05,
+    0x00,
+  ];
   const der = [
     0x30,
     message.length + 2 + oid.length,

src/rsa/rsa_js.ts

@@ -16,13 +16,9 @@ export class PureRSA {
     if (!key.e) throw "Invalid RSA key";
 
     if (options.padding === "oaep") {
-      return new RawBinary(rsa_oaep_encrypt(
-        key.length,
-        key.n,
-        key.e,
-        message,
-        options.hash,
-      ));
+      return new RawBinary(
+        rsa_oaep_encrypt(key.length, key.n, key.e, message, options.hash),
+      );
     } else if (options.padding === "pkcs1") {
       return new RawBinary(
         rsa_pkcs1_encrypt(key.length, key.n, key.e, message),
@@ -36,17 +32,9 @@ export class PureRSA {
     if (!key.d) throw "Invalid RSA key";
 
     if (options.padding === "oaep") {
-      return new RawBinary(rsa_oaep_decrypt(
-        key.length,
-        key.n,
-        key.d,
-        ciper,
-        options.hash,
-      ));
+      return new RawBinary(rsa_oaep_decrypt(key, ciper, options.hash));
     } else if (options.padding === "pkcs1") {
-      return new RawBinary(
-        rsa_pkcs1_decrypt(key.length, key.n, key.d, ciper),
-      );
+      return new RawBinary(rsa_pkcs1_decrypt(key, ciper));
     }
 
     throw "Invalid parameters";
@@ -61,9 +49,7 @@ export class PureRSA {
     if (!key.e) throw "Invalid RSA key";
 
     return rsa_pkcs1_verify(
-      key.length,
-      key.n,
-      key.e,
+      key,
       signature,
       createHash(options.hash).update(message).digest(),
     );

tests/rsa/cases/1024bit_1_ciper_oaep_sha1.txt

@@ -0,0 +1,2 @@
+k�m��8e�5�_�q�U�!���)�ҡ[�!�		��ku���όM��"Sȑ��&I܍�Ěb���:�}�p	��!m
+�/���>?S�`b���ڟ�4����%��B_����y���Su

tests/rsa/cases/1024bit_1_ciper_oaep_sha256.txt

@@ -0,0 +1 @@
+�r=Q�G�5 ����\Sa�A�%\WzQ�z��b"	ؐV���)���z[n�Ę��S[�f��",�p�W������z�g��R��ސ&�#�����q�g����\������F���Ӈݵg�

tests/rsa/cases/1024bit_1_private.pem

@@ -0,0 +1,15 @@
+-----BEGIN RSA PRIVATE KEY-----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+-----END RSA PRIVATE KEY-----

tests/rsa/cases/1024bit_1_public.pub

@@ -0,0 +1,6 @@
+-----BEGIN PUBLIC KEY-----
+MIGfMA0GCSqGSIb3DQEBAQUAA4GNADCBiQKBgQCpvfLHP+BzLb/UYDBHL0vpH6qI
+0VLTZjSDtwxIxKbhckYaizEDJ/0J3++hfAC1/Ge/7oe4nnpJJ8+I8K2I7gIJlx04
+94CrBZT9dNGq8smYgB5nW2D+AgBIKAHwM65sh0ea4UnofRWwM8UMyPG+xzFqtdQT
+L0wJEJLm5RsVxaxxXwIDAQAB
+-----END PUBLIC KEY-----

tests/rsa/cases/1024bit_2_ciper_oaep_sha1.txt

@@ -0,0 +1,2 @@
+���7��u	1���:�q�v��t
�b|ݪ-��,tR�Y:���Aܧ�q��:ƹ������3��
+�f5E�q%��h��`��g��3��h�D'������R��Iq��<��`>�|'

tests/rsa/cases/1024bit_2_ciper_pkcs1.txt

@@ -0,0 +1,2 @@
+�\�G񛑆(
+�rd�����S���ϳ�������0�Im}�U7#�A���e�D!�֬A�a�C����o��ѫ!?&$g.ב~����.?���c�3�5�x„V$�G�p��oB��o�7Q/%o

tests/rsa/cases/1024bit_2_private.pem

@@ -0,0 +1,15 @@
+-----BEGIN RSA PRIVATE KEY-----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+-----END RSA PRIVATE KEY-----

tests/rsa/cases/1024bit_2_public.pub

@@ -0,0 +1,6 @@
+-----BEGIN PUBLIC KEY-----
+MIGfMA0GCSqGSIb3DQEBAQUAA4GNADCBiQKBgQDQaYtcXuUK1GW796VloxTrr+Kf
+81XX85BxsPtfyKa+ywzRYGUrI4PO14RjkXtQV1Qlr42EngNqJGTcFtum5sWxgb1Z
+fbAWuC4HDJrX/41hzgijaMT9i07twjt+UiQg9BJ6TguSxza5/IdJHFCR/f0S1fiM
+cr4KR9mxWF6ETpi1MQIDAQAB
+-----END PUBLIC KEY-----

tests/rsa/cases/2048bit_1_private.pem

@@ -0,0 +1,27 @@
+-----BEGIN RSA PRIVATE KEY-----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+-----END RSA PRIVATE KEY-----

tests/rsa/cases/2048bit_1_public.pub

@@ -0,0 +1,9 @@
+-----BEGIN PUBLIC KEY-----
+MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEA0SQQqbw9EByOv65920q0
++qHzAuxnvLyNYZUZm01lL1zrfg8VT3NbSk+K0cNJIbEV5uoIt5fDT7i3hQP5WyA0
+1m1xULZ/fFPPkAtBSv5oeNjdJpv4a3iRWgRuKx1a0qMZNAMM37PZ90j42F3/hi2Q
++xTcBpSbA7csvqzLTIzfgAw9mx7eFJp+xxiLDfEElO9jnl8dKGcNaERkqDCcXolw
+b633qRzoc+GgYCNFUs0JzLWYWxtq9UObXFWSOirYDaz4WuMzD3sxZgy0y4xk5NxY
+iOABx1jwg/otM0T2TwgbwIN9YIHHtoHbQhPS2+tH31mbzztorXY2yioKtCizJyql
+UwIDAQAB
+-----END PUBLIC KEY-----

tests/rsa/cases/2048bit_2_ciper_oaep_sha1.txt

@@ -0,0 +1,2 @@
+K��N�4cA��3ҙx��H�������Z���n�r�PnM����;XS���%+�<��J��Ȃk���n�� �$zJ��u��$��hnW���~2P\�-�����6��.�#)�~�g�#�.u����M��O?�ð�d�m~Z��-b>rO:��yt
+e��V�~�Q������`�]�Hfε%*~��7��WM��:���%b��~rx�jSJ��V^y�y+/�%�]�2Z���&Y�$����(�y���5	��,�|�/�

tests/rsa/cases/2048bit_2_ciper_oaep_sha256.txt

@@ -0,0 +1 @@
+J٭X�[1��Zs���P���0�J�1�R9k���{���� ��4W��+��7�zk̰p�sr!?��C���*�OD�='���&��p8/��:0���2:�B�p^�\��<��>
C����a�r�����K���&n�� ��R�\Q8a���Kn��i�C��9�&��4m����Xʮ��՞0'.��=`{o�CA�CQ@�zs`2�#����`���Ɣ���F=;�<���M����ovX=��f�xn|��w

tests/rsa/cases/2048bit_2_ciper_pkcs1.txt

@@ -0,0 +1,2 @@
+P{��@�}M��s�5I��=r&�-���a񴐧�k,�_Ҭ�Hc7;����ߋ�E<���X�`/��c����4 זN�ī��
+Z�ᴚW�r�s���Hi�r�+��
/�v����,wU}�+���T�۵�~|tE�9f�NA�IX<K2mS�[mQ�g����V�<?���`>Yr8�/<��Zz�m����;�w�)M����ֺ�"'�jN?�<�S��R14�i}�A�'������������;������

tests/rsa/cases/2048bit_2_private.pem

@@ -0,0 +1,27 @@
+-----BEGIN RSA PRIVATE KEY-----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+-----END RSA PRIVATE KEY-----

tests/rsa/cases/2048bit_2_public.pub

@@ -0,0 +1,9 @@
+-----BEGIN PUBLIC KEY-----
+MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAvO45IgvaeViijFJNbmY8
+rcZL2cQwie0RKDBqxeCfgdSxOqSARmp+myLhtmCgmg2pASh/IukPkHqnUKgoj2sf
+BgBRlLsPiWb1C/7QZFPXUFGETgWEKFgomSFVVcLf27Qku3D0fRN37Wc3CONTwKI/
+NKp5lBa2b5tR46C9cI/51AhnkZgFnoAgqJDfmu1R4YuLtPuT822YE/ly+/VclxsM
+ayeArzs23GdJnR7+jqG9HcISTqueYOHX8f0x8D35Pql3QIxfwWXb2Z+0wIjprKLD
+JWWJp3m6LrMf85nFABDnQ2DoT3xgUUdh3aQrYAkvADNoPp4FOGfMpqTBEARQGB+n
+TQIDAQAB
+-----END PUBLIC KEY-----