paillier同态加密算法原理及代码实现
2024-06-06 20:41:35
paillier同态加密算法原理及代码实现
由于工程量巨大，这里先贴上代码提醒自己还有这么个事情，等寒假有时间了再慢慢补充解析
代码实现
提前生成素数打表来提升算法速度，生成素数的代码点这里
注意：待加密字符串长度需为4的倍数且长度不超过12个字符；代码在实现同态加法的部分还存在问题需要修改
#include<stdio.h>
#include<iostream>
#include<fstream>
#include<iomanip>
#include <string>
#define len1 128
#define len2 2
using namespace std;static int Primes[2800] = //前2800个素数
{3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
37, 41, 43, 47, 53, 59, 61, 67, 71, 73,
79, 83, 89, 97, 101, 103, 107, 109, 113, 127,
131, 137, 139, 149, 151, 157, 163, 167, 173, 179,
181, 191, 193, 197, 199, 211, 223, 227, 229, 233,
239, 241, 251, 257, 263, 269, 271, 277, 281, 283,
293, 307, 311, 313, 317, 331, 337, 347, 349, 353,
359, 367, 373, 379, 383, 389, 397, 401, 409, 419,
421, 431, 433, 439, 443, 449, 457, 461, 463, 467,
479, 487, 491, 499, 503, 509, 521, 523, 541, 547,
557, 563, 569, 571, 577, 587, 593, 599, 601, 607,
613, 617, 619, 631, 641, 643, 647, 653, 659, 661,
673, 677, 683, 691, 701, 709, 719, 727, 733, 739,
743, 751, 757, 761, 769, 773, 787, 797, 809, 811,
821, 823, 827, 829, 839, 853, 857, 859, 863, 877,
881, 883, 887, 907, 911, 919, 929, 937, 941, 947,
953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019,
1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087,
1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153,
1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229,
1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297,
1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381,
1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453,
1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523,
1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597,
1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663,
1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741,
1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823,
1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901,
1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993,
1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063,
2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131,
2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221,
2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293,
2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371,
2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437,
2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539,
2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621,
2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689,
2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749,
2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833,
2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909,
2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001,
3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083,
3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187,
3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259,
3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343,
3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433,
3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517,
3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581,
3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659,
3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733,
3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823,
3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911,
3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001,
4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073,
4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153,
4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241,
4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327,
4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421,
4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507,
4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591,
4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663,
4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759,
4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861,
4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943,
4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009,
5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099,
5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189,
5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281,
5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393,
5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449,
5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527,
5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641,
5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701,
5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801,
5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861,
5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953,
5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067,
6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143,
6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229,
6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311,
6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373,
6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481,
6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577,
6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679,
6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763,
6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841,
6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947,
6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001,
7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109,
7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211,
7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307,
7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417,
7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507,
7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573,
7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649,
7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727,
7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841,
7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927,
7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039,
8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117,
8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221,
8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293,
8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389,
8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513,
8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599,
8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681,
8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747,
8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837,
8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933,
8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013,
9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127,
9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203,
9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293,
9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391,
9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461,
9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539,
9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643,
9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739,
9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817,
9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901,
9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973, 10007, 10009,
10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099, 10103,
10111, 10133, 10139, 10141, 10151, 10159, 10163, 10169, 10177, 10181,
10193, 10211, 10223, 10243, 10247, 10253, 10259, 10267, 10271, 10273,
10289, 10301, 10303, 10313, 10321, 10331, 10333, 10337, 10343, 10357,
10369, 10391, 10399, 10427, 10429, 10433, 10453, 10457, 10459, 10463,
10477, 10487, 10499, 10501, 10513, 10529, 10531, 10559, 10567, 10589,
10597, 10601, 10607, 10613, 10627, 10631, 10639, 10651, 10657, 10663,
10667, 10687, 10691, 10709, 10711, 10723, 10729, 10733, 10739, 10753,
10771, 10781, 10789, 10799, 10831, 10837, 10847, 10853, 10859, 10861,
10867, 10883, 10889, 10891, 10903, 10909, 10937, 10939, 10949, 10957,
10973, 10979, 10987, 10993, 11003, 11027, 11047, 11057, 11059, 11069,
11071, 11083, 11087, 11093, 11113, 11117, 11119, 11131, 11149, 11159,
11161, 11171, 11173, 11177, 11197, 11213, 11239, 11243, 11251, 11257,
11261, 11273, 11279, 11287, 11299, 11311, 11317, 11321, 11329, 11351,
11353, 11369, 11383, 11393, 11399, 11411, 11423, 11437, 11443, 11447,
11467, 11471, 11483, 11489, 11491, 11497, 11503, 11519, 11527, 11549,
11551, 11579, 11587, 11593, 11597, 11617, 11621, 11633, 11657, 11677,
11681, 11689, 11699, 11701, 11717, 11719, 11731, 11743, 11777, 11779,
11783, 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833, 11839,
11863, 11867, 11887, 11897, 11903, 11909, 11923, 11927, 11933, 11939,
11941, 11953, 11959, 11969, 11971, 11981, 11987, 12007, 12011, 12037,
12041, 12043, 12049, 12071, 12073, 12097, 12101, 12107, 12109, 12113,
12119, 12143, 12149, 12157, 12161, 12163, 12197, 12203, 12211, 12227,
12239, 12241, 12251, 12253, 12263, 12269, 12277, 12281, 12289, 12301,
12323, 12329, 12343, 12347, 12373, 12377, 12379, 12391, 12401, 12409,
12413, 12421, 12433, 12437, 12451, 12457, 12473, 12479, 12487, 12491,
12497, 12503, 12511, 12517, 12527, 12539, 12541, 12547, 12553, 12569,
12577, 12583, 12589, 12601, 12611, 12613, 12619, 12637, 12641, 12647,
12653, 12659, 12671, 12689, 12697, 12703, 12713, 12721, 12739, 12743,
12757, 12763, 12781, 12791, 12799, 12809, 12821, 12823, 12829, 12841,
12853, 12889, 12893, 12899, 12907, 12911, 12917, 12919, 12923, 12941,
12953, 12959, 12967, 12973, 12979, 12983, 13001, 13003, 13007, 13009,
13033, 13037, 13043, 13049, 13063, 13093, 13099, 13103, 13109, 13121,
13127, 13147, 13151, 13159, 13163, 13171, 13177, 13183, 13187, 13217,
13219, 13229, 13241, 13249, 13259, 13267, 13291, 13297, 13309, 13313,
13327, 13331, 13337, 13339, 13367, 13381, 13397, 13399, 13411, 13417,
13421, 13441, 13451, 13457, 13463, 13469, 13477, 13487, 13499, 13513,
13523, 13537, 13553, 13567, 13577, 13591, 13597, 13613, 13619, 13627,
13633, 13649, 13669, 13679, 13681, 13687, 13691, 13693, 13697, 13709,
13711, 13721, 13723, 13729, 13751, 13757, 13759, 13763, 13781, 13789,
13799, 13807, 13829, 13831, 13841, 13859, 13873, 13877, 13879, 13883,
13901, 13903, 13907, 13913, 13921, 13931, 13933, 13963, 13967, 13997,
13999, 14009, 14011, 14029, 14033, 14051, 14057, 14071, 14081, 14083,
14087, 14107, 14143, 14149, 14153, 14159, 14173, 14177, 14197, 14207,
14221, 14243, 14249, 14251, 14281, 14293, 14303, 14321, 14323, 14327,
14341, 14347, 14369, 14387, 14389, 14401, 14407, 14411, 14419, 14423,
14431, 14437, 14447, 14449, 14461, 14479, 14489, 14503, 14519, 14533,
14537, 14543, 14549, 14551, 14557, 14561, 14563, 14591, 14593, 14621,
14627, 14629, 14633, 14639, 14653, 14657, 14669, 14683, 14699, 14713,
14717, 14723, 14731, 14737, 14741, 14747, 14753, 14759, 14767, 14771,
14779, 14783, 14797, 14813, 14821, 14827, 14831, 14843, 14851, 14867,
14869, 14879, 14887, 14891, 14897, 14923, 14929, 14939, 14947, 14951,
14957, 14969, 14983, 15013, 15017, 15031, 15053, 15061, 15073, 15077,
15083, 15091, 15101, 15107, 15121, 15131, 15137, 15139, 15149, 15161,
15173, 15187, 15193, 15199, 15217, 15227, 15233, 15241, 15259, 15263,
15269, 15271, 15277, 15287, 15289, 15299, 15307, 15313, 15319, 15329,
15331, 15349, 15359, 15361, 15373, 15377, 15383, 15391, 15401, 15413,
15427, 15439, 15443, 15451, 15461, 15467, 15473, 15493, 15497, 15511,
15527, 15541, 15551, 15559, 15569, 15581, 15583, 15601, 15607, 15619,
15629, 15641, 15643, 15647, 15649, 15661, 15667, 15671, 15679, 15683,
15727, 15731, 15733, 15737, 15739, 15749, 15761, 15767, 15773, 15787,
15791, 15797, 15803, 15809, 15817, 15823, 15859, 15877, 15881, 15887,
15889, 15901, 15907, 15913, 15919, 15923, 15937, 15959, 15971, 15973,
15991, 16001, 16007, 16033, 16057, 16061, 16063, 16067, 16069, 16073,
16087, 16091, 16097, 16103, 16111, 16127, 16139, 16141, 16183, 16187,
16189, 16193, 16217, 16223, 16229, 16231, 16249, 16253, 16267, 16273,
16301, 16319, 16333, 16339, 16349, 16361, 16363, 16369, 16381, 16411,
16417, 16421, 16427, 16433, 16447, 16451, 16453, 16477, 16481, 16487,
16493, 16519, 16529, 16547, 16553, 16561, 16567, 16573, 16603, 16607,
16619, 16631, 16633, 16649, 16651, 16657, 16661, 16673, 16691, 16693,
16699, 16703, 16729, 16741, 16747, 16759, 16763, 16787, 16811, 16823,
16829, 16831, 16843, 16871, 16879, 16883, 16889, 16901, 16903, 16921,
16927, 16931, 16937, 16943, 16963, 16979, 16981, 16987, 16993, 17011,
17021, 17027, 17029, 17033, 17041, 17047, 17053, 17077, 17093, 17099,
17107, 17117, 17123, 17137, 17159, 17167, 17183, 17189, 17191, 17203,
17207, 17209, 17231, 17239, 17257, 17291, 17293, 17299, 17317, 17321,
17327, 17333, 17341, 17351, 17359, 17377, 17383, 17387, 17389, 17393,
17401, 17417, 17419, 17431, 17443, 17449, 17467, 17471, 17477, 17483,
17489, 17491, 17497, 17509, 17519, 17539, 17551, 17569, 17573, 17579,
17581, 17597, 17599, 17609, 17623, 17627, 17657, 17659, 17669, 17681,
17683, 17707, 17713, 17729, 17737, 17747, 17749, 17761, 17783, 17789,
17791, 17807, 17827, 17837, 17839, 17851, 17863, 17881, 17891, 17903,
17909, 17911, 17921, 17923, 17929, 17939, 17957, 17959, 17971, 17977,
17981, 17987, 17989, 18013, 18041, 18043, 18047, 18049, 18059, 18061,
18077, 18089, 18097, 18119, 18121, 18127, 18131, 18133, 18143, 18149,
18169, 18181, 18191, 18199, 18211, 18217, 18223, 18229, 18233, 18251,
18253, 18257, 18269, 18287, 18289, 18301, 18307, 18311, 18313, 18329,
18341, 18353, 18367, 18371, 18379, 18397, 18401, 18413, 18427, 18433,
18439, 18443, 18451, 18457, 18461, 18481, 18493, 18503, 18517, 18521,
18523, 18539, 18541, 18553, 18583, 18587, 18593, 18617, 18637, 18661,
18671, 18679, 18691, 18701, 18713, 18719, 18731, 18743, 18749, 18757,
18773, 18787, 18793, 18797, 18803, 18839, 18859, 18869, 18899, 18911,
18913, 18917, 18919, 18947, 18959, 18973, 18979, 19001, 19009, 19013,
19031, 19037, 19051, 19069, 19073, 19079, 19081, 19087, 19121, 19139,
19141, 19157, 19163, 19181, 19183, 19207, 19211, 19213, 19219, 19231,
19237, 19249, 19259, 19267, 19273, 19289, 19301, 19309, 19319, 19333,
19373, 19379, 19381, 19387, 19391, 19403, 19417, 19421, 19423, 19427,
19429, 19433, 19441, 19447, 19457, 19463, 19469, 19471, 19477, 19483,
19489, 19501, 19507, 19531, 19541, 19543, 19553, 19559, 19571, 19577,
19583, 19597, 19603, 19609, 19661, 19681, 19687, 19697, 19699, 19709,
19717, 19727, 19739, 19751, 19753, 19759, 19763, 19777, 19793, 19801,
19813, 19819, 19841, 19843, 19853, 19861, 19867, 19889, 19891, 19913,
19919, 19927, 19937, 19949, 19961, 19963, 19973, 19979, 19991, 19993,
19997, 20011, 20021, 20023, 20029, 20047, 20051, 20063, 20071, 20089,
20101, 20107, 20113, 20117, 20123, 20129, 20143, 20147, 20149, 20161,
20173, 20177, 20183, 20201, 20219, 20231, 20233, 20249, 20261, 20269,
20287, 20297, 20323, 20327, 20333, 20341, 20347, 20353, 20357, 20359,
20369, 20389, 20393, 20399, 20407, 20411, 20431, 20441, 20443, 20477,
20479, 20483, 20507, 20509, 20521, 20533, 20543, 20549, 20551, 20563,
20593, 20599, 20611, 20627, 20639, 20641, 20663, 20681, 20693, 20707,
20717, 20719, 20731, 20743, 20747, 20749, 20753, 20759, 20771, 20773,
20789, 20807, 20809, 20849, 20857, 20873, 20879, 20887, 20897, 20899,
20903, 20921, 20929, 20939, 20947, 20959, 20963, 20981, 20983, 21001,
21011, 21013, 21017, 21019, 21023, 21031, 21059, 21061, 21067, 21089,
21101, 21107, 21121, 21139, 21143, 21149, 21157, 21163, 21169, 21179,
21187, 21191, 21193, 21211, 21221, 21227, 21247, 21269, 21277, 21283,
21313, 21317, 21319, 21323, 21341, 21347, 21377, 21379, 21383, 21391,
21397, 21401, 21407, 21419, 21433, 21467, 21481, 21487, 21491, 21493,
21499, 21503, 21517, 21521, 21523, 21529, 21557, 21559, 21563, 21569,
21577, 21587, 21589, 21599, 21601, 21611, 21613, 21617, 21647, 21649,
21661, 21673, 21683, 21701, 21713, 21727, 21737, 21739, 21751, 21757,
21767, 21773, 21787, 21799, 21803, 21817, 21821, 21839, 21841, 21851,
21859, 21863, 21871, 21881, 21893, 21911, 21929, 21937, 21943, 21961,
21977, 21991, 21997, 22003, 22013, 22027, 22031, 22037, 22039, 22051,
22063, 22067, 22073, 22079, 22091, 22093, 22109, 22111, 22123, 22129,
22133, 22147, 22153, 22157, 22159, 22171, 22189, 22193, 22229, 22247,
22259, 22271, 22273, 22277, 22279, 22283, 22291, 22303, 22307, 22343,
22349, 22367, 22369, 22381, 22391, 22397, 22409, 22433, 22441, 22447,
22453, 22469, 22481, 22483, 22501, 22511, 22531, 22541, 22543, 22549,
22567, 22571, 22573, 22613, 22619, 22621, 22637, 22639, 22643, 22651,
22669, 22679, 22691, 22697, 22699, 22709, 22717, 22721, 22727, 22739,
22741, 22751, 22769, 22777, 22783, 22787, 22807, 22811, 22817, 22853,
22859, 22861, 22871, 22877, 22901, 22907, 22921, 22937, 22943, 22961,
22963, 22973, 22993, 23003, 23011, 23017, 23021, 23027, 23029, 23039,
23041, 23053, 23057, 23059, 23063, 23071, 23081, 23087, 23099, 23117,
23131, 23143, 23159, 23167, 23173, 23189, 23197, 23201, 23203, 23209,
23227, 23251, 23269, 23279, 23291, 23293, 23297, 23311, 23321, 23327,
23333, 23339, 23357, 23369, 23371, 23399, 23417, 23431, 23447, 23459,
23473, 23497, 23509, 23531, 23537, 23539, 23549, 23557, 23561, 23563,
23567, 23581, 23593, 23599, 23603, 23609, 23623, 23627, 23629, 23633,
23663, 23669, 23671, 23677, 23687, 23689, 23719, 23741, 23743, 23747,
23753, 23761, 23767, 23773, 23789, 23801, 23813, 23819, 23827, 23831,
23833, 23857, 23869, 23873, 23879, 23887, 23893, 23899, 23909, 23911,
23917, 23929, 23957, 23971, 23977, 23981, 23993, 24001, 24007, 24019,
24023, 24029, 24043, 24049, 24061, 24071, 24077, 24083, 24091, 24097,
24103, 24107, 24109, 24113, 24121, 24133, 24137, 24151, 24169, 24179,
24181, 24197, 24203, 24223, 24229, 24239, 24247, 24251, 24281, 24317,
24329, 24337, 24359, 24371, 24373, 24379, 24391, 24407, 24413, 24419,
24421, 24439, 24443, 24469, 24473, 24481, 24499, 24509, 24517, 24527,
24533, 24547, 24551, 24571, 24593, 24611, 24623, 24631, 24659, 24671,
24677, 24683, 24691, 24697, 24709, 24733, 24749, 24763, 24767, 24781,
24793, 24799, 24809, 24821, 24841, 24847, 24851, 24859, 24877, 24889,
24907, 24917, 24919, 24923, 24943, 24953, 24967, 24971, 24977, 24979,
24989, 25013, 25031, 25033, 25037, 25057, 25073, 25087, 25097, 25111,
25117, 25121, 25127, 25147, 25153, 25163, 25169, 25171, 25183, 25189,
25219, 25229, 25237, 25243, 25247, 25253, 25261, 25301, 25303, 25307,
25309, 25321, 25339, 25343, 25349, 25357, 25367, 25373, 25391, 25409
};class BigNum
{public:int length; //大数长度int signal; //大数符号unsigned long array[len1]; //大数绝对值 BigNum(); //构造函数 BigNum(unsigned __int64); //构造函数用于赋初始值 BigNum(string s); //读入字符串BigNum(BigNum const& A); //复制大数，const保护了原对象的属性 ~BigNum(); //析构函数BigNum operator+(BigNum& A); //运算符+重载BigNum operator-(BigNum& A); //运算符-重载BigNum operator*(BigNum& A); //运算符*重载BigNum operator/(BigNum& A); //运算符/重载BigNum operator%(BigNum& A); //运算符%重载BigNum operator-(void); //负号重载int operator==(BigNum& A); //等于号重载int operator!=(BigNum& A); //不等号重载int Compare(BigNum const& A); //比较两大数绝对值大小void GeneratePrime(void); //产生素数int Rabin_Miller(void); //拉宾米勒算法用于素数测试void Random(int a); //随机产生一个大数void Random(BigNum A); //随机产生一个小于A的大数void print(void); //输出大数 void printS(void); //输出字符串BigNum power_mod(BigNum& A, BigNum& B); //模幂算法计算X^A mod BBigNum ex_euclid(BigNum a, BigNum b); //扩展欧几里德算法
};/*构造函数*/
BigNum::BigNum()
{length = 1;signal = 0;for (int i = 0; i < len1; i++)array[i] = 0;
}/*构造函数用于赋初始值,参数为64位无符号int*/
BigNum::BigNum(unsigned __int64 A)
{signal = 0;if (A <= 0xffffffff){length = 1;array[0] = (unsigned long)A;}else {length = 2; //长度大于1，存两位 array[1] = (unsigned long)(A >> 32);array[0] = (unsigned long)A;}for (int i = length; i < len1; i++) //补充高位 array[i] = 0;
}/*用于读入输入的要加密的字符串，大数的每一位是32位，可以存4个字符*/
BigNum::BigNum(string s)
{for (int i = 0; i < s.size(); i++) {array[i / 4] <<= 8;array[i / 4] |= s[i]; //按位或，将4个字符存到一个大数位里}length = (s.size() - 1) / 4 + 1; //数组长度
}/*用于复制大数*/
BigNum::BigNum(BigNum const& A)
{length = A.length;signal = A.signal;for (int i = 0; i < len1; i++)array[i] = A.array[i];
}/*析构函数,撤销对象占用的内存之前完成一些清理工作*/
BigNum::~BigNum()
{}/*运算符+重载，思路：两数符号相同时，和=加数当前位+被加数当前位+进位；
两位异号时，转化为减法运算*/
BigNum BigNum::operator+(BigNum& A)
{BigNum SUM, D; //B为两大数之和，D为中间变量int C = 0; //进位 unsigned __int64 sum; //存储int型的和if (signal == A.signal) //符号相同时 { SUM.signal = signal;if (length >= A.length) //长度=两数最大长度+1 SUM.length = length + 1;elseSUM.length = A.length + 1;for (int i = 0; i < SUM.length; i++) //和=加数当前位+被加数当前+进位 { sum = (unsigned __int64)A.array[i] + (unsigned __int64)array[i] + C;if (sum <= 0xffffffff)C = 0; //超过2^32-1,进位 elseC = 1;SUM.array[i] = (unsigned long)sum; //和当前位= (unsigned long)和 }while (SUM.length > 1) //消除最高位的0 if (SUM.array[SUM.length - 1] == 0)SUM.length--;else break;}else if (signal == 0){D = -A;SUM = (*this) - D;} //异号则转为减法运算 else {D = -(*this);SUM = A - D;}return SUM;
}/*运算符-重载，思路：两数同号时，比较两数大小，被减数大时，如果sub=被减数当前位-上一个
借位小于减数当前位，那么当前位有借位，差=借位*0xffffffff+sub-减数当前位+借位；减数大时，
原理一样，交换减数和被减数位置，最后负号与被减数相反；两数异号时，转化成加法运算*/
BigNum BigNum::operator-(BigNum& A)
{BigNum SUB, D; //B为两数的差，D为中间变量int F = 0, i; //借位F unsigned int sub; //暂存被减数减去上一个借位的差if (signal == A.signal) //符号相同时 { if (Compare(A) == 0 || Compare(A) == 2) //当被减数大于等于减数{ if (signal == 0)SUB.signal = 0; //符号判断 elseSUB.signal = 1;for (i = 0; i < length; i++) {sub = array[i] - F; //差=被减数当前位-上一个借位 if (sub < A.array[i])F = 1; //如果差小于减数当前位 ,有借位 elseF = 0;SUB.array[i] = 0xffffffff * F - A.array[i] + sub + F; //差的当前位 }}else //被减数小于减数,基本原理同上{ if (signal == 0) //符号判断SUB.signal = 1;elseSUB.signal = 0;for (i = 0; i < A.length; i++) {sub = A.array[i] - F;if (sub < array[i])F = 1; //向上借位elseF = 0;SUB.array[i] = 0xffffffff * F - array[i] + sub + F;}}SUB.length = i;while (SUB.length > 1) //消除最高位的0 if (SUB.array[SUB.length - 1] == 0)SUB.length--;else break;}else //如果异号,转为加法{ D = -A;SUB = (*this) + D;}return SUB;
}/*运算符*重载，思路：两数同号时，积的符号为正，否则为负，积=当前位的积+进位*/
BigNum BigNum::operator*(BigNum& A)
{BigNum PRO; //B两大数的积unsigned __int64 C = 0; //进位C unsigned __int64 pro; //暂存两数当前位的积 if (signal == A.signal) //判断符号 PRO.signal = 0;elsePRO.signal = 1;PRO.length = length + A.length; //长度不超过两数长度和 for (int i = 0; i < A.length; i++){for (int j = 0; j <= length; j++) //积=当前位的乘积+进位{ pro = (unsigned __int64)A.array[i] * (unsigned __int64)array[j] + C;if ((unsigned __int64)PRO.array[i + j] + pro > 0xffffffff) //判断有无进位C = ((unsigned __int64)PRO.array[i + j] + pro) >> 32;elseC = 0;PRO.array[i + j] += (unsigned long)pro; //当前位要加上上一次运算结果 }}while (PRO.length > 1) //消除最高位的0 if (PRO.array[PRO.length - 1] == 0)PRO.length--;else break;return PRO;
}/*运算符/重载，思路：采用试商法，取被除数最高位（如果小于除数最高位则去前两位）
和除数最高位试商，商=被除数最高位/（除数最高位+1），余数=被除数-商*除数，将余数
作为新的被除数做下次运算，每次运算完商都移位，直到余数小于等于除数，等于除数时商+1，
单独考虑除数长度为1的情况*/
BigNum BigNum::operator/(BigNum& A)
{BigNum X(*this), Y, Z, I(1);int i, len; //len移位unsigned __int64 temp, div, mul;unsigned long F = 0; //余数 if (A.length > 1) //除数长度大于1时 { while (X.Compare(A) == 0) //当被除数大于除数时{ if (X.array[X.length - 1] > A.array[A.length - 1]) //被除数最高位大{ len = X.length - A.length; //商=被除数当前位/(除数当前位+1) div = X.array[X.length - 1] / (A.array[A.length - 1] + 1);}else if (X.length > A.length) //被除数长度较长{ len = X.length - A.length - 1;temp = X.array[X.length - 1];temp = (temp << 32) + X.array[X.length - 2]; //商=被除数前两位/(除数当前位+1) if (A.array[A.length - 1] == 0xffffffff)div = (temp >> 32);else div = temp / (A.array[A.length - 1] + 1);}else //相等,加1退出{ Y = Y + I;break;}Z = BigNum(div); //暂时的商(大数) Z.length += len; //长度=原长+移位 for (i = Z.length - 1; i >= len; i--)Z.array[i] = Z.array[i - len];for (i = 0; i < len; i++)Z.array[i] = 0;Y = Y + Z; //商=原商+暂商 Z = A * Z;X = X - Z; //求余数代入下一次运算}if (X.Compare(A) == 2)Y = Y + I; //相等+1返回 if (signal != A.signal)Y.signal = 1; //判断符号，异号为负 return Y;}else //除数长度等于1 { if (X.length == 1) //被除数长度也为1 X.array[0] = X.array[0] / A.array[0];else {for (int i = X.length - 1; i >= 0; i--){div = F;div = (div << 32) + X.array[i]; //被除数=余数+当前位 X.array[i] = (unsigned long)(div / A.array[0]); //商mul = (div / A.array[0]) * A.array[0];F = (unsigned long)(div - mul); //求余,再带入下次运算 }if (X.array[X.length - 1] == 0)X.length--;}return X;}
}/*重载运算符之求模，思路同除法重载*/
BigNum BigNum::operator%(BigNum& A)
{BigNum Y(*this), Z, T;int i, len;unsigned __int64 temp, div, F = 0;if (A.length > 1) //除数长度大于1时{ while (Y.Compare(A) == 0 || Y.Compare(A) == 2) { //被除数大等于除数 div = Y.array[Y.length - 1]; // 被除数=被除数当前位 temp = A.array[A.length - 1]; // 除数=除数当前位 len = Y.length - A.length; //移位 if ((div == temp) && (len == 0)) //长度相同首位相同,余数为差{ Y = Y - A; break;}if ((div <= temp) && length) //被除数小于除数，被除数取前两位{ len--;div = (div << 32) + Y.array[Y.length - 2];}div = div / (temp + 1); //下面原理同除法重载Z = BigNum(div);if (len){Z.length += len;for (i = Z.length - 1; i >= len; i--)Z.array[i] = Z.array[i - len];for (i = 0; i < len; i++)Z.array[i] = 0;}T = A * Z;Y = Y - T;}return Y;}else //除数长度等于1时 { if (length == 1)F = array[0] % A.array[0];elsefor (int i = length - 1; i >= 0; --i){F = (F << 32) | array[i];F %= A.array[0];}return BigNum(F);}
}/*负号重载 */
BigNum BigNum::operator-(void)
{BigNum Temp = *this;Temp.signal = 1 - signal;return Temp;
}/*等于号重载 */
int BigNum::operator==(BigNum& A)
{if (signal == A.signal) //判断符号return Compare(A) == 2; //绝对值相等则返回两数相等elsereturn 0;
}/*不等号重载*/
int BigNum::operator!=(BigNum& A)
{if (signal == A.signal)return Compare(A) != 2;elsereturn 0;
}/* 比较两大数绝对值大小，1表示小于A，0表示大于A，2表示相等*/
int BigNum::Compare(BigNum const& A)
{if (length > A.length) //比较长度 return 0;else if (length < A.length)return 1;else {for (int i = length - 1; i >= 0; i--) {if (array[i] > A.array[i])return 0;else if (array[i] < A.array[i])return 1;}return 2; //相等}
}/*随机产生一个大数0<=X<=2^(32*SLEN)-1*/
void BigNum::Random(int a)
{int i;signal = a; //a传入的一般都是0 length = len2;for (i = 0; i < len2; ++i) //rand产生0-32767的数（16位长），两次生成随机数移位后的和 array[i] = rand() << 16 | rand();array[len2 - 1] |= 1 << 31; //确保大数为512位长
}/*随机产生一个小于A的大数*/
void BigNum::Random(BigNum A)
{int i;signal = 0;if (A.length > 1) //被比较数长度为1时仅生成1位 { length = A.length - 1;for (i = 0; i < length; ++i)array[i] = rand() << 16 | rand();}else //生成的位数少于被比较数 { length = 1;array[0] = rand() << 16 | rand();while (array[0] > A.array[0])array[0] = rand() << 16 | rand();}
}/*拉宾米勒算法用于素数测试，思路：1、先和前1000个素数比较，提高判断效率；2、计算
r和m使得n-1=(2^r)*m；3、随机选择a，计算y=a^m(mod n)；4、如果y=1或者y=n-1，输出素数，
结束；5、for i=1 to r-1, y=y^2(mod n)，若y=1,输出合数，结束，否则i++；6、若y!=n-1,
输出合数，结束。*/
int BigNum::Rabin_Miller(void)
{BigNum temp, X(*this), I(1), O(0), m, E(2), A, Y, T, C;int r = 0;for (int i = 0; i < 2800; i++) //被判断数先除以2800个小素数提高效率{ temp = BigNum(Primes[i]);if (X == temp)return 1; //素数if (X % temp == O) //合数 return 0;}m = *this - I; //Y=被判断数-1 while (!(m.array[0] & 1)) //Y非奇数 { r++;m = m / E;} //求得r,Y，满足X-1=(2^r)*Y for (int i = 1; i <= 5; ++i) //由于一次判断不是素数的概率小等于1/4{ int flag = 0; //所以至少要进行5次判断以确保准确性 A.Random(X); //随机生成小于被判断数的数 while (A.Compare(T) == 2) A.Random(X); //和上次避免产生相同的A T = A;Y = A.power_mod(m, X); // B=A^Y mod X if ((O.Compare((Y - I) % X) == 2) || (O.Compare((Y + I) % X) == 2))continue; //若B=1或X-1，则是素数 else {for (int j = 0; j < r; j++) {Y = (Y * Y) % X; //否则,B=b^2%X if (O.Compare((Y + I) % X) == 2) //B=1,则是合数{ flag = 1;break;}}}if (flag)continue;elsereturn 0;}return 1;
}/*产生 512 位的素数*/
void BigNum::GeneratePrime(void)
{Random(0); //确保符号为正array[0] |= 1; //确保是奇数以提高效率 while (!Rabin_Miller()) {Random(0);array[0] |= 1;}
}/*输出大数*/
void BigNum::print(void)
{cout << hex; //以十六进制打印for (int i = 1; i <= length; ++i) {if (i % 8 == 1)cout << endl; //换行，8个数组元素一行 cout << setw(8) << array[length - i] << " ";}cout << dec << endl;
}/*输出字符串，每一个array[i]存4个字符，每一次右移8位去出一个字符*/
void BigNum::printS(void)
{for (int i = 0; i <= length; ++i)for (int j = 3; j >= 0; --j) {char chr = char(array[i] >> (j * 8)); //右移8位取一个字符 if (chr) cout << chr; //是字符输出 }cout << endl;
}/*求Y，使得X^A mod B=Y */
BigNum BigNum::power_mod(BigNum& A, BigNum& B)
{BigNum X(1), N = (*this) % B;for (int i = 0; i < A.length; i++) {for (int j = 0; j < 32; j++) //转为2进制运算提高效率{ if (A.array[i] & (1 << j))X = (X * N) % B;N = (N * N) % B;}}return X;
}/*扩展欧几里德算法，求X使得AX mod B=1*/
BigNum BigNum::ex_euclid(BigNum a, BigNum b)
{BigNum X(1), U(0), O(0), A(a), B(b); //A，B为了避免修改a，b的值 while (B != O){BigNum Q(A / B), R(A % B), Temp;Temp = Q * U;Temp = X - Temp;X = U;U = Temp;A = B;B = R;}if (X.signal == 1) //负数时加上mod 的数b{ X = X + b;}*this = X; //传给当前大数 return *this;
}BigNum Gcd(BigNum a, BigNum b);//求取最大公约数
BigNum Lcm(BigNum a, BigNum b);//求取最小公倍数
BigNum Lfunction(BigNum a);//求取L函数
BigNum n, I = 1, o = 0;int main()
{BigNum p, q, I(1), e(257), t, temp, d, a, b, c, O(0), middle, g, pri_key, g1, u;//b为密文，c为解密后结果 string A;srand((unsigned)time(NULL));cout << "输入要加密的字符串" << endl;getline(cin, A);cout << endl;middle = BigNum(A);cout << "生成明文：";middle.print();cout << endl;p.GeneratePrime(); //产生素数 q.GeneratePrime();while (p == q) //判断两个素数不等 q.GeneratePrime();temp = p - I;t = q - I;t = t * temp;cout << "素数p为："; //输出素数 p.print();cout << endl;cout << "素数q为：";q.print();cout << endl;cout << "公钥n为";n = p * q;BigNum N = n * n;n.print();cout << endl;pri_key = Lcm(p - I, q - I); cout << "私钥pri_key为:";pri_key.print();cout << endl;g.GeneratePrime();cout << "公钥g为";g.print();cout << endl;BigNum middle1, r;r.GeneratePrime();BigNum temp1 = g.power_mod(middle, N), temp2 = r.power_mod(n, N);middle1 = temp1 * temp2;b = middle1.power_mod(I, N);cout << "生成密文如下:";b.print();cout << endl;u.ex_euclid(Lfunction(g.power_mod(pri_key, N)), n);c = Lfunction(b.power_mod(pri_key, N)) * u;c = c.power_mod(I, n);//解密cout << "解密后的结果为:" ;c.print();c.printS();cout << endl;system("pause");return 0;
}BigNum Gcd(BigNum a, BigNum b)
{BigNum i;do{i = a % b;a = b;b = i;} while (b != o);return a;
}BigNum Lcm(BigNum a, BigNum b)
{BigNum i;i = Gcd(a, b);i = a * b / i;return i;
}BigNum Lfunction(BigNum a)
{BigNum i;i = (a - I) / n;return i;
}
代码运行截图
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