| 1c1af145 |
1 | /* |
| 2 | * Prime generation. |
| 3 | */ |
| 4 | |
| 5 | #include <assert.h> |
| 6 | #include "ssh.h" |
| 7 | |
| 8 | /* |
| 9 | * This prime generation algorithm is pretty much cribbed from |
| 10 | * OpenSSL. The algorithm is: |
| 11 | * |
| 12 | * - invent a B-bit random number and ensure the top and bottom |
| 13 | * bits are set (so it's definitely B-bit, and it's definitely |
| 14 | * odd) |
| 15 | * |
| 16 | * - see if it's coprime to all primes below 2^16; increment it by |
| 17 | * two until it is (this shouldn't take long in general) |
| 18 | * |
| 19 | * - perform the Miller-Rabin primality test enough times to |
| 20 | * ensure the probability of it being composite is 2^-80 or |
| 21 | * less |
| 22 | * |
| 23 | * - go back to square one if any M-R test fails. |
| 24 | */ |
| 25 | |
| 26 | /* |
| 27 | * The Miller-Rabin primality test is an extension to the Fermat |
| 28 | * test. The Fermat test just checks that a^(p-1) == 1 mod p; this |
| 29 | * is vulnerable to Carmichael numbers. Miller-Rabin considers how |
| 30 | * that 1 is derived as well. |
| 31 | * |
| 32 | * Lemma: if a^2 == 1 (mod p), and p is prime, then either a == 1 |
| 33 | * or a == -1 (mod p). |
| 34 | * |
| 35 | * Proof: p divides a^2-1, i.e. p divides (a+1)(a-1). Hence, |
| 36 | * since p is prime, either p divides (a+1) or p divides (a-1). |
| 37 | * But this is the same as saying that either a is congruent to |
| 38 | * -1 mod p or a is congruent to +1 mod p. [] |
| 39 | * |
| 40 | * Comment: This fails when p is not prime. Consider p=mn, so |
| 41 | * that mn divides (a+1)(a-1). Now we could have m dividing (a+1) |
| 42 | * and n dividing (a-1), without the whole of mn dividing either. |
| 43 | * For example, consider a=10 and p=99. 99 = 9 * 11; 9 divides |
| 44 | * 10-1 and 11 divides 10+1, so a^2 is congruent to 1 mod p |
| 45 | * without a having to be congruent to either 1 or -1. |
| 46 | * |
| 47 | * So the Miller-Rabin test, as well as considering a^(p-1), |
| 48 | * considers a^((p-1)/2), a^((p-1)/4), and so on as far as it can |
| 49 | * go. In other words. we write p-1 as q * 2^k, with k as large as |
| 50 | * possible (i.e. q must be odd), and we consider the powers |
| 51 | * |
| 52 | * a^(q*2^0) a^(q*2^1) ... a^(q*2^(k-1)) a^(q*2^k) |
| 53 | * i.e. a^((n-1)/2^k) a^((n-1)/2^(k-1)) ... a^((n-1)/2) a^(n-1) |
| 54 | * |
| 55 | * If p is to be prime, the last of these must be 1. Therefore, by |
| 56 | * the above lemma, the one before it must be either 1 or -1. And |
| 57 | * _if_ it's 1, then the one before that must be either 1 or -1, |
| 58 | * and so on ... In other words, we expect to see a trailing chain |
| 59 | * of 1s preceded by a -1. (If we're unlucky, our trailing chain of |
| 60 | * 1s will be as long as the list so we'll never get to see what |
| 61 | * lies before it. This doesn't count as a test failure because it |
| 62 | * hasn't _proved_ that p is not prime.) |
| 63 | * |
| 64 | * For example, consider a=2 and p=1729. 1729 is a Carmichael |
| 65 | * number: although it's not prime, it satisfies a^(p-1) == 1 mod p |
| 66 | * for any a coprime to it. So the Fermat test wouldn't have a |
| 67 | * problem with it at all, unless we happened to stumble on an a |
| 68 | * which had a common factor. |
| 69 | * |
| 70 | * So. 1729 - 1 equals 27 * 2^6. So we look at |
| 71 | * |
| 72 | * 2^27 mod 1729 == 645 |
| 73 | * 2^108 mod 1729 == 1065 |
| 74 | * 2^216 mod 1729 == 1 |
| 75 | * 2^432 mod 1729 == 1 |
| 76 | * 2^864 mod 1729 == 1 |
| 77 | * 2^1728 mod 1729 == 1 |
| 78 | * |
| 79 | * We do have a trailing string of 1s, so the Fermat test would |
| 80 | * have been happy. But this trailing string of 1s is preceded by |
| 81 | * 1065; whereas if 1729 were prime, we'd expect to see it preceded |
| 82 | * by -1 (i.e. 1728.). Guards! Seize this impostor. |
| 83 | * |
| 84 | * (If we were unlucky, we might have tried a=16 instead of a=2; |
| 85 | * now 16^27 mod 1729 == 1, so we would have seen a long string of |
| 86 | * 1s and wouldn't have seen the thing _before_ the 1s. So, just |
| 87 | * like the Fermat test, for a given p there may well exist values |
| 88 | * of a which fail to show up its compositeness. So we try several, |
| 89 | * just like the Fermat test. The difference is that Miller-Rabin |
| 90 | * is not _in general_ fooled by Carmichael numbers.) |
| 91 | * |
| 92 | * Put simply, then, the Miller-Rabin test requires us to: |
| 93 | * |
| 94 | * 1. write p-1 as q * 2^k, with q odd |
| 95 | * 2. compute z = (a^q) mod p. |
| 96 | * 3. report success if z == 1 or z == -1. |
| 97 | * 4. square z at most k-1 times, and report success if it becomes |
| 98 | * -1 at any point. |
| 99 | * 5. report failure otherwise. |
| 100 | * |
| 101 | * (We expect z to become -1 after at most k-1 squarings, because |
| 102 | * if it became -1 after k squarings then a^(p-1) would fail to be |
| 103 | * 1. And we don't need to investigate what happens after we see a |
| 104 | * -1, because we _know_ that -1 squared is 1 modulo anything at |
| 105 | * all, so after we've seen a -1 we can be sure of seeing nothing |
| 106 | * but 1s.) |
| 107 | */ |
| 108 | |
| 109 | /* |
| 110 | * The first few odd primes. |
| 111 | * |
| 112 | * import sys |
| 113 | * def sieve(n): |
| 114 | * z = [] |
| 115 | * list = [] |
| 116 | * for i in range(n): z.append(1) |
| 117 | * for i in range(2,n): |
| 118 | * if z[i]: |
| 119 | * list.append(i) |
| 120 | * for j in range(i,n,i): z[j] = 0 |
| 121 | * return list |
| 122 | * list = sieve(65535) |
| 123 | * for i in list[1:]: sys.stdout.write("%d," % i) |
| 124 | */ |
| 125 | static const unsigned short primes[] = { |
| 126 | 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, |
| 127 | 71, 73, 79, 83, 89, 97, 101, |
| 128 | 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, |
| 129 | 179, 181, 191, 193, |
| 130 | 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, |
| 131 | 277, 281, 283, 293, |
| 132 | 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, |
| 133 | 389, 397, 401, 409, |
| 134 | 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, |
| 135 | 499, 503, 509, 521, |
| 136 | 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, |
| 137 | 617, 619, 631, 641, |
| 138 | 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, |
| 139 | 739, 743, 751, 757, |
| 140 | 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, |
| 141 | 859, 863, 877, 881, |
| 142 | 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, |
| 143 | 991, 997, 1009, |
| 144 | 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, |
| 145 | 1091, 1093, |
| 146 | 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, |
| 147 | 1193, 1201, |
| 148 | 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, |
| 149 | 1291, 1297, |
| 150 | 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, |
| 151 | 1423, 1427, |
| 152 | 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, |
| 153 | 1493, 1499, |
| 154 | 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, |
| 155 | 1601, 1607, |
| 156 | 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, |
| 157 | 1699, 1709, |
| 158 | 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, |
| 159 | 1811, 1823, |
| 160 | 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, |
| 161 | 1931, 1933, |
| 162 | 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, |
| 163 | 2029, 2039, |
| 164 | 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, |
| 165 | 2137, 2141, |
| 166 | 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, |
| 167 | 2267, 2269, |
| 168 | 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, |
| 169 | 2357, 2371, |
| 170 | 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, |
| 171 | 2459, 2467, |
| 172 | 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, |
| 173 | 2593, 2609, |
| 174 | 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, |
| 175 | 2693, 2699, |
| 176 | 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, |
| 177 | 2791, 2797, |
| 178 | 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, |
| 179 | 2903, 2909, |
| 180 | 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, |
| 181 | 3023, 3037, |
| 182 | 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, |
| 183 | 3167, 3169, |
| 184 | 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, |
| 185 | 3271, 3299, |
| 186 | 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, |
| 187 | 3373, 3389, |
| 188 | 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, |
| 189 | 3511, 3517, |
| 190 | 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, |
| 191 | 3607, 3613, |
| 192 | 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, |
| 193 | 3709, 3719, |
| 194 | 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, |
| 195 | 3833, 3847, |
| 196 | 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, |
| 197 | 3931, 3943, |
| 198 | 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, |
| 199 | 4057, 4073, |
| 200 | 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, |
| 201 | 4177, 4201, |
| 202 | 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, |
| 203 | 4283, 4289, |
| 204 | 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, |
| 205 | 4423, 4441, |
| 206 | 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, |
| 207 | 4547, 4549, |
| 208 | 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, |
| 209 | 4657, 4663, |
| 210 | 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, |
| 211 | 4789, 4793, |
| 212 | 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, |
| 213 | 4931, 4933, |
| 214 | 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, |
| 215 | 5011, 5021, |
| 216 | 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, |
| 217 | 5147, 5153, |
| 218 | 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, |
| 219 | 5279, 5281, |
| 220 | 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, |
| 221 | 5413, 5417, |
| 222 | 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, |
| 223 | 5507, 5519, |
| 224 | 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, |
| 225 | 5647, 5651, |
| 226 | 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, |
| 227 | 5743, 5749, |
| 228 | 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, |
| 229 | 5857, 5861, |
| 230 | 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, |
| 231 | 6007, 6011, |
| 232 | 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, |
| 233 | 6121, 6131, |
| 234 | 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, |
| 235 | 6247, 6257, |
| 236 | 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, |
| 237 | 6343, 6353, |
| 238 | 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, |
| 239 | 6473, 6481, |
| 240 | 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, |
| 241 | 6607, 6619, |
| 242 | 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, |
| 243 | 6733, 6737, |
| 244 | 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, |
| 245 | 6857, 6863, |
| 246 | 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, |
| 247 | 6971, 6977, |
| 248 | 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, |
| 249 | 7103, 7109, |
| 250 | 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, |
| 251 | 7229, 7237, |
| 252 | 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, |
| 253 | 7369, 7393, |
| 254 | 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, |
| 255 | 7517, 7523, |
| 256 | 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, |
| 257 | 7603, 7607, |
| 258 | 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, |
| 259 | 7723, 7727, |
| 260 | 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, |
| 261 | 7873, 7877, |
| 262 | 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, |
| 263 | 8009, 8011, |
| 264 | 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, |
| 265 | 8123, 8147, |
| 266 | 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, |
| 267 | 8263, 8269, |
| 268 | 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, |
| 269 | 8387, 8389, |
| 270 | 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, |
| 271 | 8537, 8539, |
| 272 | 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, |
| 273 | 8663, 8669, |
| 274 | 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, |
| 275 | 8753, 8761, |
| 276 | 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863, |
| 277 | 8867, 8887, |
| 278 | 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, |
| 279 | 9011, 9013, |
| 280 | 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, |
| 281 | 9151, 9157, |
| 282 | 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, |
| 283 | 9277, 9281, |
| 284 | 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, |
| 285 | 9397, 9403, |
| 286 | 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, |
| 287 | 9491, 9497, |
| 288 | 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, |
| 289 | 9631, 9643, |
| 290 | 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, |
| 291 | 9767, 9769, |
| 292 | 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, |
| 293 | 9871, 9883, |
| 294 | 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973, 10007, |
| 295 | 10009, 10037, |
| 296 | 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099, 10103, 10111, |
| 297 | 10133, 10139, |
| 298 | 10141, 10151, 10159, 10163, 10169, 10177, 10181, 10193, 10211, 10223, |
| 299 | 10243, 10247, |
| 300 | 10253, 10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321, |
| 301 | 10331, 10333, |
| 302 | 10337, 10343, 10357, 10369, 10391, 10399, 10427, 10429, 10433, 10453, |
| 303 | 10457, 10459, |
| 304 | 10463, 10477, 10487, 10499, 10501, 10513, 10529, 10531, 10559, 10567, |
| 305 | 10589, 10597, |
| 306 | 10601, 10607, 10613, 10627, 10631, 10639, 10651, 10657, 10663, 10667, |
| 307 | 10687, 10691, |
| 308 | 10709, 10711, 10723, 10729, 10733, 10739, 10753, 10771, 10781, 10789, |
| 309 | 10799, 10831, |
| 310 | 10837, 10847, 10853, 10859, 10861, 10867, 10883, 10889, 10891, 10903, |
| 311 | 10909, 10937, |
| 312 | 10939, 10949, 10957, 10973, 10979, 10987, 10993, 11003, 11027, 11047, |
| 313 | 11057, 11059, |
| 314 | 11069, 11071, 11083, 11087, 11093, 11113, 11117, 11119, 11131, 11149, |
| 315 | 11159, 11161, |
| 316 | 11171, 11173, 11177, 11197, 11213, 11239, 11243, 11251, 11257, 11261, |
| 317 | 11273, 11279, |
| 318 | 11287, 11299, 11311, 11317, 11321, 11329, 11351, 11353, 11369, 11383, |
| 319 | 11393, 11399, |
| 320 | 11411, 11423, 11437, 11443, 11447, 11467, 11471, 11483, 11489, 11491, |
| 321 | 11497, 11503, |
| 322 | 11519, 11527, 11549, 11551, 11579, 11587, 11593, 11597, 11617, 11621, |
| 323 | 11633, 11657, |
| 324 | 11677, 11681, 11689, 11699, 11701, 11717, 11719, 11731, 11743, 11777, |
| 325 | 11779, 11783, |
| 326 | 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833, 11839, 11863, |
| 327 | 11867, 11887, |
| 328 | 11897, 11903, 11909, 11923, 11927, 11933, 11939, 11941, 11953, 11959, |
| 329 | 11969, 11971, |
| 330 | 11981, 11987, 12007, 12011, 12037, 12041, 12043, 12049, 12071, 12073, |
| 331 | 12097, 12101, |
| 332 | 12107, 12109, 12113, 12119, 12143, 12149, 12157, 12161, 12163, 12197, |
| 333 | 12203, 12211, |
| 334 | 12227, 12239, 12241, 12251, 12253, 12263, 12269, 12277, 12281, 12289, |
| 335 | 12301, 12323, |
| 336 | 12329, 12343, 12347, 12373, 12377, 12379, 12391, 12401, 12409, 12413, |
| 337 | 12421, 12433, |
| 338 | 12437, 12451, 12457, 12473, 12479, 12487, 12491, 12497, 12503, 12511, |
| 339 | 12517, 12527, |
| 340 | 12539, 12541, 12547, 12553, 12569, 12577, 12583, 12589, 12601, 12611, |
| 341 | 12613, 12619, |
| 342 | 12637, 12641, 12647, 12653, 12659, 12671, 12689, 12697, 12703, 12713, |
| 343 | 12721, 12739, |
| 344 | 12743, 12757, 12763, 12781, 12791, 12799, 12809, 12821, 12823, 12829, |
| 345 | 12841, 12853, |
| 346 | 12889, 12893, 12899, 12907, 12911, 12917, 12919, 12923, 12941, 12953, |
| 347 | 12959, 12967, |
| 348 | 12973, 12979, 12983, 13001, 13003, 13007, 13009, 13033, 13037, 13043, |
| 349 | 13049, 13063, |
| 350 | 13093, 13099, 13103, 13109, 13121, 13127, 13147, 13151, 13159, 13163, |
| 351 | 13171, 13177, |
| 352 | 13183, 13187, 13217, 13219, 13229, 13241, 13249, 13259, 13267, 13291, |
| 353 | 13297, 13309, |
| 354 | 13313, 13327, 13331, 13337, 13339, 13367, 13381, 13397, 13399, 13411, |
| 355 | 13417, 13421, |
| 356 | 13441, 13451, 13457, 13463, 13469, 13477, 13487, 13499, 13513, 13523, |
| 357 | 13537, 13553, |
| 358 | 13567, 13577, 13591, 13597, 13613, 13619, 13627, 13633, 13649, 13669, |
| 359 | 13679, 13681, |
| 360 | 13687, 13691, 13693, 13697, 13709, 13711, 13721, 13723, 13729, 13751, |
| 361 | 13757, 13759, |
| 362 | 13763, 13781, 13789, 13799, 13807, 13829, 13831, 13841, 13859, 13873, |
| 363 | 13877, 13879, |
| 364 | 13883, 13901, 13903, 13907, 13913, 13921, 13931, 13933, 13963, 13967, |
| 365 | 13997, 13999, |
| 366 | 14009, 14011, 14029, 14033, 14051, 14057, 14071, 14081, 14083, 14087, |
| 367 | 14107, 14143, |
| 368 | 14149, 14153, 14159, 14173, 14177, 14197, 14207, 14221, 14243, 14249, |
| 369 | 14251, 14281, |
| 370 | 14293, 14303, 14321, 14323, 14327, 14341, 14347, 14369, 14387, 14389, |
| 371 | 14401, 14407, |
| 372 | 14411, 14419, 14423, 14431, 14437, 14447, 14449, 14461, 14479, 14489, |
| 373 | 14503, 14519, |
| 374 | 14533, 14537, 14543, 14549, 14551, 14557, 14561, 14563, 14591, 14593, |
| 375 | 14621, 14627, |
| 376 | 14629, 14633, 14639, 14653, 14657, 14669, 14683, 14699, 14713, 14717, |
| 377 | 14723, 14731, |
| 378 | 14737, 14741, 14747, 14753, 14759, 14767, 14771, 14779, 14783, 14797, |
| 379 | 14813, 14821, |
| 380 | 14827, 14831, 14843, 14851, 14867, 14869, 14879, 14887, 14891, 14897, |
| 381 | 14923, 14929, |
| 382 | 14939, 14947, 14951, 14957, 14969, 14983, 15013, 15017, 15031, 15053, |
| 383 | 15061, 15073, |
| 384 | 15077, 15083, 15091, 15101, 15107, 15121, 15131, 15137, 15139, 15149, |
| 385 | 15161, 15173, |
| 386 | 15187, 15193, 15199, 15217, 15227, 15233, 15241, 15259, 15263, 15269, |
| 387 | 15271, 15277, |
| 388 | 15287, 15289, 15299, 15307, 15313, 15319, 15329, 15331, 15349, 15359, |
| 389 | 15361, 15373, |
| 390 | 15377, 15383, 15391, 15401, 15413, 15427, 15439, 15443, 15451, 15461, |
| 391 | 15467, 15473, |
| 392 | 15493, 15497, 15511, 15527, 15541, 15551, 15559, 15569, 15581, 15583, |
| 393 | 15601, 15607, |
| 394 | 15619, 15629, 15641, 15643, 15647, 15649, 15661, 15667, 15671, 15679, |
| 395 | 15683, 15727, |
| 396 | 15731, 15733, 15737, 15739, 15749, 15761, 15767, 15773, 15787, 15791, |
| 397 | 15797, 15803, |
| 398 | 15809, 15817, 15823, 15859, 15877, 15881, 15887, 15889, 15901, 15907, |
| 399 | 15913, 15919, |
| 400 | 15923, 15937, 15959, 15971, 15973, 15991, 16001, 16007, 16033, 16057, |
| 401 | 16061, 16063, |
| 402 | 16067, 16069, 16073, 16087, 16091, 16097, 16103, 16111, 16127, 16139, |
| 403 | 16141, 16183, |
| 404 | 16187, 16189, 16193, 16217, 16223, 16229, 16231, 16249, 16253, 16267, |
| 405 | 16273, 16301, |
| 406 | 16319, 16333, 16339, 16349, 16361, 16363, 16369, 16381, 16411, 16417, |
| 407 | 16421, 16427, |
| 408 | 16433, 16447, 16451, 16453, 16477, 16481, 16487, 16493, 16519, 16529, |
| 409 | 16547, 16553, |
| 410 | 16561, 16567, 16573, 16603, 16607, 16619, 16631, 16633, 16649, 16651, |
| 411 | 16657, 16661, |
| 412 | 16673, 16691, 16693, 16699, 16703, 16729, 16741, 16747, 16759, 16763, |
| 413 | 16787, 16811, |
| 414 | 16823, 16829, 16831, 16843, 16871, 16879, 16883, 16889, 16901, 16903, |
| 415 | 16921, 16927, |
| 416 | 16931, 16937, 16943, 16963, 16979, 16981, 16987, 16993, 17011, 17021, |
| 417 | 17027, 17029, |
| 418 | 17033, 17041, 17047, 17053, 17077, 17093, 17099, 17107, 17117, 17123, |
| 419 | 17137, 17159, |
| 420 | 17167, 17183, 17189, 17191, 17203, 17207, 17209, 17231, 17239, 17257, |
| 421 | 17291, 17293, |
| 422 | 17299, 17317, 17321, 17327, 17333, 17341, 17351, 17359, 17377, 17383, |
| 423 | 17387, 17389, |
| 424 | 17393, 17401, 17417, 17419, 17431, 17443, 17449, 17467, 17471, 17477, |
| 425 | 17483, 17489, |
| 426 | 17491, 17497, 17509, 17519, 17539, 17551, 17569, 17573, 17579, 17581, |
| 427 | 17597, 17599, |
| 428 | 17609, 17623, 17627, 17657, 17659, 17669, 17681, 17683, 17707, 17713, |
| 429 | 17729, 17737, |
| 430 | 17747, 17749, 17761, 17783, 17789, 17791, 17807, 17827, 17837, 17839, |
| 431 | 17851, 17863, |
| 432 | 17881, 17891, 17903, 17909, 17911, 17921, 17923, 17929, 17939, 17957, |
| 433 | 17959, 17971, |
| 434 | 17977, 17981, 17987, 17989, 18013, 18041, 18043, 18047, 18049, 18059, |
| 435 | 18061, 18077, |
| 436 | 18089, 18097, 18119, 18121, 18127, 18131, 18133, 18143, 18149, 18169, |
| 437 | 18181, 18191, |
| 438 | 18199, 18211, 18217, 18223, 18229, 18233, 18251, 18253, 18257, 18269, |
| 439 | 18287, 18289, |
| 440 | 18301, 18307, 18311, 18313, 18329, 18341, 18353, 18367, 18371, 18379, |
| 441 | 18397, 18401, |
| 442 | 18413, 18427, 18433, 18439, 18443, 18451, 18457, 18461, 18481, 18493, |
| 443 | 18503, 18517, |
| 444 | 18521, 18523, 18539, 18541, 18553, 18583, 18587, 18593, 18617, 18637, |
| 445 | 18661, 18671, |
| 446 | 18679, 18691, 18701, 18713, 18719, 18731, 18743, 18749, 18757, 18773, |
| 447 | 18787, 18793, |
| 448 | 18797, 18803, 18839, 18859, 18869, 18899, 18911, 18913, 18917, 18919, |
| 449 | 18947, 18959, |
| 450 | 18973, 18979, 19001, 19009, 19013, 19031, 19037, 19051, 19069, 19073, |
| 451 | 19079, 19081, |
| 452 | 19087, 19121, 19139, 19141, 19157, 19163, 19181, 19183, 19207, 19211, |
| 453 | 19213, 19219, |
| 454 | 19231, 19237, 19249, 19259, 19267, 19273, 19289, 19301, 19309, 19319, |
| 455 | 19333, 19373, |
| 456 | 19379, 19381, 19387, 19391, 19403, 19417, 19421, 19423, 19427, 19429, |
| 457 | 19433, 19441, |
| 458 | 19447, 19457, 19463, 19469, 19471, 19477, 19483, 19489, 19501, 19507, |
| 459 | 19531, 19541, |
| 460 | 19543, 19553, 19559, 19571, 19577, 19583, 19597, 19603, 19609, 19661, |
| 461 | 19681, 19687, |
| 462 | 19697, 19699, 19709, 19717, 19727, 19739, 19751, 19753, 19759, 19763, |
| 463 | 19777, 19793, |
| 464 | 19801, 19813, 19819, 19841, 19843, 19853, 19861, 19867, 19889, 19891, |
| 465 | 19913, 19919, |
| 466 | 19927, 19937, 19949, 19961, 19963, 19973, 19979, 19991, 19993, 19997, |
| 467 | 20011, 20021, |
| 468 | 20023, 20029, 20047, 20051, 20063, 20071, 20089, 20101, 20107, 20113, |
| 469 | 20117, 20123, |
| 470 | 20129, 20143, 20147, 20149, 20161, 20173, 20177, 20183, 20201, 20219, |
| 471 | 20231, 20233, |
| 472 | 20249, 20261, 20269, 20287, 20297, 20323, 20327, 20333, 20341, 20347, |
| 473 | 20353, 20357, |
| 474 | 20359, 20369, 20389, 20393, 20399, 20407, 20411, 20431, 20441, 20443, |
| 475 | 20477, 20479, |
| 476 | 20483, 20507, 20509, 20521, 20533, 20543, 20549, 20551, 20563, 20593, |
| 477 | 20599, 20611, |
| 478 | 20627, 20639, 20641, 20663, 20681, 20693, 20707, 20717, 20719, 20731, |
| 479 | 20743, 20747, |
| 480 | 20749, 20753, 20759, 20771, 20773, 20789, 20807, 20809, 20849, 20857, |
| 481 | 20873, 20879, |
| 482 | 20887, 20897, 20899, 20903, 20921, 20929, 20939, 20947, 20959, 20963, |
| 483 | 20981, 20983, |
| 484 | 21001, 21011, 21013, 21017, 21019, 21023, 21031, 21059, 21061, 21067, |
| 485 | 21089, 21101, |
| 486 | 21107, 21121, 21139, 21143, 21149, 21157, 21163, 21169, 21179, 21187, |
| 487 | 21191, 21193, |
| 488 | 21211, 21221, 21227, 21247, 21269, 21277, 21283, 21313, 21317, 21319, |
| 489 | 21323, 21341, |
| 490 | 21347, 21377, 21379, 21383, 21391, 21397, 21401, 21407, 21419, 21433, |
| 491 | 21467, 21481, |
| 492 | 21487, 21491, 21493, 21499, 21503, 21517, 21521, 21523, 21529, 21557, |
| 493 | 21559, 21563, |
| 494 | 21569, 21577, 21587, 21589, 21599, 21601, 21611, 21613, 21617, 21647, |
| 495 | 21649, 21661, |
| 496 | 21673, 21683, 21701, 21713, 21727, 21737, 21739, 21751, 21757, 21767, |
| 497 | 21773, 21787, |
| 498 | 21799, 21803, 21817, 21821, 21839, 21841, 21851, 21859, 21863, 21871, |
| 499 | 21881, 21893, |
| 500 | 21911, 21929, 21937, 21943, 21961, 21977, 21991, 21997, 22003, 22013, |
| 501 | 22027, 22031, |
| 502 | 22037, 22039, 22051, 22063, 22067, 22073, 22079, 22091, 22093, 22109, |
| 503 | 22111, 22123, |
| 504 | 22129, 22133, 22147, 22153, 22157, 22159, 22171, 22189, 22193, 22229, |
| 505 | 22247, 22259, |
| 506 | 22271, 22273, 22277, 22279, 22283, 22291, 22303, 22307, 22343, 22349, |
| 507 | 22367, 22369, |
| 508 | 22381, 22391, 22397, 22409, 22433, 22441, 22447, 22453, 22469, 22481, |
| 509 | 22483, 22501, |
| 510 | 22511, 22531, 22541, 22543, 22549, 22567, 22571, 22573, 22613, 22619, |
| 511 | 22621, 22637, |
| 512 | 22639, 22643, 22651, 22669, 22679, 22691, 22697, 22699, 22709, 22717, |
| 513 | 22721, 22727, |
| 514 | 22739, 22741, 22751, 22769, 22777, 22783, 22787, 22807, 22811, 22817, |
| 515 | 22853, 22859, |
| 516 | 22861, 22871, 22877, 22901, 22907, 22921, 22937, 22943, 22961, 22963, |
| 517 | 22973, 22993, |
| 518 | 23003, 23011, 23017, 23021, 23027, 23029, 23039, 23041, 23053, 23057, |
| 519 | 23059, 23063, |
| 520 | 23071, 23081, 23087, 23099, 23117, 23131, 23143, 23159, 23167, 23173, |
| 521 | 23189, 23197, |
| 522 | 23201, 23203, 23209, 23227, 23251, 23269, 23279, 23291, 23293, 23297, |
| 523 | 23311, 23321, |
| 524 | 23327, 23333, 23339, 23357, 23369, 23371, 23399, 23417, 23431, 23447, |
| 525 | 23459, 23473, |
| 526 | 23497, 23509, 23531, 23537, 23539, 23549, 23557, 23561, 23563, 23567, |
| 527 | 23581, 23593, |
| 528 | 23599, 23603, 23609, 23623, 23627, 23629, 23633, 23663, 23669, 23671, |
| 529 | 23677, 23687, |
| 530 | 23689, 23719, 23741, 23743, 23747, 23753, 23761, 23767, 23773, 23789, |
| 531 | 23801, 23813, |
| 532 | 23819, 23827, 23831, 23833, 23857, 23869, 23873, 23879, 23887, 23893, |
| 533 | 23899, 23909, |
| 534 | 23911, 23917, 23929, 23957, 23971, 23977, 23981, 23993, 24001, 24007, |
| 535 | 24019, 24023, |
| 536 | 24029, 24043, 24049, 24061, 24071, 24077, 24083, 24091, 24097, 24103, |
| 537 | 24107, 24109, |
| 538 | 24113, 24121, 24133, 24137, 24151, 24169, 24179, 24181, 24197, 24203, |
| 539 | 24223, 24229, |
| 540 | 24239, 24247, 24251, 24281, 24317, 24329, 24337, 24359, 24371, 24373, |
| 541 | 24379, 24391, |
| 542 | 24407, 24413, 24419, 24421, 24439, 24443, 24469, 24473, 24481, 24499, |
| 543 | 24509, 24517, |
| 544 | 24527, 24533, 24547, 24551, 24571, 24593, 24611, 24623, 24631, 24659, |
| 545 | 24671, 24677, |
| 546 | 24683, 24691, 24697, 24709, 24733, 24749, 24763, 24767, 24781, 24793, |
| 547 | 24799, 24809, |
| 548 | 24821, 24841, 24847, 24851, 24859, 24877, 24889, 24907, 24917, 24919, |
| 549 | 24923, 24943, |
| 550 | 24953, 24967, 24971, 24977, 24979, 24989, 25013, 25031, 25033, 25037, |
| 551 | 25057, 25073, |
| 552 | 25087, 25097, 25111, 25117, 25121, 25127, 25147, 25153, 25163, 25169, |
| 553 | 25171, 25183, |
| 554 | 25189, 25219, 25229, 25237, 25243, 25247, 25253, 25261, 25301, 25303, |
| 555 | 25307, 25309, |
| 556 | 25321, 25339, 25343, 25349, 25357, 25367, 25373, 25391, 25409, 25411, |
| 557 | 25423, 25439, |
| 558 | 25447, 25453, 25457, 25463, 25469, 25471, 25523, 25537, 25541, 25561, |
| 559 | 25577, 25579, |
| 560 | 25583, 25589, 25601, 25603, 25609, 25621, 25633, 25639, 25643, 25657, |
| 561 | 25667, 25673, |
| 562 | 25679, 25693, 25703, 25717, 25733, 25741, 25747, 25759, 25763, 25771, |
| 563 | 25793, 25799, |
| 564 | 25801, 25819, 25841, 25847, 25849, 25867, 25873, 25889, 25903, 25913, |
| 565 | 25919, 25931, |
| 566 | 25933, 25939, 25943, 25951, 25969, 25981, 25997, 25999, 26003, 26017, |
| 567 | 26021, 26029, |
| 568 | 26041, 26053, 26083, 26099, 26107, 26111, 26113, 26119, 26141, 26153, |
| 569 | 26161, 26171, |
| 570 | 26177, 26183, 26189, 26203, 26209, 26227, 26237, 26249, 26251, 26261, |
| 571 | 26263, 26267, |
| 572 | 26293, 26297, 26309, 26317, 26321, 26339, 26347, 26357, 26371, 26387, |
| 573 | 26393, 26399, |
| 574 | 26407, 26417, 26423, 26431, 26437, 26449, 26459, 26479, 26489, 26497, |
| 575 | 26501, 26513, |
| 576 | 26539, 26557, 26561, 26573, 26591, 26597, 26627, 26633, 26641, 26647, |
| 577 | 26669, 26681, |
| 578 | 26683, 26687, 26693, 26699, 26701, 26711, 26713, 26717, 26723, 26729, |
| 579 | 26731, 26737, |
| 580 | 26759, 26777, 26783, 26801, 26813, 26821, 26833, 26839, 26849, 26861, |
| 581 | 26863, 26879, |
| 582 | 26881, 26891, 26893, 26903, 26921, 26927, 26947, 26951, 26953, 26959, |
| 583 | 26981, 26987, |
| 584 | 26993, 27011, 27017, 27031, 27043, 27059, 27061, 27067, 27073, 27077, |
| 585 | 27091, 27103, |
| 586 | 27107, 27109, 27127, 27143, 27179, 27191, 27197, 27211, 27239, 27241, |
| 587 | 27253, 27259, |
| 588 | 27271, 27277, 27281, 27283, 27299, 27329, 27337, 27361, 27367, 27397, |
| 589 | 27407, 27409, |
| 590 | 27427, 27431, 27437, 27449, 27457, 27479, 27481, 27487, 27509, 27527, |
| 591 | 27529, 27539, |
| 592 | 27541, 27551, 27581, 27583, 27611, 27617, 27631, 27647, 27653, 27673, |
| 593 | 27689, 27691, |
| 594 | 27697, 27701, 27733, 27737, 27739, 27743, 27749, 27751, 27763, 27767, |
| 595 | 27773, 27779, |
| 596 | 27791, 27793, 27799, 27803, 27809, 27817, 27823, 27827, 27847, 27851, |
| 597 | 27883, 27893, |
| 598 | 27901, 27917, 27919, 27941, 27943, 27947, 27953, 27961, 27967, 27983, |
| 599 | 27997, 28001, |
| 600 | 28019, 28027, 28031, 28051, 28057, 28069, 28081, 28087, 28097, 28099, |
| 601 | 28109, 28111, |
| 602 | 28123, 28151, 28163, 28181, 28183, 28201, 28211, 28219, 28229, 28277, |
| 603 | 28279, 28283, |
| 604 | 28289, 28297, 28307, 28309, 28319, 28349, 28351, 28387, 28393, 28403, |
| 605 | 28409, 28411, |
| 606 | 28429, 28433, 28439, 28447, 28463, 28477, 28493, 28499, 28513, 28517, |
| 607 | 28537, 28541, |
| 608 | 28547, 28549, 28559, 28571, 28573, 28579, 28591, 28597, 28603, 28607, |
| 609 | 28619, 28621, |
| 610 | 28627, 28631, 28643, 28649, 28657, 28661, 28663, 28669, 28687, 28697, |
| 611 | 28703, 28711, |
| 612 | 28723, 28729, 28751, 28753, 28759, 28771, 28789, 28793, 28807, 28813, |
| 613 | 28817, 28837, |
| 614 | 28843, 28859, 28867, 28871, 28879, 28901, 28909, 28921, 28927, 28933, |
| 615 | 28949, 28961, |
| 616 | 28979, 29009, 29017, 29021, 29023, 29027, 29033, 29059, 29063, 29077, |
| 617 | 29101, 29123, |
| 618 | 29129, 29131, 29137, 29147, 29153, 29167, 29173, 29179, 29191, 29201, |
| 619 | 29207, 29209, |
| 620 | 29221, 29231, 29243, 29251, 29269, 29287, 29297, 29303, 29311, 29327, |
| 621 | 29333, 29339, |
| 622 | 29347, 29363, 29383, 29387, 29389, 29399, 29401, 29411, 29423, 29429, |
| 623 | 29437, 29443, |
| 624 | 29453, 29473, 29483, 29501, 29527, 29531, 29537, 29567, 29569, 29573, |
| 625 | 29581, 29587, |
| 626 | 29599, 29611, 29629, 29633, 29641, 29663, 29669, 29671, 29683, 29717, |
| 627 | 29723, 29741, |
| 628 | 29753, 29759, 29761, 29789, 29803, 29819, 29833, 29837, 29851, 29863, |
| 629 | 29867, 29873, |
| 630 | 29879, 29881, 29917, 29921, 29927, 29947, 29959, 29983, 29989, 30011, |
| 631 | 30013, 30029, |
| 632 | 30047, 30059, 30071, 30089, 30091, 30097, 30103, 30109, 30113, 30119, |
| 633 | 30133, 30137, |
| 634 | 30139, 30161, 30169, 30181, 30187, 30197, 30203, 30211, 30223, 30241, |
| 635 | 30253, 30259, |
| 636 | 30269, 30271, 30293, 30307, 30313, 30319, 30323, 30341, 30347, 30367, |
| 637 | 30389, 30391, |
| 638 | 30403, 30427, 30431, 30449, 30467, 30469, 30491, 30493, 30497, 30509, |
| 639 | 30517, 30529, |
| 640 | 30539, 30553, 30557, 30559, 30577, 30593, 30631, 30637, 30643, 30649, |
| 641 | 30661, 30671, |
| 642 | 30677, 30689, 30697, 30703, 30707, 30713, 30727, 30757, 30763, 30773, |
| 643 | 30781, 30803, |
| 644 | 30809, 30817, 30829, 30839, 30841, 30851, 30853, 30859, 30869, 30871, |
| 645 | 30881, 30893, |
| 646 | 30911, 30931, 30937, 30941, 30949, 30971, 30977, 30983, 31013, 31019, |
| 647 | 31033, 31039, |
| 648 | 31051, 31063, 31069, 31079, 31081, 31091, 31121, 31123, 31139, 31147, |
| 649 | 31151, 31153, |
| 650 | 31159, 31177, 31181, 31183, 31189, 31193, 31219, 31223, 31231, 31237, |
| 651 | 31247, 31249, |
| 652 | 31253, 31259, 31267, 31271, 31277, 31307, 31319, 31321, 31327, 31333, |
| 653 | 31337, 31357, |
| 654 | 31379, 31387, 31391, 31393, 31397, 31469, 31477, 31481, 31489, 31511, |
| 655 | 31513, 31517, |
| 656 | 31531, 31541, 31543, 31547, 31567, 31573, 31583, 31601, 31607, 31627, |
| 657 | 31643, 31649, |
| 658 | 31657, 31663, 31667, 31687, 31699, 31721, 31723, 31727, 31729, 31741, |
| 659 | 31751, 31769, |
| 660 | 31771, 31793, 31799, 31817, 31847, 31849, 31859, 31873, 31883, 31891, |
| 661 | 31907, 31957, |
| 662 | 31963, 31973, 31981, 31991, 32003, 32009, 32027, 32029, 32051, 32057, |
| 663 | 32059, 32063, |
| 664 | 32069, 32077, 32083, 32089, 32099, 32117, 32119, 32141, 32143, 32159, |
| 665 | 32173, 32183, |
| 666 | 32189, 32191, 32203, 32213, 32233, 32237, 32251, 32257, 32261, 32297, |
| 667 | 32299, 32303, |
| 668 | 32309, 32321, 32323, 32327, 32341, 32353, 32359, 32363, 32369, 32371, |
| 669 | 32377, 32381, |
| 670 | 32401, 32411, 32413, 32423, 32429, 32441, 32443, 32467, 32479, 32491, |
| 671 | 32497, 32503, |
| 672 | 32507, 32531, 32533, 32537, 32561, 32563, 32569, 32573, 32579, 32587, |
| 673 | 32603, 32609, |
| 674 | 32611, 32621, 32633, 32647, 32653, 32687, 32693, 32707, 32713, 32717, |
| 675 | 32719, 32749, |
| 676 | 32771, 32779, 32783, 32789, 32797, 32801, 32803, 32831, 32833, 32839, |
| 677 | 32843, 32869, |
| 678 | 32887, 32909, 32911, 32917, 32933, 32939, 32941, 32957, 32969, 32971, |
| 679 | 32983, 32987, |
| 680 | 32993, 32999, 33013, 33023, 33029, 33037, 33049, 33053, 33071, 33073, |
| 681 | 33083, 33091, |
| 682 | 33107, 33113, 33119, 33149, 33151, 33161, 33179, 33181, 33191, 33199, |
| 683 | 33203, 33211, |
| 684 | 33223, 33247, 33287, 33289, 33301, 33311, 33317, 33329, 33331, 33343, |
| 685 | 33347, 33349, |
| 686 | 33353, 33359, 33377, 33391, 33403, 33409, 33413, 33427, 33457, 33461, |
| 687 | 33469, 33479, |
| 688 | 33487, 33493, 33503, 33521, 33529, 33533, 33547, 33563, 33569, 33577, |
| 689 | 33581, 33587, |
| 690 | 33589, 33599, 33601, 33613, 33617, 33619, 33623, 33629, 33637, 33641, |
| 691 | 33647, 33679, |
| 692 | 33703, 33713, 33721, 33739, 33749, 33751, 33757, 33767, 33769, 33773, |
| 693 | 33791, 33797, |
| 694 | 33809, 33811, 33827, 33829, 33851, 33857, 33863, 33871, 33889, 33893, |
| 695 | 33911, 33923, |
| 696 | 33931, 33937, 33941, 33961, 33967, 33997, 34019, 34031, 34033, 34039, |
| 697 | 34057, 34061, |
| 698 | 34123, 34127, 34129, 34141, 34147, 34157, 34159, 34171, 34183, 34211, |
| 699 | 34213, 34217, |
| 700 | 34231, 34253, 34259, 34261, 34267, 34273, 34283, 34297, 34301, 34303, |
| 701 | 34313, 34319, |
| 702 | 34327, 34337, 34351, 34361, 34367, 34369, 34381, 34403, 34421, 34429, |
| 703 | 34439, 34457, |
| 704 | 34469, 34471, 34483, 34487, 34499, 34501, 34511, 34513, 34519, 34537, |
| 705 | 34543, 34549, |
| 706 | 34583, 34589, 34591, 34603, 34607, 34613, 34631, 34649, 34651, 34667, |
| 707 | 34673, 34679, |
| 708 | 34687, 34693, 34703, 34721, 34729, 34739, 34747, 34757, 34759, 34763, |
| 709 | 34781, 34807, |
| 710 | 34819, 34841, 34843, 34847, 34849, 34871, 34877, 34883, 34897, 34913, |
| 711 | 34919, 34939, |
| 712 | 34949, 34961, 34963, 34981, 35023, 35027, 35051, 35053, 35059, 35069, |
| 713 | 35081, 35083, |
| 714 | 35089, 35099, 35107, 35111, 35117, 35129, 35141, 35149, 35153, 35159, |
| 715 | 35171, 35201, |
| 716 | 35221, 35227, 35251, 35257, 35267, 35279, 35281, 35291, 35311, 35317, |
| 717 | 35323, 35327, |
| 718 | 35339, 35353, 35363, 35381, 35393, 35401, 35407, 35419, 35423, 35437, |
| 719 | 35447, 35449, |
| 720 | 35461, 35491, 35507, 35509, 35521, 35527, 35531, 35533, 35537, 35543, |
| 721 | 35569, 35573, |
| 722 | 35591, 35593, 35597, 35603, 35617, 35671, 35677, 35729, 35731, 35747, |
| 723 | 35753, 35759, |
| 724 | 35771, 35797, 35801, 35803, 35809, 35831, 35837, 35839, 35851, 35863, |
| 725 | 35869, 35879, |
| 726 | 35897, 35899, 35911, 35923, 35933, 35951, 35963, 35969, 35977, 35983, |
| 727 | 35993, 35999, |
| 728 | 36007, 36011, 36013, 36017, 36037, 36061, 36067, 36073, 36083, 36097, |
| 729 | 36107, 36109, |
| 730 | 36131, 36137, 36151, 36161, 36187, 36191, 36209, 36217, 36229, 36241, |
| 731 | 36251, 36263, |
| 732 | 36269, 36277, 36293, 36299, 36307, 36313, 36319, 36341, 36343, 36353, |
| 733 | 36373, 36383, |
| 734 | 36389, 36433, 36451, 36457, 36467, 36469, 36473, 36479, 36493, 36497, |
| 735 | 36523, 36527, |
| 736 | 36529, 36541, 36551, 36559, 36563, 36571, 36583, 36587, 36599, 36607, |
| 737 | 36629, 36637, |
| 738 | 36643, 36653, 36671, 36677, 36683, 36691, 36697, 36709, 36713, 36721, |
| 739 | 36739, 36749, |
| 740 | 36761, 36767, 36779, 36781, 36787, 36791, 36793, 36809, 36821, 36833, |
| 741 | 36847, 36857, |
| 742 | 36871, 36877, 36887, 36899, 36901, 36913, 36919, 36923, 36929, 36931, |
| 743 | 36943, 36947, |
| 744 | 36973, 36979, 36997, 37003, 37013, 37019, 37021, 37039, 37049, 37057, |
| 745 | 37061, 37087, |
| 746 | 37097, 37117, 37123, 37139, 37159, 37171, 37181, 37189, 37199, 37201, |
| 747 | 37217, 37223, |
| 748 | 37243, 37253, 37273, 37277, 37307, 37309, 37313, 37321, 37337, 37339, |
| 749 | 37357, 37361, |
| 750 | 37363, 37369, 37379, 37397, 37409, 37423, 37441, 37447, 37463, 37483, |
| 751 | 37489, 37493, |
| 752 | 37501, 37507, 37511, 37517, 37529, 37537, 37547, 37549, 37561, 37567, |
| 753 | 37571, 37573, |
| 754 | 37579, 37589, 37591, 37607, 37619, 37633, 37643, 37649, 37657, 37663, |
| 755 | 37691, 37693, |
| 756 | 37699, 37717, 37747, 37781, 37783, 37799, 37811, 37813, 37831, 37847, |
| 757 | 37853, 37861, |
| 758 | 37871, 37879, 37889, 37897, 37907, 37951, 37957, 37963, 37967, 37987, |
| 759 | 37991, 37993, |
| 760 | 37997, 38011, 38039, 38047, 38053, 38069, 38083, 38113, 38119, 38149, |
| 761 | 38153, 38167, |
| 762 | 38177, 38183, 38189, 38197, 38201, 38219, 38231, 38237, 38239, 38261, |
| 763 | 38273, 38281, |
| 764 | 38287, 38299, 38303, 38317, 38321, 38327, 38329, 38333, 38351, 38371, |
| 765 | 38377, 38393, |
| 766 | 38431, 38447, 38449, 38453, 38459, 38461, 38501, 38543, 38557, 38561, |
| 767 | 38567, 38569, |
| 768 | 38593, 38603, 38609, 38611, 38629, 38639, 38651, 38653, 38669, 38671, |
| 769 | 38677, 38693, |
| 770 | 38699, 38707, 38711, 38713, 38723, 38729, 38737, 38747, 38749, 38767, |
| 771 | 38783, 38791, |
| 772 | 38803, 38821, 38833, 38839, 38851, 38861, 38867, 38873, 38891, 38903, |
| 773 | 38917, 38921, |
| 774 | 38923, 38933, 38953, 38959, 38971, 38977, 38993, 39019, 39023, 39041, |
| 775 | 39043, 39047, |
| 776 | 39079, 39089, 39097, 39103, 39107, 39113, 39119, 39133, 39139, 39157, |
| 777 | 39161, 39163, |
| 778 | 39181, 39191, 39199, 39209, 39217, 39227, 39229, 39233, 39239, 39241, |
| 779 | 39251, 39293, |
| 780 | 39301, 39313, 39317, 39323, 39341, 39343, 39359, 39367, 39371, 39373, |
| 781 | 39383, 39397, |
| 782 | 39409, 39419, 39439, 39443, 39451, 39461, 39499, 39503, 39509, 39511, |
| 783 | 39521, 39541, |
| 784 | 39551, 39563, 39569, 39581, 39607, 39619, 39623, 39631, 39659, 39667, |
| 785 | 39671, 39679, |
| 786 | 39703, 39709, 39719, 39727, 39733, 39749, 39761, 39769, 39779, 39791, |
| 787 | 39799, 39821, |
| 788 | 39827, 39829, 39839, 39841, 39847, 39857, 39863, 39869, 39877, 39883, |
| 789 | 39887, 39901, |
| 790 | 39929, 39937, 39953, 39971, 39979, 39983, 39989, 40009, 40013, 40031, |
| 791 | 40037, 40039, |
| 792 | 40063, 40087, 40093, 40099, 40111, 40123, 40127, 40129, 40151, 40153, |
| 793 | 40163, 40169, |
| 794 | 40177, 40189, 40193, 40213, 40231, 40237, 40241, 40253, 40277, 40283, |
| 795 | 40289, 40343, |
| 796 | 40351, 40357, 40361, 40387, 40423, 40427, 40429, 40433, 40459, 40471, |
| 797 | 40483, 40487, |
| 798 | 40493, 40499, 40507, 40519, 40529, 40531, 40543, 40559, 40577, 40583, |
| 799 | 40591, 40597, |
| 800 | 40609, 40627, 40637, 40639, 40693, 40697, 40699, 40709, 40739, 40751, |
| 801 | 40759, 40763, |
| 802 | 40771, 40787, 40801, 40813, 40819, 40823, 40829, 40841, 40847, 40849, |
| 803 | 40853, 40867, |
| 804 | 40879, 40883, 40897, 40903, 40927, 40933, 40939, 40949, 40961, 40973, |
| 805 | 40993, 41011, |
| 806 | 41017, 41023, 41039, 41047, 41051, 41057, 41077, 41081, 41113, 41117, |
| 807 | 41131, 41141, |
| 808 | 41143, 41149, 41161, 41177, 41179, 41183, 41189, 41201, 41203, 41213, |
| 809 | 41221, 41227, |
| 810 | 41231, 41233, 41243, 41257, 41263, 41269, 41281, 41299, 41333, 41341, |
| 811 | 41351, 41357, |
| 812 | 41381, 41387, 41389, 41399, 41411, 41413, 41443, 41453, 41467, 41479, |
| 813 | 41491, 41507, |
| 814 | 41513, 41519, 41521, 41539, 41543, 41549, 41579, 41593, 41597, 41603, |
| 815 | 41609, 41611, |
| 816 | 41617, 41621, 41627, 41641, 41647, 41651, 41659, 41669, 41681, 41687, |
| 817 | 41719, 41729, |
| 818 | 41737, 41759, 41761, 41771, 41777, 41801, 41809, 41813, 41843, 41849, |
| 819 | 41851, 41863, |
| 820 | 41879, 41887, 41893, 41897, 41903, 41911, 41927, 41941, 41947, 41953, |
| 821 | 41957, 41959, |
| 822 | 41969, 41981, 41983, 41999, 42013, 42017, 42019, 42023, 42043, 42061, |
| 823 | 42071, 42073, |
| 824 | 42083, 42089, 42101, 42131, 42139, 42157, 42169, 42179, 42181, 42187, |
| 825 | 42193, 42197, |
| 826 | 42209, 42221, 42223, 42227, 42239, 42257, 42281, 42283, 42293, 42299, |
| 827 | 42307, 42323, |
| 828 | 42331, 42337, 42349, 42359, 42373, 42379, 42391, 42397, 42403, 42407, |
| 829 | 42409, 42433, |
| 830 | 42437, 42443, 42451, 42457, 42461, 42463, 42467, 42473, 42487, 42491, |
| 831 | 42499, 42509, |
| 832 | 42533, 42557, 42569, 42571, 42577, 42589, 42611, 42641, 42643, 42649, |
| 833 | 42667, 42677, |
| 834 | 42683, 42689, 42697, 42701, 42703, 42709, 42719, 42727, 42737, 42743, |
| 835 | 42751, 42767, |
| 836 | 42773, 42787, 42793, 42797, 42821, 42829, 42839, 42841, 42853, 42859, |
| 837 | 42863, 42899, |
| 838 | 42901, 42923, 42929, 42937, 42943, 42953, 42961, 42967, 42979, 42989, |
| 839 | 43003, 43013, |
| 840 | 43019, 43037, 43049, 43051, 43063, 43067, 43093, 43103, 43117, 43133, |
| 841 | 43151, 43159, |
| 842 | 43177, 43189, 43201, 43207, 43223, 43237, 43261, 43271, 43283, 43291, |
| 843 | 43313, 43319, |
| 844 | 43321, 43331, 43391, 43397, 43399, 43403, 43411, 43427, 43441, 43451, |
| 845 | 43457, 43481, |
| 846 | 43487, 43499, 43517, 43541, 43543, 43573, 43577, 43579, 43591, 43597, |
| 847 | 43607, 43609, |
| 848 | 43613, 43627, 43633, 43649, 43651, 43661, 43669, 43691, 43711, 43717, |
| 849 | 43721, 43753, |
| 850 | 43759, 43777, 43781, 43783, 43787, 43789, 43793, 43801, 43853, 43867, |
| 851 | 43889, 43891, |
| 852 | 43913, 43933, 43943, 43951, 43961, 43963, 43969, 43973, 43987, 43991, |
| 853 | 43997, 44017, |
| 854 | 44021, 44027, 44029, 44041, 44053, 44059, 44071, 44087, 44089, 44101, |
| 855 | 44111, 44119, |
| 856 | 44123, 44129, 44131, 44159, 44171, 44179, 44189, 44201, 44203, 44207, |
| 857 | 44221, 44249, |
| 858 | 44257, 44263, 44267, 44269, 44273, 44279, 44281, 44293, 44351, 44357, |
| 859 | 44371, 44381, |
| 860 | 44383, 44389, 44417, 44449, 44453, 44483, 44491, 44497, 44501, 44507, |
| 861 | 44519, 44531, |
| 862 | 44533, 44537, 44543, 44549, 44563, 44579, 44587, 44617, 44621, 44623, |
| 863 | 44633, 44641, |
| 864 | 44647, 44651, 44657, 44683, 44687, 44699, 44701, 44711, 44729, 44741, |
| 865 | 44753, 44771, |
| 866 | 44773, 44777, 44789, 44797, 44809, 44819, 44839, 44843, 44851, 44867, |
| 867 | 44879, 44887, |
| 868 | 44893, 44909, 44917, 44927, 44939, 44953, 44959, 44963, 44971, 44983, |
| 869 | 44987, 45007, |
| 870 | 45013, 45053, 45061, 45077, 45083, 45119, 45121, 45127, 45131, 45137, |
| 871 | 45139, 45161, |
| 872 | 45179, 45181, 45191, 45197, 45233, 45247, 45259, 45263, 45281, 45289, |
| 873 | 45293, 45307, |
| 874 | 45317, 45319, 45329, 45337, 45341, 45343, 45361, 45377, 45389, 45403, |
| 875 | 45413, 45427, |
| 876 | 45433, 45439, 45481, 45491, 45497, 45503, 45523, 45533, 45541, 45553, |
| 877 | 45557, 45569, |
| 878 | 45587, 45589, 45599, 45613, 45631, 45641, 45659, 45667, 45673, 45677, |
| 879 | 45691, 45697, |
| 880 | 45707, 45737, 45751, 45757, 45763, 45767, 45779, 45817, 45821, 45823, |
| 881 | 45827, 45833, |
| 882 | 45841, 45853, 45863, 45869, 45887, 45893, 45943, 45949, 45953, 45959, |
| 883 | 45971, 45979, |
| 884 | 45989, 46021, 46027, 46049, 46051, 46061, 46073, 46091, 46093, 46099, |
| 885 | 46103, 46133, |
| 886 | 46141, 46147, 46153, 46171, 46181, 46183, 46187, 46199, 46219, 46229, |
| 887 | 46237, 46261, |
| 888 | 46271, 46273, 46279, 46301, 46307, 46309, 46327, 46337, 46349, 46351, |
| 889 | 46381, 46399, |
| 890 | 46411, 46439, 46441, 46447, 46451, 46457, 46471, 46477, 46489, 46499, |
| 891 | 46507, 46511, |
| 892 | 46523, 46549, 46559, 46567, 46573, 46589, 46591, 46601, 46619, 46633, |
| 893 | 46639, 46643, |
| 894 | 46649, 46663, 46679, 46681, 46687, 46691, 46703, 46723, 46727, 46747, |
| 895 | 46751, 46757, |
| 896 | 46769, 46771, 46807, 46811, 46817, 46819, 46829, 46831, 46853, 46861, |
| 897 | 46867, 46877, |
| 898 | 46889, 46901, 46919, 46933, 46957, 46993, 46997, 47017, 47041, 47051, |
| 899 | 47057, 47059, |
| 900 | 47087, 47093, 47111, 47119, 47123, 47129, 47137, 47143, 47147, 47149, |
| 901 | 47161, 47189, |
| 902 | 47207, 47221, 47237, 47251, 47269, 47279, 47287, 47293, 47297, 47303, |
| 903 | 47309, 47317, |
| 904 | 47339, 47351, 47353, 47363, 47381, 47387, 47389, 47407, 47417, 47419, |
| 905 | 47431, 47441, |
| 906 | 47459, 47491, 47497, 47501, 47507, 47513, 47521, 47527, 47533, 47543, |
| 907 | 47563, 47569, |
| 908 | 47581, 47591, 47599, 47609, 47623, 47629, 47639, 47653, 47657, 47659, |
| 909 | 47681, 47699, |
| 910 | 47701, 47711, 47713, 47717, 47737, 47741, 47743, 47777, 47779, 47791, |
| 911 | 47797, 47807, |
| 912 | 47809, 47819, 47837, 47843, 47857, 47869, 47881, 47903, 47911, 47917, |
| 913 | 47933, 47939, |
| 914 | 47947, 47951, 47963, 47969, 47977, 47981, 48017, 48023, 48029, 48049, |
| 915 | 48073, 48079, |
| 916 | 48091, 48109, 48119, 48121, 48131, 48157, 48163, 48179, 48187, 48193, |
| 917 | 48197, 48221, |
| 918 | 48239, 48247, 48259, 48271, 48281, 48299, 48311, 48313, 48337, 48341, |
| 919 | 48353, 48371, |
| 920 | 48383, 48397, 48407, 48409, 48413, 48437, 48449, 48463, 48473, 48479, |
| 921 | 48481, 48487, |
| 922 | 48491, 48497, 48523, 48527, 48533, 48539, 48541, 48563, 48571, 48589, |
| 923 | 48593, 48611, |
| 924 | 48619, 48623, 48647, 48649, 48661, 48673, 48677, 48679, 48731, 48733, |
| 925 | 48751, 48757, |
| 926 | 48761, 48767, 48779, 48781, 48787, 48799, 48809, 48817, 48821, 48823, |
| 927 | 48847, 48857, |
| 928 | 48859, 48869, 48871, 48883, 48889, 48907, 48947, 48953, 48973, 48989, |
| 929 | 48991, 49003, |
| 930 | 49009, 49019, 49031, 49033, 49037, 49043, 49057, 49069, 49081, 49103, |
| 931 | 49109, 49117, |
| 932 | 49121, 49123, 49139, 49157, 49169, 49171, 49177, 49193, 49199, 49201, |
| 933 | 49207, 49211, |
| 934 | 49223, 49253, 49261, 49277, 49279, 49297, 49307, 49331, 49333, 49339, |
| 935 | 49363, 49367, |
| 936 | 49369, 49391, 49393, 49409, 49411, 49417, 49429, 49433, 49451, 49459, |
| 937 | 49463, 49477, |
| 938 | 49481, 49499, 49523, 49529, 49531, 49537, 49547, 49549, 49559, 49597, |
| 939 | 49603, 49613, |
| 940 | 49627, 49633, 49639, 49663, 49667, 49669, 49681, 49697, 49711, 49727, |
| 941 | 49739, 49741, |
| 942 | 49747, 49757, 49783, 49787, 49789, 49801, 49807, 49811, 49823, 49831, |
| 943 | 49843, 49853, |
| 944 | 49871, 49877, 49891, 49919, 49921, 49927, 49937, 49939, 49943, 49957, |
| 945 | 49991, 49993, |
| 946 | 49999, 50021, 50023, 50033, 50047, 50051, 50053, 50069, 50077, 50087, |
| 947 | 50093, 50101, |
| 948 | 50111, 50119, 50123, 50129, 50131, 50147, 50153, 50159, 50177, 50207, |
| 949 | 50221, 50227, |
| 950 | 50231, 50261, 50263, 50273, 50287, 50291, 50311, 50321, 50329, 50333, |
| 951 | 50341, 50359, |
| 952 | 50363, 50377, 50383, 50387, 50411, 50417, 50423, 50441, 50459, 50461, |
| 953 | 50497, 50503, |
| 954 | 50513, 50527, 50539, 50543, 50549, 50551, 50581, 50587, 50591, 50593, |
| 955 | 50599, 50627, |
| 956 | 50647, 50651, 50671, 50683, 50707, 50723, 50741, 50753, 50767, 50773, |
| 957 | 50777, 50789, |
| 958 | 50821, 50833, 50839, 50849, 50857, 50867, 50873, 50891, 50893, 50909, |
| 959 | 50923, 50929, |
| 960 | 50951, 50957, 50969, 50971, 50989, 50993, 51001, 51031, 51043, 51047, |
| 961 | 51059, 51061, |
| 962 | 51071, 51109, 51131, 51133, 51137, 51151, 51157, 51169, 51193, 51197, |
| 963 | 51199, 51203, |
| 964 | 51217, 51229, 51239, 51241, 51257, 51263, 51283, 51287, 51307, 51329, |
| 965 | 51341, 51343, |
| 966 | 51347, 51349, 51361, 51383, 51407, 51413, 51419, 51421, 51427, 51431, |
| 967 | 51437, 51439, |
| 968 | 51449, 51461, 51473, 51479, 51481, 51487, 51503, 51511, 51517, 51521, |
| 969 | 51539, 51551, |
| 970 | 51563, 51577, 51581, 51593, 51599, 51607, 51613, 51631, 51637, 51647, |
| 971 | 51659, 51673, |
| 972 | 51679, 51683, 51691, 51713, 51719, 51721, 51749, 51767, 51769, 51787, |
| 973 | 51797, 51803, |
| 974 | 51817, 51827, 51829, 51839, 51853, 51859, 51869, 51871, 51893, 51899, |
| 975 | 51907, 51913, |
| 976 | 51929, 51941, 51949, 51971, 51973, 51977, 51991, 52009, 52021, 52027, |
| 977 | 52051, 52057, |
| 978 | 52067, 52069, 52081, 52103, 52121, 52127, 52147, 52153, 52163, 52177, |
| 979 | 52181, 52183, |
| 980 | 52189, 52201, 52223, 52237, 52249, 52253, 52259, 52267, 52289, 52291, |
| 981 | 52301, 52313, |
| 982 | 52321, 52361, 52363, 52369, 52379, 52387, 52391, 52433, 52453, 52457, |
| 983 | 52489, 52501, |
| 984 | 52511, 52517, 52529, 52541, 52543, 52553, 52561, 52567, 52571, 52579, |
| 985 | 52583, 52609, |
| 986 | 52627, 52631, 52639, 52667, 52673, 52691, 52697, 52709, 52711, 52721, |
| 987 | 52727, 52733, |
| 988 | 52747, 52757, 52769, 52783, 52807, 52813, 52817, 52837, 52859, 52861, |
| 989 | 52879, 52883, |
| 990 | 52889, 52901, 52903, 52919, 52937, 52951, 52957, 52963, 52967, 52973, |
| 991 | 52981, 52999, |
| 992 | 53003, 53017, 53047, 53051, 53069, 53077, 53087, 53089, 53093, 53101, |
| 993 | 53113, 53117, |
| 994 | 53129, 53147, 53149, 53161, 53171, 53173, 53189, 53197, 53201, 53231, |
| 995 | 53233, 53239, |
| 996 | 53267, 53269, 53279, 53281, 53299, 53309, 53323, 53327, 53353, 53359, |
| 997 | 53377, 53381, |
| 998 | 53401, 53407, 53411, 53419, 53437, 53441, 53453, 53479, 53503, 53507, |
| 999 | 53527, 53549, |
| 1000 | 53551, 53569, 53591, 53593, 53597, 53609, 53611, 53617, 53623, 53629, |
| 1001 | 53633, 53639, |
| 1002 | 53653, 53657, 53681, 53693, 53699, 53717, 53719, 53731, 53759, 53773, |
| 1003 | 53777, 53783, |
| 1004 | 53791, 53813, 53819, 53831, 53849, 53857, 53861, 53881, 53887, 53891, |
| 1005 | 53897, 53899, |
| 1006 | 53917, 53923, 53927, 53939, 53951, 53959, 53987, 53993, 54001, 54011, |
| 1007 | 54013, 54037, |
| 1008 | 54049, 54059, 54083, 54091, 54101, 54121, 54133, 54139, 54151, 54163, |
| 1009 | 54167, 54181, |
| 1010 | 54193, 54217, 54251, 54269, 54277, 54287, 54293, 54311, 54319, 54323, |
| 1011 | 54331, 54347, |
| 1012 | 54361, 54367, 54371, 54377, 54401, 54403, 54409, 54413, 54419, 54421, |
| 1013 | 54437, 54443, |
| 1014 | 54449, 54469, 54493, 54497, 54499, 54503, 54517, 54521, 54539, 54541, |
| 1015 | 54547, 54559, |
| 1016 | 54563, 54577, 54581, 54583, 54601, 54617, 54623, 54629, 54631, 54647, |
| 1017 | 54667, 54673, |
| 1018 | 54679, 54709, 54713, 54721, 54727, 54751, 54767, 54773, 54779, 54787, |
| 1019 | 54799, 54829, |
| 1020 | 54833, 54851, 54869, 54877, 54881, 54907, 54917, 54919, 54941, 54949, |
| 1021 | 54959, 54973, |
| 1022 | 54979, 54983, 55001, 55009, 55021, 55049, 55051, 55057, 55061, 55073, |
| 1023 | 55079, 55103, |
| 1024 | 55109, 55117, 55127, 55147, 55163, 55171, 55201, 55207, 55213, 55217, |
| 1025 | 55219, 55229, |
| 1026 | 55243, 55249, 55259, 55291, 55313, 55331, 55333, 55337, 55339, 55343, |
| 1027 | 55351, 55373, |
| 1028 | 55381, 55399, 55411, 55439, 55441, 55457, 55469, 55487, 55501, 55511, |
| 1029 | 55529, 55541, |
| 1030 | 55547, 55579, 55589, 55603, 55609, 55619, 55621, 55631, 55633, 55639, |
| 1031 | 55661, 55663, |
| 1032 | 55667, 55673, 55681, 55691, 55697, 55711, 55717, 55721, 55733, 55763, |
| 1033 | 55787, 55793, |
| 1034 | 55799, 55807, 55813, 55817, 55819, 55823, 55829, 55837, 55843, 55849, |
| 1035 | 55871, 55889, |
| 1036 | 55897, 55901, 55903, 55921, 55927, 55931, 55933, 55949, 55967, 55987, |
| 1037 | 55997, 56003, |
| 1038 | 56009, 56039, 56041, 56053, 56081, 56087, 56093, 56099, 56101, 56113, |
| 1039 | 56123, 56131, |
| 1040 | 56149, 56167, 56171, 56179, 56197, 56207, 56209, 56237, 56239, 56249, |
| 1041 | 56263, 56267, |
| 1042 | 56269, 56299, 56311, 56333, 56359, 56369, 56377, 56383, 56393, 56401, |
| 1043 | 56417, 56431, |
| 1044 | 56437, 56443, 56453, 56467, 56473, 56477, 56479, 56489, 56501, 56503, |
| 1045 | 56509, 56519, |
| 1046 | 56527, 56531, 56533, 56543, 56569, 56591, 56597, 56599, 56611, 56629, |
| 1047 | 56633, 56659, |
| 1048 | 56663, 56671, 56681, 56687, 56701, 56711, 56713, 56731, 56737, 56747, |
| 1049 | 56767, 56773, |
| 1050 | 56779, 56783, 56807, 56809, 56813, 56821, 56827, 56843, 56857, 56873, |
| 1051 | 56891, 56893, |
| 1052 | 56897, 56909, 56911, 56921, 56923, 56929, 56941, 56951, 56957, 56963, |
| 1053 | 56983, 56989, |
| 1054 | 56993, 56999, 57037, 57041, 57047, 57059, 57073, 57077, 57089, 57097, |
| 1055 | 57107, 57119, |
| 1056 | 57131, 57139, 57143, 57149, 57163, 57173, 57179, 57191, 57193, 57203, |
| 1057 | 57221, 57223, |
| 1058 | 57241, 57251, 57259, 57269, 57271, 57283, 57287, 57301, 57329, 57331, |
| 1059 | 57347, 57349, |
| 1060 | 57367, 57373, 57383, 57389, 57397, 57413, 57427, 57457, 57467, 57487, |
| 1061 | 57493, 57503, |
| 1062 | 57527, 57529, 57557, 57559, 57571, 57587, 57593, 57601, 57637, 57641, |
| 1063 | 57649, 57653, |
| 1064 | 57667, 57679, 57689, 57697, 57709, 57713, 57719, 57727, 57731, 57737, |
| 1065 | 57751, 57773, |
| 1066 | 57781, 57787, 57791, 57793, 57803, 57809, 57829, 57839, 57847, 57853, |
| 1067 | 57859, 57881, |
| 1068 | 57899, 57901, 57917, 57923, 57943, 57947, 57973, 57977, 57991, 58013, |
| 1069 | 58027, 58031, |
| 1070 | 58043, 58049, 58057, 58061, 58067, 58073, 58099, 58109, 58111, 58129, |
| 1071 | 58147, 58151, |
| 1072 | 58153, 58169, 58171, 58189, 58193, 58199, 58207, 58211, 58217, 58229, |
| 1073 | 58231, 58237, |
| 1074 | 58243, 58271, 58309, 58313, 58321, 58337, 58363, 58367, 58369, 58379, |
| 1075 | 58391, 58393, |
| 1076 | 58403, 58411, 58417, 58427, 58439, 58441, 58451, 58453, 58477, 58481, |
| 1077 | 58511, 58537, |
| 1078 | 58543, 58549, 58567, 58573, 58579, 58601, 58603, 58613, 58631, 58657, |
| 1079 | 58661, 58679, |
| 1080 | 58687, 58693, 58699, 58711, 58727, 58733, 58741, 58757, 58763, 58771, |
| 1081 | 58787, 58789, |
| 1082 | 58831, 58889, 58897, 58901, 58907, 58909, 58913, 58921, 58937, 58943, |
| 1083 | 58963, 58967, |
| 1084 | 58979, 58991, 58997, 59009, 59011, 59021, 59023, 59029, 59051, 59053, |
| 1085 | 59063, 59069, |
| 1086 | 59077, 59083, 59093, 59107, 59113, 59119, 59123, 59141, 59149, 59159, |
| 1087 | 59167, 59183, |
| 1088 | 59197, 59207, 59209, 59219, 59221, 59233, 59239, 59243, 59263, 59273, |
| 1089 | 59281, 59333, |
| 1090 | 59341, 59351, 59357, 59359, 59369, 59377, 59387, 59393, 59399, 59407, |
| 1091 | 59417, 59419, |
| 1092 | 59441, 59443, 59447, 59453, 59467, 59471, 59473, 59497, 59509, 59513, |
| 1093 | 59539, 59557, |
| 1094 | 59561, 59567, 59581, 59611, 59617, 59621, 59627, 59629, 59651, 59659, |
| 1095 | 59663, 59669, |
| 1096 | 59671, 59693, 59699, 59707, 59723, 59729, 59743, 59747, 59753, 59771, |
| 1097 | 59779, 59791, |
| 1098 | 59797, 59809, 59833, 59863, 59879, 59887, 59921, 59929, 59951, 59957, |
| 1099 | 59971, 59981, |
| 1100 | 59999, 60013, 60017, 60029, 60037, 60041, 60077, 60083, 60089, 60091, |
| 1101 | 60101, 60103, |
| 1102 | 60107, 60127, 60133, 60139, 60149, 60161, 60167, 60169, 60209, 60217, |
| 1103 | 60223, 60251, |
| 1104 | 60257, 60259, 60271, 60289, 60293, 60317, 60331, 60337, 60343, 60353, |
| 1105 | 60373, 60383, |
| 1106 | 60397, 60413, 60427, 60443, 60449, 60457, 60493, 60497, 60509, 60521, |
| 1107 | 60527, 60539, |
| 1108 | 60589, 60601, 60607, 60611, 60617, 60623, 60631, 60637, 60647, 60649, |
| 1109 | 60659, 60661, |
| 1110 | 60679, 60689, 60703, 60719, 60727, 60733, 60737, 60757, 60761, 60763, |
| 1111 | 60773, 60779, |
| 1112 | 60793, 60811, 60821, 60859, 60869, 60887, 60889, 60899, 60901, 60913, |
| 1113 | 60917, 60919, |
| 1114 | 60923, 60937, 60943, 60953, 60961, 61001, 61007, 61027, 61031, 61043, |
| 1115 | 61051, 61057, |
| 1116 | 61091, 61099, 61121, 61129, 61141, 61151, 61153, 61169, 61211, 61223, |
| 1117 | 61231, 61253, |
| 1118 | 61261, 61283, 61291, 61297, 61331, 61333, 61339, 61343, 61357, 61363, |
| 1119 | 61379, 61381, |
| 1120 | 61403, 61409, 61417, 61441, 61463, 61469, 61471, 61483, 61487, 61493, |
| 1121 | 61507, 61511, |
| 1122 | 61519, 61543, 61547, 61553, 61559, 61561, 61583, 61603, 61609, 61613, |
| 1123 | 61627, 61631, |
| 1124 | 61637, 61643, 61651, 61657, 61667, 61673, 61681, 61687, 61703, 61717, |
| 1125 | 61723, 61729, |
| 1126 | 61751, 61757, 61781, 61813, 61819, 61837, 61843, 61861, 61871, 61879, |
| 1127 | 61909, 61927, |
| 1128 | 61933, 61949, 61961, 61967, 61979, 61981, 61987, 61991, 62003, 62011, |
| 1129 | 62017, 62039, |
| 1130 | 62047, 62053, 62057, 62071, 62081, 62099, 62119, 62129, 62131, 62137, |
| 1131 | 62141, 62143, |
| 1132 | 62171, 62189, 62191, 62201, 62207, 62213, 62219, 62233, 62273, 62297, |
| 1133 | 62299, 62303, |
| 1134 | 62311, 62323, 62327, 62347, 62351, 62383, 62401, 62417, 62423, 62459, |
| 1135 | 62467, 62473, |
| 1136 | 62477, 62483, 62497, 62501, 62507, 62533, 62539, 62549, 62563, 62581, |
| 1137 | 62591, 62597, |
| 1138 | 62603, 62617, 62627, 62633, 62639, 62653, 62659, 62683, 62687, 62701, |
| 1139 | 62723, 62731, |
| 1140 | 62743, 62753, 62761, 62773, 62791, 62801, 62819, 62827, 62851, 62861, |
| 1141 | 62869, 62873, |
| 1142 | 62897, 62903, 62921, 62927, 62929, 62939, 62969, 62971, 62981, 62983, |
| 1143 | 62987, 62989, |
| 1144 | 63029, 63031, 63059, 63067, 63073, 63079, 63097, 63103, 63113, 63127, |
| 1145 | 63131, 63149, |
| 1146 | 63179, 63197, 63199, 63211, 63241, 63247, 63277, 63281, 63299, 63311, |
| 1147 | 63313, 63317, |
| 1148 | 63331, 63337, 63347, 63353, 63361, 63367, 63377, 63389, 63391, 63397, |
| 1149 | 63409, 63419, |
| 1150 | 63421, 63439, 63443, 63463, 63467, 63473, 63487, 63493, 63499, 63521, |
| 1151 | 63527, 63533, |
| 1152 | 63541, 63559, 63577, 63587, 63589, 63599, 63601, 63607, 63611, 63617, |
| 1153 | 63629, 63647, |
| 1154 | 63649, 63659, 63667, 63671, 63689, 63691, 63697, 63703, 63709, 63719, |
| 1155 | 63727, 63737, |
| 1156 | 63743, 63761, 63773, 63781, 63793, 63799, 63803, 63809, 63823, 63839, |
| 1157 | 63841, 63853, |
| 1158 | 63857, 63863, 63901, 63907, 63913, 63929, 63949, 63977, 63997, 64007, |
| 1159 | 64013, 64019, |
| 1160 | 64033, 64037, 64063, 64067, 64081, 64091, 64109, 64123, 64151, 64153, |
| 1161 | 64157, 64171, |
| 1162 | 64187, 64189, 64217, 64223, 64231, 64237, 64271, 64279, 64283, 64301, |
| 1163 | 64303, 64319, |
| 1164 | 64327, 64333, 64373, 64381, 64399, 64403, 64433, 64439, 64451, 64453, |
| 1165 | 64483, 64489, |
| 1166 | 64499, 64513, 64553, 64567, 64577, 64579, 64591, 64601, 64609, 64613, |
| 1167 | 64621, 64627, |
| 1168 | 64633, 64661, 64663, 64667, 64679, 64693, 64709, 64717, 64747, 64763, |
| 1169 | 64781, 64783, |
| 1170 | 64793, 64811, 64817, 64849, 64853, 64871, 64877, 64879, 64891, 64901, |
| 1171 | 64919, 64921, |
| 1172 | 64927, 64937, 64951, 64969, 64997, 65003, 65011, 65027, 65029, 65033, |
| 1173 | 65053, 65063, |
| 1174 | 65071, 65089, 65099, 65101, 65111, 65119, 65123, 65129, 65141, 65147, |
| 1175 | 65167, 65171, |
| 1176 | 65173, 65179, 65183, 65203, 65213, 65239, 65257, 65267, 65269, 65287, |
| 1177 | 65293, 65309, |
| 1178 | 65323, 65327, 65353, 65357, 65371, 65381, 65393, 65407, 65413, 65419, |
| 1179 | 65423, 65437, |
| 1180 | 65447, 65449, 65479, 65497, 65519, 65521, |
| 1181 | }; |
| 1182 | |
| 1183 | #define NPRIMES (sizeof(primes) / sizeof(*primes)) |
| 1184 | |
| 1185 | /* |
| 1186 | * Generate a prime. We can deal with various extra properties of |
| 1187 | * the prime: |
| 1188 | * |
| 1189 | * - to speed up use in RSA, we can arrange to select a prime with |
| 1190 | * the property (prime % modulus) != residue. |
| 1191 | * |
| 1192 | * - for use in DSA, we can arrange to select a prime which is one |
| 1193 | * more than a multiple of a dirty great bignum. In this case |
| 1194 | * `bits' gives the size of the factor by which we _multiply_ |
| 1195 | * that bignum, rather than the size of the whole number. |
| 1196 | */ |
| 1197 | Bignum primegen(int bits, int modulus, int residue, Bignum factor, |
| 1198 | int phase, progfn_t pfn, void *pfnparam) |
| 1199 | { |
| 1200 | int i, k, v, byte, bitsleft, check, checks; |
| 1201 | unsigned long delta; |
| 1202 | unsigned long moduli[NPRIMES + 1]; |
| 1203 | unsigned long residues[NPRIMES + 1]; |
| 1204 | unsigned long multipliers[NPRIMES + 1]; |
| 1205 | Bignum p, pm1, q, wqp, wqp2; |
| 1206 | int progress = 0; |
| 1207 | |
| 1208 | byte = 0; |
| 1209 | bitsleft = 0; |
| 1210 | |
| 1211 | STARTOVER: |
| 1212 | |
| 1213 | pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress); |
| 1214 | |
| 1215 | /* |
| 1216 | * Generate a k-bit random number with top and bottom bits set. |
| 1217 | * Alternatively, if `factor' is nonzero, generate a k-bit |
| 1218 | * random number with the top bit set and the bottom bit clear, |
| 1219 | * multiply it by `factor', and add one. |
| 1220 | */ |
| 1221 | p = bn_power_2(bits - 1); |
| 1222 | for (i = 0; i < bits; i++) { |
| 1223 | if (i == 0 || i == bits - 1) |
| 1224 | v = (i != 0 || !factor) ? 1 : 0; |
| 1225 | else { |
| 1226 | if (bitsleft <= 0) |
| 1227 | bitsleft = 8, byte = random_byte(); |
| 1228 | v = byte & 1; |
| 1229 | byte >>= 1; |
| 1230 | bitsleft--; |
| 1231 | } |
| 1232 | bignum_set_bit(p, i, v); |
| 1233 | } |
| 1234 | if (factor) { |
| 1235 | Bignum tmp = p; |
| 1236 | p = bigmul(tmp, factor); |
| 1237 | freebn(tmp); |
| 1238 | assert(bignum_bit(p, 0) == 0); |
| 1239 | bignum_set_bit(p, 0, 1); |
| 1240 | } |
| 1241 | |
| 1242 | /* |
| 1243 | * Ensure this random number is coprime to the first few |
| 1244 | * primes, by repeatedly adding either 2 or 2*factor to it |
| 1245 | * until it is. |
| 1246 | */ |
| 1247 | for (i = 0; i < NPRIMES; i++) { |
| 1248 | moduli[i] = primes[i]; |
| 1249 | residues[i] = bignum_mod_short(p, primes[i]); |
| 1250 | if (factor) |
| 1251 | multipliers[i] = bignum_mod_short(factor, primes[i]); |
| 1252 | else |
| 1253 | multipliers[i] = 1; |
| 1254 | } |
| 1255 | moduli[NPRIMES] = modulus; |
| 1256 | residues[NPRIMES] = (bignum_mod_short(p, (unsigned short) modulus) |
| 1257 | + modulus - residue); |
| 1258 | if (factor) |
| 1259 | multipliers[NPRIMES] = bignum_mod_short(factor, modulus); |
| 1260 | else |
| 1261 | multipliers[NPRIMES] = 1; |
| 1262 | delta = 0; |
| 1263 | while (1) { |
| 1264 | for (i = 0; i < (sizeof(moduli) / sizeof(*moduli)); i++) |
| 1265 | if (!((residues[i] + delta * multipliers[i]) % moduli[i])) |
| 1266 | break; |
| 1267 | if (i < (sizeof(moduli) / sizeof(*moduli))) { /* we broke */ |
| 1268 | delta += 2; |
| 1269 | if (delta > 65536) { |
| 1270 | freebn(p); |
| 1271 | goto STARTOVER; |
| 1272 | } |
| 1273 | continue; |
| 1274 | } |
| 1275 | break; |
| 1276 | } |
| 1277 | q = p; |
| 1278 | if (factor) { |
| 1279 | Bignum tmp; |
| 1280 | tmp = bignum_from_long(delta); |
| 1281 | p = bigmuladd(tmp, factor, q); |
| 1282 | freebn(tmp); |
| 1283 | } else { |
| 1284 | p = bignum_add_long(q, delta); |
| 1285 | } |
| 1286 | freebn(q); |
| 1287 | |
| 1288 | /* |
| 1289 | * Now apply the Miller-Rabin primality test a few times. First |
| 1290 | * work out how many checks are needed. |
| 1291 | */ |
| 1292 | checks = 27; |
| 1293 | if (bits >= 150) |
| 1294 | checks = 18; |
| 1295 | if (bits >= 200) |
| 1296 | checks = 15; |
| 1297 | if (bits >= 250) |
| 1298 | checks = 12; |
| 1299 | if (bits >= 300) |
| 1300 | checks = 9; |
| 1301 | if (bits >= 350) |
| 1302 | checks = 8; |
| 1303 | if (bits >= 400) |
| 1304 | checks = 7; |
| 1305 | if (bits >= 450) |
| 1306 | checks = 6; |
| 1307 | if (bits >= 550) |
| 1308 | checks = 5; |
| 1309 | if (bits >= 650) |
| 1310 | checks = 4; |
| 1311 | if (bits >= 850) |
| 1312 | checks = 3; |
| 1313 | if (bits >= 1300) |
| 1314 | checks = 2; |
| 1315 | |
| 1316 | /* |
| 1317 | * Next, write p-1 as q*2^k. |
| 1318 | */ |
| 1319 | for (k = 0; bignum_bit(p, k) == !k; k++) |
| 1320 | continue; /* find first 1 bit in p-1 */ |
| 1321 | q = bignum_rshift(p, k); |
| 1322 | /* And store p-1 itself, which we'll need. */ |
| 1323 | pm1 = copybn(p); |
| 1324 | decbn(pm1); |
| 1325 | |
| 1326 | /* |
| 1327 | * Now, for each check ... |
| 1328 | */ |
| 1329 | for (check = 0; check < checks; check++) { |
| 1330 | Bignum w; |
| 1331 | |
| 1332 | /* |
| 1333 | * Invent a random number between 1 and p-1 inclusive. |
| 1334 | */ |
| 1335 | while (1) { |
| 1336 | w = bn_power_2(bits - 1); |
| 1337 | for (i = 0; i < bits; i++) { |
| 1338 | if (bitsleft <= 0) |
| 1339 | bitsleft = 8, byte = random_byte(); |
| 1340 | v = byte & 1; |
| 1341 | byte >>= 1; |
| 1342 | bitsleft--; |
| 1343 | bignum_set_bit(w, i, v); |
| 1344 | } |
| 1345 | bn_restore_invariant(w); |
| 1346 | if (bignum_cmp(w, p) >= 0 || bignum_cmp(w, Zero) == 0) { |
| 1347 | freebn(w); |
| 1348 | continue; |
| 1349 | } |
| 1350 | break; |
| 1351 | } |
| 1352 | |
| 1353 | pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress); |
| 1354 | |
| 1355 | /* |
| 1356 | * Compute w^q mod p. |
| 1357 | */ |
| 1358 | wqp = modpow(w, q, p); |
| 1359 | freebn(w); |
| 1360 | |
| 1361 | /* |
| 1362 | * See if this is 1, or if it is -1, or if it becomes -1 |
| 1363 | * when squared at most k-1 times. |
| 1364 | */ |
| 1365 | if (bignum_cmp(wqp, One) == 0 || bignum_cmp(wqp, pm1) == 0) { |
| 1366 | freebn(wqp); |
| 1367 | continue; |
| 1368 | } |
| 1369 | for (i = 0; i < k - 1; i++) { |
| 1370 | wqp2 = modmul(wqp, wqp, p); |
| 1371 | freebn(wqp); |
| 1372 | wqp = wqp2; |
| 1373 | if (bignum_cmp(wqp, pm1) == 0) |
| 1374 | break; |
| 1375 | } |
| 1376 | if (i < k - 1) { |
| 1377 | freebn(wqp); |
| 1378 | continue; |
| 1379 | } |
| 1380 | |
| 1381 | /* |
| 1382 | * It didn't. Therefore, w is a witness for the |
| 1383 | * compositeness of p. |
| 1384 | */ |
| 1385 | freebn(wqp); |
| 1386 | freebn(p); |
| 1387 | freebn(pm1); |
| 1388 | freebn(q); |
| 1389 | goto STARTOVER; |
| 1390 | } |
| 1391 | |
| 1392 | /* |
| 1393 | * We have a prime! |
| 1394 | */ |
| 1395 | freebn(q); |
| 1396 | freebn(pm1); |
| 1397 | return p; |
| 1398 | } |
projects / people / ms / putty.git / blame