| 9400cf6f |
1 | /* |
| 2 | * Prime generation. |
| 3 | */ |
| 4 | |
| 5 | #include "ssh.h" |
| 6 | |
| 7 | /* |
| b73c0b37 |
8 | * This prime generation algorithm is pretty much cribbed from |
| 9 | * OpenSSL. The algorithm is: |
| 10 | * |
| 11 | * - invent a B-bit random number and ensure the top and bottom |
| 12 | * bits are set (so it's definitely B-bit, and it's definitely |
| 13 | * odd) |
| 14 | * |
| 15 | * - see if it's coprime to all primes below 2^16; increment it by |
| 16 | * two until it is (this shouldn't take long in general) |
| 17 | * |
| 18 | * - perform the Miller-Rabin primality test enough times to |
| 19 | * ensure the probability of it being composite is 2^-80 or |
| 20 | * less |
| 21 | * |
| 22 | * - go back to square one if any M-R test fails. |
| 23 | */ |
| 24 | |
| 25 | /* |
| 26 | * The Miller-Rabin primality test is an extension to the Fermat |
| 27 | * test. The Fermat test just checks that a^(n-1) == 1 mod n; this |
| 28 | * is vulnerable to Carmichael numbers. Miller-Rabin makes use of |
| 29 | * the fact that if p is truly prime and a^K == 1 mod p for even K, |
| 30 | * then a^(K/2) must be congruent to either 1 or -1. In Hence, we |
| 31 | * write n-1 as q * 2^k, with odd q, and then we compute a^q, a^2q, |
| 32 | * a^4q, a^8q, ..., a^(n-1) mod n. If n is prime, the last of these |
| 33 | * must be 1, and the last one that _isn't_ 1 must be -1. So we |
| 34 | * expect to see either a^q congruent to 1, or a^q congruent to -1, |
| 35 | * or a^q to become congruent to -1 after squaring at most k-1 |
| 36 | * times. |
| 37 | * |
| 38 | * For example, consider a=2 and n=1729 (a Carmichael number). |
| 39 | * 2^1728 mod 1729 is 1, so the Fermat test would have no problem |
| 40 | * with this. But Miller-Rabin looks at 2^(1728/2), 2^(1728/4), |
| 41 | * ..., 2^(1728/64) as well. Now 2^(1728/64) == 645, 2^(1728/32) == |
| 42 | * 1065, 2^(1728/16) == 1. Hang on! The value before the first 1 |
| 43 | * was 1065, and we expected 1728 (i.e. -1). Guards! Seize this |
| 44 | * impostor. |
| 45 | * |
| 46 | * (It doesn't work for all bases. Try a=932 and n=1729, and even |
| 47 | * the Miller-Rabin test can't tell the difference, because |
| 48 | * 932^(1728/64) is already 1 and so we don't get to see what |
| 49 | * happens before the first 1. But there isn't any class of numbers |
| 50 | * which give false positives on Miller-Rabin for _all_ bases, so |
| 51 | * by trying several bases we probabilistically rule out Carmichael |
| 52 | * numbers as well as everything else composite.) |
| 53 | */ |
| 54 | |
| 55 | /* |
| 9400cf6f |
56 | * The first few odd primes. |
| 57 | * |
| 58 | * import sys |
| 59 | * def sieve(n): |
| 60 | * z = [] |
| 61 | * list = [] |
| 62 | * for i in range(n): z.append(1) |
| 63 | * for i in range(2,n): |
| 64 | * if z[i]: |
| 65 | * list.append(i) |
| 66 | * for j in range(i,n,i): z[j] = 0 |
| 67 | * return list |
| 68 | * list = sieve(65535) |
| 69 | * for i in list[1:]: sys.stdout.write("%d," % i) |
| 70 | */ |
| 71 | static const unsigned short primes[] = { |
| 72 | 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, |
| 73 | 103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193, |
| 74 | 197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293, |
| 75 | 307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409, |
| 76 | 419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521, |
| 77 | 523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641, |
| 78 | 643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757, |
| 79 | 761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881, |
| 80 | 883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009, |
| 81 | 1013,1019,1021,1031,1033,1039,1049,1051,1061,1063,1069,1087,1091,1093, |
| 82 | 1097,1103,1109,1117,1123,1129,1151,1153,1163,1171,1181,1187,1193,1201, |
| 83 | 1213,1217,1223,1229,1231,1237,1249,1259,1277,1279,1283,1289,1291,1297, |
| 84 | 1301,1303,1307,1319,1321,1327,1361,1367,1373,1381,1399,1409,1423,1427, |
| 85 | 1429,1433,1439,1447,1451,1453,1459,1471,1481,1483,1487,1489,1493,1499, |
| 86 | 1511,1523,1531,1543,1549,1553,1559,1567,1571,1579,1583,1597,1601,1607, |
| 87 | 1609,1613,1619,1621,1627,1637,1657,1663,1667,1669,1693,1697,1699,1709, |
| 88 | 1721,1723,1733,1741,1747,1753,1759,1777,1783,1787,1789,1801,1811,1823, |
| 89 | 1831,1847,1861,1867,1871,1873,1877,1879,1889,1901,1907,1913,1931,1933, |
| 90 | 1949,1951,1973,1979,1987,1993,1997,1999,2003,2011,2017,2027,2029,2039, |
| 91 | 2053,2063,2069,2081,2083,2087,2089,2099,2111,2113,2129,2131,2137,2141, |
| 92 | 2143,2153,2161,2179,2203,2207,2213,2221,2237,2239,2243,2251,2267,2269, |
| 93 | 2273,2281,2287,2293,2297,2309,2311,2333,2339,2341,2347,2351,2357,2371, |
| 94 | 2377,2381,2383,2389,2393,2399,2411,2417,2423,2437,2441,2447,2459,2467, |
| 95 | 2473,2477,2503,2521,2531,2539,2543,2549,2551,2557,2579,2591,2593,2609, |
| 96 | 2617,2621,2633,2647,2657,2659,2663,2671,2677,2683,2687,2689,2693,2699, |
| 97 | 2707,2711,2713,2719,2729,2731,2741,2749,2753,2767,2777,2789,2791,2797, |
| 98 | 2801,2803,2819,2833,2837,2843,2851,2857,2861,2879,2887,2897,2903,2909, |
| 99 | 2917,2927,2939,2953,2957,2963,2969,2971,2999,3001,3011,3019,3023,3037, |
| 100 | 3041,3049,3061,3067,3079,3083,3089,3109,3119,3121,3137,3163,3167,3169, |
| 101 | 3181,3187,3191,3203,3209,3217,3221,3229,3251,3253,3257,3259,3271,3299, |
| 102 | 3301,3307,3313,3319,3323,3329,3331,3343,3347,3359,3361,3371,3373,3389, |
| 103 | 3391,3407,3413,3433,3449,3457,3461,3463,3467,3469,3491,3499,3511,3517, |
| 104 | 3527,3529,3533,3539,3541,3547,3557,3559,3571,3581,3583,3593,3607,3613, |
| 105 | 3617,3623,3631,3637,3643,3659,3671,3673,3677,3691,3697,3701,3709,3719, |
| 106 | 3727,3733,3739,3761,3767,3769,3779,3793,3797,3803,3821,3823,3833,3847, |
| 107 | 3851,3853,3863,3877,3881,3889,3907,3911,3917,3919,3923,3929,3931,3943, |
| 108 | 3947,3967,3989,4001,4003,4007,4013,4019,4021,4027,4049,4051,4057,4073, |
| 109 | 4079,4091,4093,4099,4111,4127,4129,4133,4139,4153,4157,4159,4177,4201, |
| 110 | 4211,4217,4219,4229,4231,4241,4243,4253,4259,4261,4271,4273,4283,4289, |
| 111 | 4297,4327,4337,4339,4349,4357,4363,4373,4391,4397,4409,4421,4423,4441, |
| 112 | 4447,4451,4457,4463,4481,4483,4493,4507,4513,4517,4519,4523,4547,4549, |
| 113 | 4561,4567,4583,4591,4597,4603,4621,4637,4639,4643,4649,4651,4657,4663, |
| 114 | 4673,4679,4691,4703,4721,4723,4729,4733,4751,4759,4783,4787,4789,4793, |
| 115 | 4799,4801,4813,4817,4831,4861,4871,4877,4889,4903,4909,4919,4931,4933, |
| 116 | 4937,4943,4951,4957,4967,4969,4973,4987,4993,4999,5003,5009,5011,5021, |
| 117 | 5023,5039,5051,5059,5077,5081,5087,5099,5101,5107,5113,5119,5147,5153, |
| 118 | 5167,5171,5179,5189,5197,5209,5227,5231,5233,5237,5261,5273,5279,5281, |
| 119 | 5297,5303,5309,5323,5333,5347,5351,5381,5387,5393,5399,5407,5413,5417, |
| 120 | 5419,5431,5437,5441,5443,5449,5471,5477,5479,5483,5501,5503,5507,5519, |
| 121 | 5521,5527,5531,5557,5563,5569,5573,5581,5591,5623,5639,5641,5647,5651, |
| 122 | 5653,5657,5659,5669,5683,5689,5693,5701,5711,5717,5737,5741,5743,5749, |
| 123 | 5779,5783,5791,5801,5807,5813,5821,5827,5839,5843,5849,5851,5857,5861, |
| 124 | 5867,5869,5879,5881,5897,5903,5923,5927,5939,5953,5981,5987,6007,6011, |
| 125 | 6029,6037,6043,6047,6053,6067,6073,6079,6089,6091,6101,6113,6121,6131, |
| 126 | 6133,6143,6151,6163,6173,6197,6199,6203,6211,6217,6221,6229,6247,6257, |
| 127 | 6263,6269,6271,6277,6287,6299,6301,6311,6317,6323,6329,6337,6343,6353, |
| 128 | 6359,6361,6367,6373,6379,6389,6397,6421,6427,6449,6451,6469,6473,6481, |
| 129 | 6491,6521,6529,6547,6551,6553,6563,6569,6571,6577,6581,6599,6607,6619, |
| 130 | 6637,6653,6659,6661,6673,6679,6689,6691,6701,6703,6709,6719,6733,6737, |
| 131 | 6761,6763,6779,6781,6791,6793,6803,6823,6827,6829,6833,6841,6857,6863, |
| 132 | 6869,6871,6883,6899,6907,6911,6917,6947,6949,6959,6961,6967,6971,6977, |
| 133 | 6983,6991,6997,7001,7013,7019,7027,7039,7043,7057,7069,7079,7103,7109, |
| 134 | 7121,7127,7129,7151,7159,7177,7187,7193,7207,7211,7213,7219,7229,7237, |
| 135 | 7243,7247,7253,7283,7297,7307,7309,7321,7331,7333,7349,7351,7369,7393, |
| 136 | 7411,7417,7433,7451,7457,7459,7477,7481,7487,7489,7499,7507,7517,7523, |
| 137 | 7529,7537,7541,7547,7549,7559,7561,7573,7577,7583,7589,7591,7603,7607, |
| 138 | 7621,7639,7643,7649,7669,7673,7681,7687,7691,7699,7703,7717,7723,7727, |
| 139 | 7741,7753,7757,7759,7789,7793,7817,7823,7829,7841,7853,7867,7873,7877, |
| 140 | 7879,7883,7901,7907,7919,7927,7933,7937,7949,7951,7963,7993,8009,8011, |
| 141 | 8017,8039,8053,8059,8069,8081,8087,8089,8093,8101,8111,8117,8123,8147, |
| 142 | 8161,8167,8171,8179,8191,8209,8219,8221,8231,8233,8237,8243,8263,8269, |
| 143 | 8273,8287,8291,8293,8297,8311,8317,8329,8353,8363,8369,8377,8387,8389, |
| 144 | 8419,8423,8429,8431,8443,8447,8461,8467,8501,8513,8521,8527,8537,8539, |
| 145 | 8543,8563,8573,8581,8597,8599,8609,8623,8627,8629,8641,8647,8663,8669, |
| 146 | 8677,8681,8689,8693,8699,8707,8713,8719,8731,8737,8741,8747,8753,8761, |
| 147 | 8779,8783,8803,8807,8819,8821,8831,8837,8839,8849,8861,8863,8867,8887, |
| 148 | 8893,8923,8929,8933,8941,8951,8963,8969,8971,8999,9001,9007,9011,9013, |
| 149 | 9029,9041,9043,9049,9059,9067,9091,9103,9109,9127,9133,9137,9151,9157, |
| 150 | 9161,9173,9181,9187,9199,9203,9209,9221,9227,9239,9241,9257,9277,9281, |
| 151 | 9283,9293,9311,9319,9323,9337,9341,9343,9349,9371,9377,9391,9397,9403, |
| 152 | 9413,9419,9421,9431,9433,9437,9439,9461,9463,9467,9473,9479,9491,9497, |
| 153 | 9511,9521,9533,9539,9547,9551,9587,9601,9613,9619,9623,9629,9631,9643, |
| 154 | 9649,9661,9677,9679,9689,9697,9719,9721,9733,9739,9743,9749,9767,9769, |
| 155 | 9781,9787,9791,9803,9811,9817,9829,9833,9839,9851,9857,9859,9871,9883, |
| 156 | 9887,9901,9907,9923,9929,9931,9941,9949,9967,9973,10007,10009,10037, |
| 157 | 10039,10061,10067,10069,10079,10091,10093,10099,10103,10111,10133,10139, |
| 158 | 10141,10151,10159,10163,10169,10177,10181,10193,10211,10223,10243,10247, |
| 159 | 10253,10259,10267,10271,10273,10289,10301,10303,10313,10321,10331,10333, |
| 160 | 10337,10343,10357,10369,10391,10399,10427,10429,10433,10453,10457,10459, |
| 161 | 10463,10477,10487,10499,10501,10513,10529,10531,10559,10567,10589,10597, |
| 162 | 10601,10607,10613,10627,10631,10639,10651,10657,10663,10667,10687,10691, |
| 163 | 10709,10711,10723,10729,10733,10739,10753,10771,10781,10789,10799,10831, |
| 164 | 10837,10847,10853,10859,10861,10867,10883,10889,10891,10903,10909,10937, |
| 165 | 10939,10949,10957,10973,10979,10987,10993,11003,11027,11047,11057,11059, |
| 166 | 11069,11071,11083,11087,11093,11113,11117,11119,11131,11149,11159,11161, |
| 167 | 11171,11173,11177,11197,11213,11239,11243,11251,11257,11261,11273,11279, |
| 168 | 11287,11299,11311,11317,11321,11329,11351,11353,11369,11383,11393,11399, |
| 169 | 11411,11423,11437,11443,11447,11467,11471,11483,11489,11491,11497,11503, |
| 170 | 11519,11527,11549,11551,11579,11587,11593,11597,11617,11621,11633,11657, |
| 171 | 11677,11681,11689,11699,11701,11717,11719,11731,11743,11777,11779,11783, |
| 172 | 11789,11801,11807,11813,11821,11827,11831,11833,11839,11863,11867,11887, |
| 173 | 11897,11903,11909,11923,11927,11933,11939,11941,11953,11959,11969,11971, |
| 174 | 11981,11987,12007,12011,12037,12041,12043,12049,12071,12073,12097,12101, |
| 175 | 12107,12109,12113,12119,12143,12149,12157,12161,12163,12197,12203,12211, |
| 176 | 12227,12239,12241,12251,12253,12263,12269,12277,12281,12289,12301,12323, |
| 177 | 12329,12343,12347,12373,12377,12379,12391,12401,12409,12413,12421,12433, |
| 178 | 12437,12451,12457,12473,12479,12487,12491,12497,12503,12511,12517,12527, |
| 179 | 12539,12541,12547,12553,12569,12577,12583,12589,12601,12611,12613,12619, |
| 180 | 12637,12641,12647,12653,12659,12671,12689,12697,12703,12713,12721,12739, |
| 181 | 12743,12757,12763,12781,12791,12799,12809,12821,12823,12829,12841,12853, |
| 182 | 12889,12893,12899,12907,12911,12917,12919,12923,12941,12953,12959,12967, |
| 183 | 12973,12979,12983,13001,13003,13007,13009,13033,13037,13043,13049,13063, |
| 184 | 13093,13099,13103,13109,13121,13127,13147,13151,13159,13163,13171,13177, |
| 185 | 13183,13187,13217,13219,13229,13241,13249,13259,13267,13291,13297,13309, |
| 186 | 13313,13327,13331,13337,13339,13367,13381,13397,13399,13411,13417,13421, |
| 187 | 13441,13451,13457,13463,13469,13477,13487,13499,13513,13523,13537,13553, |
| 188 | 13567,13577,13591,13597,13613,13619,13627,13633,13649,13669,13679,13681, |
| 189 | 13687,13691,13693,13697,13709,13711,13721,13723,13729,13751,13757,13759, |
| 190 | 13763,13781,13789,13799,13807,13829,13831,13841,13859,13873,13877,13879, |
| 191 | 13883,13901,13903,13907,13913,13921,13931,13933,13963,13967,13997,13999, |
| 192 | 14009,14011,14029,14033,14051,14057,14071,14081,14083,14087,14107,14143, |
| 193 | 14149,14153,14159,14173,14177,14197,14207,14221,14243,14249,14251,14281, |
| 194 | 14293,14303,14321,14323,14327,14341,14347,14369,14387,14389,14401,14407, |
| 195 | 14411,14419,14423,14431,14437,14447,14449,14461,14479,14489,14503,14519, |
| 196 | 14533,14537,14543,14549,14551,14557,14561,14563,14591,14593,14621,14627, |
| 197 | 14629,14633,14639,14653,14657,14669,14683,14699,14713,14717,14723,14731, |
| 198 | 14737,14741,14747,14753,14759,14767,14771,14779,14783,14797,14813,14821, |
| 199 | 14827,14831,14843,14851,14867,14869,14879,14887,14891,14897,14923,14929, |
| 200 | 14939,14947,14951,14957,14969,14983,15013,15017,15031,15053,15061,15073, |
| 201 | 15077,15083,15091,15101,15107,15121,15131,15137,15139,15149,15161,15173, |
| 202 | 15187,15193,15199,15217,15227,15233,15241,15259,15263,15269,15271,15277, |
| 203 | 15287,15289,15299,15307,15313,15319,15329,15331,15349,15359,15361,15373, |
| 204 | 15377,15383,15391,15401,15413,15427,15439,15443,15451,15461,15467,15473, |
| 205 | 15493,15497,15511,15527,15541,15551,15559,15569,15581,15583,15601,15607, |
| 206 | 15619,15629,15641,15643,15647,15649,15661,15667,15671,15679,15683,15727, |
| 207 | 15731,15733,15737,15739,15749,15761,15767,15773,15787,15791,15797,15803, |
| 208 | 15809,15817,15823,15859,15877,15881,15887,15889,15901,15907,15913,15919, |
| 209 | 15923,15937,15959,15971,15973,15991,16001,16007,16033,16057,16061,16063, |
| 210 | 16067,16069,16073,16087,16091,16097,16103,16111,16127,16139,16141,16183, |
| 211 | 16187,16189,16193,16217,16223,16229,16231,16249,16253,16267,16273,16301, |
| 212 | 16319,16333,16339,16349,16361,16363,16369,16381,16411,16417,16421,16427, |
| 213 | 16433,16447,16451,16453,16477,16481,16487,16493,16519,16529,16547,16553, |
| 214 | 16561,16567,16573,16603,16607,16619,16631,16633,16649,16651,16657,16661, |
| 215 | 16673,16691,16693,16699,16703,16729,16741,16747,16759,16763,16787,16811, |
| 216 | 16823,16829,16831,16843,16871,16879,16883,16889,16901,16903,16921,16927, |
| 217 | 16931,16937,16943,16963,16979,16981,16987,16993,17011,17021,17027,17029, |
| 218 | 17033,17041,17047,17053,17077,17093,17099,17107,17117,17123,17137,17159, |
| 219 | 17167,17183,17189,17191,17203,17207,17209,17231,17239,17257,17291,17293, |
| 220 | 17299,17317,17321,17327,17333,17341,17351,17359,17377,17383,17387,17389, |
| 221 | 17393,17401,17417,17419,17431,17443,17449,17467,17471,17477,17483,17489, |
| 222 | 17491,17497,17509,17519,17539,17551,17569,17573,17579,17581,17597,17599, |
| 223 | 17609,17623,17627,17657,17659,17669,17681,17683,17707,17713,17729,17737, |
| 224 | 17747,17749,17761,17783,17789,17791,17807,17827,17837,17839,17851,17863, |
| 225 | 17881,17891,17903,17909,17911,17921,17923,17929,17939,17957,17959,17971, |
| 226 | 17977,17981,17987,17989,18013,18041,18043,18047,18049,18059,18061,18077, |
| 227 | 18089,18097,18119,18121,18127,18131,18133,18143,18149,18169,18181,18191, |
| 228 | 18199,18211,18217,18223,18229,18233,18251,18253,18257,18269,18287,18289, |
| 229 | 18301,18307,18311,18313,18329,18341,18353,18367,18371,18379,18397,18401, |
| 230 | 18413,18427,18433,18439,18443,18451,18457,18461,18481,18493,18503,18517, |
| 231 | 18521,18523,18539,18541,18553,18583,18587,18593,18617,18637,18661,18671, |
| 232 | 18679,18691,18701,18713,18719,18731,18743,18749,18757,18773,18787,18793, |
| 233 | 18797,18803,18839,18859,18869,18899,18911,18913,18917,18919,18947,18959, |
| 234 | 18973,18979,19001,19009,19013,19031,19037,19051,19069,19073,19079,19081, |
| 235 | 19087,19121,19139,19141,19157,19163,19181,19183,19207,19211,19213,19219, |
| 236 | 19231,19237,19249,19259,19267,19273,19289,19301,19309,19319,19333,19373, |
| 237 | 19379,19381,19387,19391,19403,19417,19421,19423,19427,19429,19433,19441, |
| 238 | 19447,19457,19463,19469,19471,19477,19483,19489,19501,19507,19531,19541, |
| 239 | 19543,19553,19559,19571,19577,19583,19597,19603,19609,19661,19681,19687, |
| 240 | 19697,19699,19709,19717,19727,19739,19751,19753,19759,19763,19777,19793, |
| 241 | 19801,19813,19819,19841,19843,19853,19861,19867,19889,19891,19913,19919, |
| 242 | 19927,19937,19949,19961,19963,19973,19979,19991,19993,19997,20011,20021, |
| 243 | 20023,20029,20047,20051,20063,20071,20089,20101,20107,20113,20117,20123, |
| 244 | 20129,20143,20147,20149,20161,20173,20177,20183,20201,20219,20231,20233, |
| 245 | 20249,20261,20269,20287,20297,20323,20327,20333,20341,20347,20353,20357, |
| 246 | 20359,20369,20389,20393,20399,20407,20411,20431,20441,20443,20477,20479, |
| 247 | 20483,20507,20509,20521,20533,20543,20549,20551,20563,20593,20599,20611, |
| 248 | 20627,20639,20641,20663,20681,20693,20707,20717,20719,20731,20743,20747, |
| 249 | 20749,20753,20759,20771,20773,20789,20807,20809,20849,20857,20873,20879, |
| 250 | 20887,20897,20899,20903,20921,20929,20939,20947,20959,20963,20981,20983, |
| 251 | 21001,21011,21013,21017,21019,21023,21031,21059,21061,21067,21089,21101, |
| 252 | 21107,21121,21139,21143,21149,21157,21163,21169,21179,21187,21191,21193, |
| 253 | 21211,21221,21227,21247,21269,21277,21283,21313,21317,21319,21323,21341, |
| 254 | 21347,21377,21379,21383,21391,21397,21401,21407,21419,21433,21467,21481, |
| 255 | 21487,21491,21493,21499,21503,21517,21521,21523,21529,21557,21559,21563, |
| 256 | 21569,21577,21587,21589,21599,21601,21611,21613,21617,21647,21649,21661, |
| 257 | 21673,21683,21701,21713,21727,21737,21739,21751,21757,21767,21773,21787, |
| 258 | 21799,21803,21817,21821,21839,21841,21851,21859,21863,21871,21881,21893, |
| 259 | 21911,21929,21937,21943,21961,21977,21991,21997,22003,22013,22027,22031, |
| 260 | 22037,22039,22051,22063,22067,22073,22079,22091,22093,22109,22111,22123, |
| 261 | 22129,22133,22147,22153,22157,22159,22171,22189,22193,22229,22247,22259, |
| 262 | 22271,22273,22277,22279,22283,22291,22303,22307,22343,22349,22367,22369, |
| 263 | 22381,22391,22397,22409,22433,22441,22447,22453,22469,22481,22483,22501, |
| 264 | 22511,22531,22541,22543,22549,22567,22571,22573,22613,22619,22621,22637, |
| 265 | 22639,22643,22651,22669,22679,22691,22697,22699,22709,22717,22721,22727, |
| 266 | 22739,22741,22751,22769,22777,22783,22787,22807,22811,22817,22853,22859, |
| 267 | 22861,22871,22877,22901,22907,22921,22937,22943,22961,22963,22973,22993, |
| 268 | 23003,23011,23017,23021,23027,23029,23039,23041,23053,23057,23059,23063, |
| 269 | 23071,23081,23087,23099,23117,23131,23143,23159,23167,23173,23189,23197, |
| 270 | 23201,23203,23209,23227,23251,23269,23279,23291,23293,23297,23311,23321, |
| 271 | 23327,23333,23339,23357,23369,23371,23399,23417,23431,23447,23459,23473, |
| 272 | 23497,23509,23531,23537,23539,23549,23557,23561,23563,23567,23581,23593, |
| 273 | 23599,23603,23609,23623,23627,23629,23633,23663,23669,23671,23677,23687, |
| 274 | 23689,23719,23741,23743,23747,23753,23761,23767,23773,23789,23801,23813, |
| 275 | 23819,23827,23831,23833,23857,23869,23873,23879,23887,23893,23899,23909, |
| 276 | 23911,23917,23929,23957,23971,23977,23981,23993,24001,24007,24019,24023, |
| 277 | 24029,24043,24049,24061,24071,24077,24083,24091,24097,24103,24107,24109, |
| 278 | 24113,24121,24133,24137,24151,24169,24179,24181,24197,24203,24223,24229, |
| 279 | 24239,24247,24251,24281,24317,24329,24337,24359,24371,24373,24379,24391, |
| 280 | 24407,24413,24419,24421,24439,24443,24469,24473,24481,24499,24509,24517, |
| 281 | 24527,24533,24547,24551,24571,24593,24611,24623,24631,24659,24671,24677, |
| 282 | 24683,24691,24697,24709,24733,24749,24763,24767,24781,24793,24799,24809, |
| 283 | 24821,24841,24847,24851,24859,24877,24889,24907,24917,24919,24923,24943, |
| 284 | 24953,24967,24971,24977,24979,24989,25013,25031,25033,25037,25057,25073, |
| 285 | 25087,25097,25111,25117,25121,25127,25147,25153,25163,25169,25171,25183, |
| 286 | 25189,25219,25229,25237,25243,25247,25253,25261,25301,25303,25307,25309, |
| 287 | 25321,25339,25343,25349,25357,25367,25373,25391,25409,25411,25423,25439, |
| 288 | 25447,25453,25457,25463,25469,25471,25523,25537,25541,25561,25577,25579, |
| 289 | 25583,25589,25601,25603,25609,25621,25633,25639,25643,25657,25667,25673, |
| 290 | 25679,25693,25703,25717,25733,25741,25747,25759,25763,25771,25793,25799, |
| 291 | 25801,25819,25841,25847,25849,25867,25873,25889,25903,25913,25919,25931, |
| 292 | 25933,25939,25943,25951,25969,25981,25997,25999,26003,26017,26021,26029, |
| 293 | 26041,26053,26083,26099,26107,26111,26113,26119,26141,26153,26161,26171, |
| 294 | 26177,26183,26189,26203,26209,26227,26237,26249,26251,26261,26263,26267, |
| 295 | 26293,26297,26309,26317,26321,26339,26347,26357,26371,26387,26393,26399, |
| 296 | 26407,26417,26423,26431,26437,26449,26459,26479,26489,26497,26501,26513, |
| 297 | 26539,26557,26561,26573,26591,26597,26627,26633,26641,26647,26669,26681, |
| 298 | 26683,26687,26693,26699,26701,26711,26713,26717,26723,26729,26731,26737, |
| 299 | 26759,26777,26783,26801,26813,26821,26833,26839,26849,26861,26863,26879, |
| 300 | 26881,26891,26893,26903,26921,26927,26947,26951,26953,26959,26981,26987, |
| 301 | 26993,27011,27017,27031,27043,27059,27061,27067,27073,27077,27091,27103, |
| 302 | 27107,27109,27127,27143,27179,27191,27197,27211,27239,27241,27253,27259, |
| 303 | 27271,27277,27281,27283,27299,27329,27337,27361,27367,27397,27407,27409, |
| 304 | 27427,27431,27437,27449,27457,27479,27481,27487,27509,27527,27529,27539, |
| 305 | 27541,27551,27581,27583,27611,27617,27631,27647,27653,27673,27689,27691, |
| 306 | 27697,27701,27733,27737,27739,27743,27749,27751,27763,27767,27773,27779, |
| 307 | 27791,27793,27799,27803,27809,27817,27823,27827,27847,27851,27883,27893, |
| 308 | 27901,27917,27919,27941,27943,27947,27953,27961,27967,27983,27997,28001, |
| 309 | 28019,28027,28031,28051,28057,28069,28081,28087,28097,28099,28109,28111, |
| 310 | 28123,28151,28163,28181,28183,28201,28211,28219,28229,28277,28279,28283, |
| 311 | 28289,28297,28307,28309,28319,28349,28351,28387,28393,28403,28409,28411, |
| 312 | 28429,28433,28439,28447,28463,28477,28493,28499,28513,28517,28537,28541, |
| 313 | 28547,28549,28559,28571,28573,28579,28591,28597,28603,28607,28619,28621, |
| 314 | 28627,28631,28643,28649,28657,28661,28663,28669,28687,28697,28703,28711, |
| 315 | 28723,28729,28751,28753,28759,28771,28789,28793,28807,28813,28817,28837, |
| 316 | 28843,28859,28867,28871,28879,28901,28909,28921,28927,28933,28949,28961, |
| 317 | 28979,29009,29017,29021,29023,29027,29033,29059,29063,29077,29101,29123, |
| 318 | 29129,29131,29137,29147,29153,29167,29173,29179,29191,29201,29207,29209, |
| 319 | 29221,29231,29243,29251,29269,29287,29297,29303,29311,29327,29333,29339, |
| 320 | 29347,29363,29383,29387,29389,29399,29401,29411,29423,29429,29437,29443, |
| 321 | 29453,29473,29483,29501,29527,29531,29537,29567,29569,29573,29581,29587, |
| 322 | 29599,29611,29629,29633,29641,29663,29669,29671,29683,29717,29723,29741, |
| 323 | 29753,29759,29761,29789,29803,29819,29833,29837,29851,29863,29867,29873, |
| 324 | 29879,29881,29917,29921,29927,29947,29959,29983,29989,30011,30013,30029, |
| 325 | 30047,30059,30071,30089,30091,30097,30103,30109,30113,30119,30133,30137, |
| 326 | 30139,30161,30169,30181,30187,30197,30203,30211,30223,30241,30253,30259, |
| 327 | 30269,30271,30293,30307,30313,30319,30323,30341,30347,30367,30389,30391, |
| 328 | 30403,30427,30431,30449,30467,30469,30491,30493,30497,30509,30517,30529, |
| 329 | 30539,30553,30557,30559,30577,30593,30631,30637,30643,30649,30661,30671, |
| 330 | 30677,30689,30697,30703,30707,30713,30727,30757,30763,30773,30781,30803, |
| 331 | 30809,30817,30829,30839,30841,30851,30853,30859,30869,30871,30881,30893, |
| 332 | 30911,30931,30937,30941,30949,30971,30977,30983,31013,31019,31033,31039, |
| 333 | 31051,31063,31069,31079,31081,31091,31121,31123,31139,31147,31151,31153, |
| 334 | 31159,31177,31181,31183,31189,31193,31219,31223,31231,31237,31247,31249, |
| 335 | 31253,31259,31267,31271,31277,31307,31319,31321,31327,31333,31337,31357, |
| 336 | 31379,31387,31391,31393,31397,31469,31477,31481,31489,31511,31513,31517, |
| 337 | 31531,31541,31543,31547,31567,31573,31583,31601,31607,31627,31643,31649, |
| 338 | 31657,31663,31667,31687,31699,31721,31723,31727,31729,31741,31751,31769, |
| 339 | 31771,31793,31799,31817,31847,31849,31859,31873,31883,31891,31907,31957, |
| 340 | 31963,31973,31981,31991,32003,32009,32027,32029,32051,32057,32059,32063, |
| 341 | 32069,32077,32083,32089,32099,32117,32119,32141,32143,32159,32173,32183, |
| 342 | 32189,32191,32203,32213,32233,32237,32251,32257,32261,32297,32299,32303, |
| 343 | 32309,32321,32323,32327,32341,32353,32359,32363,32369,32371,32377,32381, |
| 344 | 32401,32411,32413,32423,32429,32441,32443,32467,32479,32491,32497,32503, |
| 345 | 32507,32531,32533,32537,32561,32563,32569,32573,32579,32587,32603,32609, |
| 346 | 32611,32621,32633,32647,32653,32687,32693,32707,32713,32717,32719,32749, |
| 347 | 32771,32779,32783,32789,32797,32801,32803,32831,32833,32839,32843,32869, |
| 348 | 32887,32909,32911,32917,32933,32939,32941,32957,32969,32971,32983,32987, |
| 349 | 32993,32999,33013,33023,33029,33037,33049,33053,33071,33073,33083,33091, |
| 350 | 33107,33113,33119,33149,33151,33161,33179,33181,33191,33199,33203,33211, |
| 351 | 33223,33247,33287,33289,33301,33311,33317,33329,33331,33343,33347,33349, |
| 352 | 33353,33359,33377,33391,33403,33409,33413,33427,33457,33461,33469,33479, |
| 353 | 33487,33493,33503,33521,33529,33533,33547,33563,33569,33577,33581,33587, |
| 354 | 33589,33599,33601,33613,33617,33619,33623,33629,33637,33641,33647,33679, |
| 355 | 33703,33713,33721,33739,33749,33751,33757,33767,33769,33773,33791,33797, |
| 356 | 33809,33811,33827,33829,33851,33857,33863,33871,33889,33893,33911,33923, |
| 357 | 33931,33937,33941,33961,33967,33997,34019,34031,34033,34039,34057,34061, |
| 358 | 34123,34127,34129,34141,34147,34157,34159,34171,34183,34211,34213,34217, |
| 359 | 34231,34253,34259,34261,34267,34273,34283,34297,34301,34303,34313,34319, |
| 360 | 34327,34337,34351,34361,34367,34369,34381,34403,34421,34429,34439,34457, |
| 361 | 34469,34471,34483,34487,34499,34501,34511,34513,34519,34537,34543,34549, |
| 362 | 34583,34589,34591,34603,34607,34613,34631,34649,34651,34667,34673,34679, |
| 363 | 34687,34693,34703,34721,34729,34739,34747,34757,34759,34763,34781,34807, |
| 364 | 34819,34841,34843,34847,34849,34871,34877,34883,34897,34913,34919,34939, |
| 365 | 34949,34961,34963,34981,35023,35027,35051,35053,35059,35069,35081,35083, |
| 366 | 35089,35099,35107,35111,35117,35129,35141,35149,35153,35159,35171,35201, |
| 367 | 35221,35227,35251,35257,35267,35279,35281,35291,35311,35317,35323,35327, |
| 368 | 35339,35353,35363,35381,35393,35401,35407,35419,35423,35437,35447,35449, |
| 369 | 35461,35491,35507,35509,35521,35527,35531,35533,35537,35543,35569,35573, |
| 370 | 35591,35593,35597,35603,35617,35671,35677,35729,35731,35747,35753,35759, |
| 371 | 35771,35797,35801,35803,35809,35831,35837,35839,35851,35863,35869,35879, |
| 372 | 35897,35899,35911,35923,35933,35951,35963,35969,35977,35983,35993,35999, |
| 373 | 36007,36011,36013,36017,36037,36061,36067,36073,36083,36097,36107,36109, |
| 374 | 36131,36137,36151,36161,36187,36191,36209,36217,36229,36241,36251,36263, |
| 375 | 36269,36277,36293,36299,36307,36313,36319,36341,36343,36353,36373,36383, |
| 376 | 36389,36433,36451,36457,36467,36469,36473,36479,36493,36497,36523,36527, |
| 377 | 36529,36541,36551,36559,36563,36571,36583,36587,36599,36607,36629,36637, |
| 378 | 36643,36653,36671,36677,36683,36691,36697,36709,36713,36721,36739,36749, |
| 379 | 36761,36767,36779,36781,36787,36791,36793,36809,36821,36833,36847,36857, |
| 380 | 36871,36877,36887,36899,36901,36913,36919,36923,36929,36931,36943,36947, |
| 381 | 36973,36979,36997,37003,37013,37019,37021,37039,37049,37057,37061,37087, |
| 382 | 37097,37117,37123,37139,37159,37171,37181,37189,37199,37201,37217,37223, |
| 383 | 37243,37253,37273,37277,37307,37309,37313,37321,37337,37339,37357,37361, |
| 384 | 37363,37369,37379,37397,37409,37423,37441,37447,37463,37483,37489,37493, |
| 385 | 37501,37507,37511,37517,37529,37537,37547,37549,37561,37567,37571,37573, |
| 386 | 37579,37589,37591,37607,37619,37633,37643,37649,37657,37663,37691,37693, |
| 387 | 37699,37717,37747,37781,37783,37799,37811,37813,37831,37847,37853,37861, |
| 388 | 37871,37879,37889,37897,37907,37951,37957,37963,37967,37987,37991,37993, |
| 389 | 37997,38011,38039,38047,38053,38069,38083,38113,38119,38149,38153,38167, |
| 390 | 38177,38183,38189,38197,38201,38219,38231,38237,38239,38261,38273,38281, |
| 391 | 38287,38299,38303,38317,38321,38327,38329,38333,38351,38371,38377,38393, |
| 392 | 38431,38447,38449,38453,38459,38461,38501,38543,38557,38561,38567,38569, |
| 393 | 38593,38603,38609,38611,38629,38639,38651,38653,38669,38671,38677,38693, |
| 394 | 38699,38707,38711,38713,38723,38729,38737,38747,38749,38767,38783,38791, |
| 395 | 38803,38821,38833,38839,38851,38861,38867,38873,38891,38903,38917,38921, |
| 396 | 38923,38933,38953,38959,38971,38977,38993,39019,39023,39041,39043,39047, |
| 397 | 39079,39089,39097,39103,39107,39113,39119,39133,39139,39157,39161,39163, |
| 398 | 39181,39191,39199,39209,39217,39227,39229,39233,39239,39241,39251,39293, |
| 399 | 39301,39313,39317,39323,39341,39343,39359,39367,39371,39373,39383,39397, |
| 400 | 39409,39419,39439,39443,39451,39461,39499,39503,39509,39511,39521,39541, |
| 401 | 39551,39563,39569,39581,39607,39619,39623,39631,39659,39667,39671,39679, |
| 402 | 39703,39709,39719,39727,39733,39749,39761,39769,39779,39791,39799,39821, |
| 403 | 39827,39829,39839,39841,39847,39857,39863,39869,39877,39883,39887,39901, |
| 404 | 39929,39937,39953,39971,39979,39983,39989,40009,40013,40031,40037,40039, |
| 405 | 40063,40087,40093,40099,40111,40123,40127,40129,40151,40153,40163,40169, |
| 406 | 40177,40189,40193,40213,40231,40237,40241,40253,40277,40283,40289,40343, |
| 407 | 40351,40357,40361,40387,40423,40427,40429,40433,40459,40471,40483,40487, |
| 408 | 40493,40499,40507,40519,40529,40531,40543,40559,40577,40583,40591,40597, |
| 409 | 40609,40627,40637,40639,40693,40697,40699,40709,40739,40751,40759,40763, |
| 410 | 40771,40787,40801,40813,40819,40823,40829,40841,40847,40849,40853,40867, |
| 411 | 40879,40883,40897,40903,40927,40933,40939,40949,40961,40973,40993,41011, |
| 412 | 41017,41023,41039,41047,41051,41057,41077,41081,41113,41117,41131,41141, |
| 413 | 41143,41149,41161,41177,41179,41183,41189,41201,41203,41213,41221,41227, |
| 414 | 41231,41233,41243,41257,41263,41269,41281,41299,41333,41341,41351,41357, |
| 415 | 41381,41387,41389,41399,41411,41413,41443,41453,41467,41479,41491,41507, |
| 416 | 41513,41519,41521,41539,41543,41549,41579,41593,41597,41603,41609,41611, |
| 417 | 41617,41621,41627,41641,41647,41651,41659,41669,41681,41687,41719,41729, |
| 418 | 41737,41759,41761,41771,41777,41801,41809,41813,41843,41849,41851,41863, |
| 419 | 41879,41887,41893,41897,41903,41911,41927,41941,41947,41953,41957,41959, |
| 420 | 41969,41981,41983,41999,42013,42017,42019,42023,42043,42061,42071,42073, |
| 421 | 42083,42089,42101,42131,42139,42157,42169,42179,42181,42187,42193,42197, |
| 422 | 42209,42221,42223,42227,42239,42257,42281,42283,42293,42299,42307,42323, |
| 423 | 42331,42337,42349,42359,42373,42379,42391,42397,42403,42407,42409,42433, |
| 424 | 42437,42443,42451,42457,42461,42463,42467,42473,42487,42491,42499,42509, |
| 425 | 42533,42557,42569,42571,42577,42589,42611,42641,42643,42649,42667,42677, |
| 426 | 42683,42689,42697,42701,42703,42709,42719,42727,42737,42743,42751,42767, |
| 427 | 42773,42787,42793,42797,42821,42829,42839,42841,42853,42859,42863,42899, |
| 428 | 42901,42923,42929,42937,42943,42953,42961,42967,42979,42989,43003,43013, |
| 429 | 43019,43037,43049,43051,43063,43067,43093,43103,43117,43133,43151,43159, |
| 430 | 43177,43189,43201,43207,43223,43237,43261,43271,43283,43291,43313,43319, |
| 431 | 43321,43331,43391,43397,43399,43403,43411,43427,43441,43451,43457,43481, |
| 432 | 43487,43499,43517,43541,43543,43573,43577,43579,43591,43597,43607,43609, |
| 433 | 43613,43627,43633,43649,43651,43661,43669,43691,43711,43717,43721,43753, |
| 434 | 43759,43777,43781,43783,43787,43789,43793,43801,43853,43867,43889,43891, |
| 435 | 43913,43933,43943,43951,43961,43963,43969,43973,43987,43991,43997,44017, |
| 436 | 44021,44027,44029,44041,44053,44059,44071,44087,44089,44101,44111,44119, |
| 437 | 44123,44129,44131,44159,44171,44179,44189,44201,44203,44207,44221,44249, |
| 438 | 44257,44263,44267,44269,44273,44279,44281,44293,44351,44357,44371,44381, |
| 439 | 44383,44389,44417,44449,44453,44483,44491,44497,44501,44507,44519,44531, |
| 440 | 44533,44537,44543,44549,44563,44579,44587,44617,44621,44623,44633,44641, |
| 441 | 44647,44651,44657,44683,44687,44699,44701,44711,44729,44741,44753,44771, |
| 442 | 44773,44777,44789,44797,44809,44819,44839,44843,44851,44867,44879,44887, |
| 443 | 44893,44909,44917,44927,44939,44953,44959,44963,44971,44983,44987,45007, |
| 444 | 45013,45053,45061,45077,45083,45119,45121,45127,45131,45137,45139,45161, |
| 445 | 45179,45181,45191,45197,45233,45247,45259,45263,45281,45289,45293,45307, |
| 446 | 45317,45319,45329,45337,45341,45343,45361,45377,45389,45403,45413,45427, |
| 447 | 45433,45439,45481,45491,45497,45503,45523,45533,45541,45553,45557,45569, |
| 448 | 45587,45589,45599,45613,45631,45641,45659,45667,45673,45677,45691,45697, |
| 449 | 45707,45737,45751,45757,45763,45767,45779,45817,45821,45823,45827,45833, |
| 450 | 45841,45853,45863,45869,45887,45893,45943,45949,45953,45959,45971,45979, |
| 451 | 45989,46021,46027,46049,46051,46061,46073,46091,46093,46099,46103,46133, |
| 452 | 46141,46147,46153,46171,46181,46183,46187,46199,46219,46229,46237,46261, |
| 453 | 46271,46273,46279,46301,46307,46309,46327,46337,46349,46351,46381,46399, |
| 454 | 46411,46439,46441,46447,46451,46457,46471,46477,46489,46499,46507,46511, |
| 455 | 46523,46549,46559,46567,46573,46589,46591,46601,46619,46633,46639,46643, |
| 456 | 46649,46663,46679,46681,46687,46691,46703,46723,46727,46747,46751,46757, |
| 457 | 46769,46771,46807,46811,46817,46819,46829,46831,46853,46861,46867,46877, |
| 458 | 46889,46901,46919,46933,46957,46993,46997,47017,47041,47051,47057,47059, |
| 459 | 47087,47093,47111,47119,47123,47129,47137,47143,47147,47149,47161,47189, |
| 460 | 47207,47221,47237,47251,47269,47279,47287,47293,47297,47303,47309,47317, |
| 461 | 47339,47351,47353,47363,47381,47387,47389,47407,47417,47419,47431,47441, |
| 462 | 47459,47491,47497,47501,47507,47513,47521,47527,47533,47543,47563,47569, |
| 463 | 47581,47591,47599,47609,47623,47629,47639,47653,47657,47659,47681,47699, |
| 464 | 47701,47711,47713,47717,47737,47741,47743,47777,47779,47791,47797,47807, |
| 465 | 47809,47819,47837,47843,47857,47869,47881,47903,47911,47917,47933,47939, |
| 466 | 47947,47951,47963,47969,47977,47981,48017,48023,48029,48049,48073,48079, |
| 467 | 48091,48109,48119,48121,48131,48157,48163,48179,48187,48193,48197,48221, |
| 468 | 48239,48247,48259,48271,48281,48299,48311,48313,48337,48341,48353,48371, |
| 469 | 48383,48397,48407,48409,48413,48437,48449,48463,48473,48479,48481,48487, |
| 470 | 48491,48497,48523,48527,48533,48539,48541,48563,48571,48589,48593,48611, |
| 471 | 48619,48623,48647,48649,48661,48673,48677,48679,48731,48733,48751,48757, |
| 472 | 48761,48767,48779,48781,48787,48799,48809,48817,48821,48823,48847,48857, |
| 473 | 48859,48869,48871,48883,48889,48907,48947,48953,48973,48989,48991,49003, |
| 474 | 49009,49019,49031,49033,49037,49043,49057,49069,49081,49103,49109,49117, |
| 475 | 49121,49123,49139,49157,49169,49171,49177,49193,49199,49201,49207,49211, |
| 476 | 49223,49253,49261,49277,49279,49297,49307,49331,49333,49339,49363,49367, |
| 477 | 49369,49391,49393,49409,49411,49417,49429,49433,49451,49459,49463,49477, |
| 478 | 49481,49499,49523,49529,49531,49537,49547,49549,49559,49597,49603,49613, |
| 479 | 49627,49633,49639,49663,49667,49669,49681,49697,49711,49727,49739,49741, |
| 480 | 49747,49757,49783,49787,49789,49801,49807,49811,49823,49831,49843,49853, |
| 481 | 49871,49877,49891,49919,49921,49927,49937,49939,49943,49957,49991,49993, |
| 482 | 49999,50021,50023,50033,50047,50051,50053,50069,50077,50087,50093,50101, |
| 483 | 50111,50119,50123,50129,50131,50147,50153,50159,50177,50207,50221,50227, |
| 484 | 50231,50261,50263,50273,50287,50291,50311,50321,50329,50333,50341,50359, |
| 485 | 50363,50377,50383,50387,50411,50417,50423,50441,50459,50461,50497,50503, |
| 486 | 50513,50527,50539,50543,50549,50551,50581,50587,50591,50593,50599,50627, |
| 487 | 50647,50651,50671,50683,50707,50723,50741,50753,50767,50773,50777,50789, |
| 488 | 50821,50833,50839,50849,50857,50867,50873,50891,50893,50909,50923,50929, |
| 489 | 50951,50957,50969,50971,50989,50993,51001,51031,51043,51047,51059,51061, |
| 490 | 51071,51109,51131,51133,51137,51151,51157,51169,51193,51197,51199,51203, |
| 491 | 51217,51229,51239,51241,51257,51263,51283,51287,51307,51329,51341,51343, |
| 492 | 51347,51349,51361,51383,51407,51413,51419,51421,51427,51431,51437,51439, |
| 493 | 51449,51461,51473,51479,51481,51487,51503,51511,51517,51521,51539,51551, |
| 494 | 51563,51577,51581,51593,51599,51607,51613,51631,51637,51647,51659,51673, |
| 495 | 51679,51683,51691,51713,51719,51721,51749,51767,51769,51787,51797,51803, |
| 496 | 51817,51827,51829,51839,51853,51859,51869,51871,51893,51899,51907,51913, |
| 497 | 51929,51941,51949,51971,51973,51977,51991,52009,52021,52027,52051,52057, |
| 498 | 52067,52069,52081,52103,52121,52127,52147,52153,52163,52177,52181,52183, |
| 499 | 52189,52201,52223,52237,52249,52253,52259,52267,52289,52291,52301,52313, |
| 500 | 52321,52361,52363,52369,52379,52387,52391,52433,52453,52457,52489,52501, |
| 501 | 52511,52517,52529,52541,52543,52553,52561,52567,52571,52579,52583,52609, |
| 502 | 52627,52631,52639,52667,52673,52691,52697,52709,52711,52721,52727,52733, |
| 503 | 52747,52757,52769,52783,52807,52813,52817,52837,52859,52861,52879,52883, |
| 504 | 52889,52901,52903,52919,52937,52951,52957,52963,52967,52973,52981,52999, |
| 505 | 53003,53017,53047,53051,53069,53077,53087,53089,53093,53101,53113,53117, |
| 506 | 53129,53147,53149,53161,53171,53173,53189,53197,53201,53231,53233,53239, |
| 507 | 53267,53269,53279,53281,53299,53309,53323,53327,53353,53359,53377,53381, |
| 508 | 53401,53407,53411,53419,53437,53441,53453,53479,53503,53507,53527,53549, |
| 509 | 53551,53569,53591,53593,53597,53609,53611,53617,53623,53629,53633,53639, |
| 510 | 53653,53657,53681,53693,53699,53717,53719,53731,53759,53773,53777,53783, |
| 511 | 53791,53813,53819,53831,53849,53857,53861,53881,53887,53891,53897,53899, |
| 512 | 53917,53923,53927,53939,53951,53959,53987,53993,54001,54011,54013,54037, |
| 513 | 54049,54059,54083,54091,54101,54121,54133,54139,54151,54163,54167,54181, |
| 514 | 54193,54217,54251,54269,54277,54287,54293,54311,54319,54323,54331,54347, |
| 515 | 54361,54367,54371,54377,54401,54403,54409,54413,54419,54421,54437,54443, |
| 516 | 54449,54469,54493,54497,54499,54503,54517,54521,54539,54541,54547,54559, |
| 517 | 54563,54577,54581,54583,54601,54617,54623,54629,54631,54647,54667,54673, |
| 518 | 54679,54709,54713,54721,54727,54751,54767,54773,54779,54787,54799,54829, |
| 519 | 54833,54851,54869,54877,54881,54907,54917,54919,54941,54949,54959,54973, |
| 520 | 54979,54983,55001,55009,55021,55049,55051,55057,55061,55073,55079,55103, |
| 521 | 55109,55117,55127,55147,55163,55171,55201,55207,55213,55217,55219,55229, |
| 522 | 55243,55249,55259,55291,55313,55331,55333,55337,55339,55343,55351,55373, |
| 523 | 55381,55399,55411,55439,55441,55457,55469,55487,55501,55511,55529,55541, |
| 524 | 55547,55579,55589,55603,55609,55619,55621,55631,55633,55639,55661,55663, |
| 525 | 55667,55673,55681,55691,55697,55711,55717,55721,55733,55763,55787,55793, |
| 526 | 55799,55807,55813,55817,55819,55823,55829,55837,55843,55849,55871,55889, |
| 527 | 55897,55901,55903,55921,55927,55931,55933,55949,55967,55987,55997,56003, |
| 528 | 56009,56039,56041,56053,56081,56087,56093,56099,56101,56113,56123,56131, |
| 529 | 56149,56167,56171,56179,56197,56207,56209,56237,56239,56249,56263,56267, |
| 530 | 56269,56299,56311,56333,56359,56369,56377,56383,56393,56401,56417,56431, |
| 531 | 56437,56443,56453,56467,56473,56477,56479,56489,56501,56503,56509,56519, |
| 532 | 56527,56531,56533,56543,56569,56591,56597,56599,56611,56629,56633,56659, |
| 533 | 56663,56671,56681,56687,56701,56711,56713,56731,56737,56747,56767,56773, |
| 534 | 56779,56783,56807,56809,56813,56821,56827,56843,56857,56873,56891,56893, |
| 535 | 56897,56909,56911,56921,56923,56929,56941,56951,56957,56963,56983,56989, |
| 536 | 56993,56999,57037,57041,57047,57059,57073,57077,57089,57097,57107,57119, |
| 537 | 57131,57139,57143,57149,57163,57173,57179,57191,57193,57203,57221,57223, |
| 538 | 57241,57251,57259,57269,57271,57283,57287,57301,57329,57331,57347,57349, |
| 539 | 57367,57373,57383,57389,57397,57413,57427,57457,57467,57487,57493,57503, |
| 540 | 57527,57529,57557,57559,57571,57587,57593,57601,57637,57641,57649,57653, |
| 541 | 57667,57679,57689,57697,57709,57713,57719,57727,57731,57737,57751,57773, |
| 542 | 57781,57787,57791,57793,57803,57809,57829,57839,57847,57853,57859,57881, |
| 543 | 57899,57901,57917,57923,57943,57947,57973,57977,57991,58013,58027,58031, |
| 544 | 58043,58049,58057,58061,58067,58073,58099,58109,58111,58129,58147,58151, |
| 545 | 58153,58169,58171,58189,58193,58199,58207,58211,58217,58229,58231,58237, |
| 546 | 58243,58271,58309,58313,58321,58337,58363,58367,58369,58379,58391,58393, |
| 547 | 58403,58411,58417,58427,58439,58441,58451,58453,58477,58481,58511,58537, |
| 548 | 58543,58549,58567,58573,58579,58601,58603,58613,58631,58657,58661,58679, |
| 549 | 58687,58693,58699,58711,58727,58733,58741,58757,58763,58771,58787,58789, |
| 550 | 58831,58889,58897,58901,58907,58909,58913,58921,58937,58943,58963,58967, |
| 551 | 58979,58991,58997,59009,59011,59021,59023,59029,59051,59053,59063,59069, |
| 552 | 59077,59083,59093,59107,59113,59119,59123,59141,59149,59159,59167,59183, |
| 553 | 59197,59207,59209,59219,59221,59233,59239,59243,59263,59273,59281,59333, |
| 554 | 59341,59351,59357,59359,59369,59377,59387,59393,59399,59407,59417,59419, |
| 555 | 59441,59443,59447,59453,59467,59471,59473,59497,59509,59513,59539,59557, |
| 556 | 59561,59567,59581,59611,59617,59621,59627,59629,59651,59659,59663,59669, |
| 557 | 59671,59693,59699,59707,59723,59729,59743,59747,59753,59771,59779,59791, |
| 558 | 59797,59809,59833,59863,59879,59887,59921,59929,59951,59957,59971,59981, |
| 559 | 59999,60013,60017,60029,60037,60041,60077,60083,60089,60091,60101,60103, |
| 560 | 60107,60127,60133,60139,60149,60161,60167,60169,60209,60217,60223,60251, |
| 561 | 60257,60259,60271,60289,60293,60317,60331,60337,60343,60353,60373,60383, |
| 562 | 60397,60413,60427,60443,60449,60457,60493,60497,60509,60521,60527,60539, |
| 563 | 60589,60601,60607,60611,60617,60623,60631,60637,60647,60649,60659,60661, |
| 564 | 60679,60689,60703,60719,60727,60733,60737,60757,60761,60763,60773,60779, |
| 565 | 60793,60811,60821,60859,60869,60887,60889,60899,60901,60913,60917,60919, |
| 566 | 60923,60937,60943,60953,60961,61001,61007,61027,61031,61043,61051,61057, |
| 567 | 61091,61099,61121,61129,61141,61151,61153,61169,61211,61223,61231,61253, |
| 568 | 61261,61283,61291,61297,61331,61333,61339,61343,61357,61363,61379,61381, |
| 569 | 61403,61409,61417,61441,61463,61469,61471,61483,61487,61493,61507,61511, |
| 570 | 61519,61543,61547,61553,61559,61561,61583,61603,61609,61613,61627,61631, |
| 571 | 61637,61643,61651,61657,61667,61673,61681,61687,61703,61717,61723,61729, |
| 572 | 61751,61757,61781,61813,61819,61837,61843,61861,61871,61879,61909,61927, |
| 573 | 61933,61949,61961,61967,61979,61981,61987,61991,62003,62011,62017,62039, |
| 574 | 62047,62053,62057,62071,62081,62099,62119,62129,62131,62137,62141,62143, |
| 575 | 62171,62189,62191,62201,62207,62213,62219,62233,62273,62297,62299,62303, |
| 576 | 62311,62323,62327,62347,62351,62383,62401,62417,62423,62459,62467,62473, |
| 577 | 62477,62483,62497,62501,62507,62533,62539,62549,62563,62581,62591,62597, |
| 578 | 62603,62617,62627,62633,62639,62653,62659,62683,62687,62701,62723,62731, |
| 579 | 62743,62753,62761,62773,62791,62801,62819,62827,62851,62861,62869,62873, |
| 580 | 62897,62903,62921,62927,62929,62939,62969,62971,62981,62983,62987,62989, |
| 581 | 63029,63031,63059,63067,63073,63079,63097,63103,63113,63127,63131,63149, |
| 582 | 63179,63197,63199,63211,63241,63247,63277,63281,63299,63311,63313,63317, |
| 583 | 63331,63337,63347,63353,63361,63367,63377,63389,63391,63397,63409,63419, |
| 584 | 63421,63439,63443,63463,63467,63473,63487,63493,63499,63521,63527,63533, |
| 585 | 63541,63559,63577,63587,63589,63599,63601,63607,63611,63617,63629,63647, |
| 586 | 63649,63659,63667,63671,63689,63691,63697,63703,63709,63719,63727,63737, |
| 587 | 63743,63761,63773,63781,63793,63799,63803,63809,63823,63839,63841,63853, |
| 588 | 63857,63863,63901,63907,63913,63929,63949,63977,63997,64007,64013,64019, |
| 589 | 64033,64037,64063,64067,64081,64091,64109,64123,64151,64153,64157,64171, |
| 590 | 64187,64189,64217,64223,64231,64237,64271,64279,64283,64301,64303,64319, |
| 591 | 64327,64333,64373,64381,64399,64403,64433,64439,64451,64453,64483,64489, |
| 592 | 64499,64513,64553,64567,64577,64579,64591,64601,64609,64613,64621,64627, |
| 593 | 64633,64661,64663,64667,64679,64693,64709,64717,64747,64763,64781,64783, |
| 594 | 64793,64811,64817,64849,64853,64871,64877,64879,64891,64901,64919,64921, |
| 595 | 64927,64937,64951,64969,64997,65003,65011,65027,65029,65033,65053,65063, |
| 596 | 65071,65089,65099,65101,65111,65119,65123,65129,65141,65147,65167,65171, |
| 597 | 65173,65179,65183,65203,65213,65239,65257,65267,65269,65287,65293,65309, |
| 598 | 65323,65327,65353,65357,65371,65381,65393,65407,65413,65419,65423,65437, |
| 599 | 65447,65449,65479,65497,65519,65521, |
| 600 | }; |
| 601 | |
| 602 | #define NPRIMES (sizeof(primes) / sizeof(*primes)) |
| 603 | |
| 604 | /* |
| 605 | * Generate a prime. We arrange to select a prime with the property |
| 606 | * (prime % modulus) != residue (to speed up use in RSA). |
| 607 | */ |
| 608 | Bignum primegen(int bits, int modulus, int residue, |
| 609 | int phase, progfn_t pfn, void *pfnparam) { |
| 610 | int i, k, v, byte, bitsleft, check, checks; |
| 611 | unsigned long delta, moduli[NPRIMES+1], residues[NPRIMES+1]; |
| b73c0b37 |
612 | Bignum p, pm1, q, wqp, wqp2; |
| 9400cf6f |
613 | int progress = 0; |
| 614 | |
| 615 | byte = 0; bitsleft = 0; |
| 616 | |
| 617 | STARTOVER: |
| 618 | |
| 619 | pfn(pfnparam, phase, ++progress); |
| 620 | |
| 621 | /* |
| 622 | * Generate a k-bit random number with top and bottom bits set. |
| 623 | */ |
| 624 | p = newbn((bits+15)/16); |
| 625 | for (i = 0; i < bits; i++) { |
| 626 | if (i == 0 || i == bits-1) |
| 627 | v = 1; |
| 628 | else { |
| 629 | if (bitsleft <= 0) |
| 630 | bitsleft = 8; byte = random_byte(); |
| 631 | v = byte & 1; |
| 632 | byte >>= 1; |
| 633 | bitsleft--; |
| 634 | } |
| 635 | bignum_set_bit(p, i, v); |
| 636 | } |
| 637 | |
| 638 | /* |
| 639 | * Ensure this random number is coprime to the first few |
| 640 | * primes, by repeatedly adding 2 to it until it is. |
| 641 | */ |
| 642 | for (i = 0; i < NPRIMES; i++) { |
| 643 | moduli[i] = primes[i]; |
| 644 | residues[i] = bignum_mod_short(p, primes[i]); |
| 645 | } |
| 646 | moduli[NPRIMES] = modulus; |
| b73c0b37 |
647 | residues[NPRIMES] = (bignum_mod_short(p, (unsigned short)modulus) |
| 648 | + modulus - residue); |
| 9400cf6f |
649 | delta = 0; |
| 650 | while (1) { |
| 651 | for (i = 0; i < (sizeof(moduli) / sizeof(*moduli)); i++) |
| 652 | if (!((residues[i] + delta) % moduli[i])) |
| 653 | break; |
| 654 | if (i < (sizeof(moduli) / sizeof(*moduli))) {/* we broke */ |
| 655 | delta += 2; |
| 656 | if (delta < 2) { |
| 657 | freebn(p); |
| 658 | goto STARTOVER; |
| 659 | } |
| 660 | continue; |
| 661 | } |
| 662 | break; |
| 663 | } |
| 664 | q = p; |
| 665 | p = bignum_add_long(q, delta); |
| 666 | freebn(q); |
| 667 | |
| 668 | /* |
| b73c0b37 |
669 | * Now apply the Miller-Rabin primality test a few times. First |
| 670 | * work out how many checks are needed. |
| 9400cf6f |
671 | */ |
| 672 | checks = 27; |
| 673 | if (bits >= 150) checks = 18; |
| 674 | if (bits >= 200) checks = 15; |
| 675 | if (bits >= 250) checks = 12; |
| 676 | if (bits >= 300) checks = 9; |
| 677 | if (bits >= 350) checks = 8; |
| 678 | if (bits >= 400) checks = 7; |
| 679 | if (bits >= 450) checks = 6; |
| 680 | if (bits >= 550) checks = 5; |
| 681 | if (bits >= 650) checks = 4; |
| 682 | if (bits >= 850) checks = 3; |
| 683 | if (bits >= 1300) checks = 2; |
| 684 | |
| 685 | /* |
| 686 | * Next, write p-1 as q*2^k. |
| 687 | */ |
| 688 | for (k = 0; bignum_bit(p, k) == !k; k++); /* find first 1 bit in p-1 */ |
| 689 | q = bignum_rshift(p, k); |
| b73c0b37 |
690 | /* And store p-1 itself, which we'll need. */ |
| 691 | pm1 = copybn(p); |
| 692 | decbn(pm1); |
| 9400cf6f |
693 | |
| 694 | /* |
| 695 | * Now, for each check ... |
| 696 | */ |
| 697 | for (check = 0; check < checks; check++) { |
| 698 | Bignum w; |
| 699 | |
| 700 | /* |
| 701 | * Invent a random number between 1 and p-1 inclusive. |
| 702 | */ |
| 703 | while (1) { |
| 704 | w = newbn((bits+15)/16); |
| 705 | for (i = 0; i < bits; i++) { |
| 706 | if (bitsleft <= 0) |
| 707 | bitsleft = 8; byte = random_byte(); |
| 708 | v = byte & 1; |
| 709 | byte >>= 1; |
| 710 | bitsleft--; |
| 711 | bignum_set_bit(w, i, v); |
| 712 | } |
| 713 | if (bignum_cmp(w, p) >= 0 || bignum_cmp(w, Zero) == 0) { |
| 714 | freebn(w); |
| 715 | continue; |
| 716 | } |
| 717 | break; |
| 718 | } |
| 719 | |
| 720 | pfn(pfnparam, phase, ++progress); |
| 721 | |
| 722 | /* |
| 723 | * Compute w^q mod p. |
| 724 | */ |
| 59600f67 |
725 | wqp = modpow(w, q, p); |
| 9400cf6f |
726 | freebn(w); |
| 727 | |
| 728 | /* |
| b73c0b37 |
729 | * See if this is 1, or if it is -1, or if it becomes -1 |
| 730 | * when squared at most k-1 times. |
| 9400cf6f |
731 | */ |
| b73c0b37 |
732 | if (bignum_cmp(wqp, One) == 0 || bignum_cmp(wqp, pm1) == 0) { |
| 9400cf6f |
733 | freebn(wqp); |
| 734 | continue; |
| 735 | } |
| b73c0b37 |
736 | for (i = 0; i < k-1; i++) { |
| 59600f67 |
737 | wqp2 = modmul(wqp, wqp, p); |
| 9400cf6f |
738 | freebn(wqp); |
| 739 | wqp = wqp2; |
| b73c0b37 |
740 | if (bignum_cmp(wqp, pm1) == 0) |
| 9400cf6f |
741 | break; |
| 742 | } |
| b73c0b37 |
743 | if (i < k-1) { |
| 9400cf6f |
744 | freebn(wqp); |
| 745 | continue; |
| 746 | } |
| 747 | |
| 748 | /* |
| 749 | * It didn't. Therefore, w is a witness for the |
| 750 | * compositeness of p. |
| 751 | */ |
| 752 | freebn(p); |
| b73c0b37 |
753 | freebn(pm1); |
| 9400cf6f |
754 | freebn(q); |
| 755 | goto STARTOVER; |
| 756 | } |
| 757 | |
| 758 | /* |
| 759 | * We have a prime! |
| 760 | */ |
| 761 | freebn(q); |
| b73c0b37 |
762 | freebn(pm1); |
| 9400cf6f |
763 | return p; |
| 764 | } |
projects / u / mdw / putty / blame