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1 | /* |
| 2 | * Prime generation. |
| 3 | */ |
| 4 | |
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5 | #include <assert.h> |
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6 | #include "ssh.h" |
| 7 | |
| 8 | /* |
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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 |
| 506ec8d9 |
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.) |
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107 | */ |
| 108 | |
| 109 | /* |
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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[] = { |
| 22cc5f54 |
126 | 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, |
| 127 | 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, |
| 128 | 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, |
| 129 | 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, |
| 130 | 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, |
| 131 | 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, |
| 132 | 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, |
| 133 | 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, |
| 134 | 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, |
| 135 | 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, |
| 136 | 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, |
| 137 | 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, |
| 138 | 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, |
| 139 | 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, |
| 140 | 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, |
| 141 | 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, |
| 142 | 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, |
| 143 | 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, |
| 144 | 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, |
| 145 | 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, |
| 146 | 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, |
| 147 | 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, |
| 148 | 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, |
| 149 | 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, |
| 150 | 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, |
| 151 | 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, |
| 152 | 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, |
| 153 | 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, |
| 154 | 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, |
| 155 | 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, |
| 156 | 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, |
| 157 | 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, |
| 158 | 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, |
| 159 | 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, |
| 160 | 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, |
| 161 | 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, |
| 162 | 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, |
| 163 | 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, |
| 164 | 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, |
| 165 | 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, |
| 166 | 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, |
| 167 | 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, |
| 168 | 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, |
| 169 | 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, |
| 170 | 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, |
| 171 | 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, |
| 172 | 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, |
| 173 | 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, |
| 174 | 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, |
| 175 | 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, |
| 176 | 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, |
| 177 | 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, |
| 178 | 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, |
| 179 | 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, |
| 180 | 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, |
| 181 | 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, |
| 182 | 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, |
| 183 | 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, |
| 184 | 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, |
| 185 | 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, |
| 186 | 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, |
| 187 | 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, |
| 188 | 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, |
| 189 | 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, |
| 190 | 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, |
| 191 | 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, |
| 192 | 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, |
| 193 | 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, |
| 194 | 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, |
| 195 | 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, |
| 196 | 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, |
| 197 | 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, |
| 198 | 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, |
| 199 | 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, |
| 200 | 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, |
| 201 | 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, |
| 202 | 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, |
| 203 | 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, |
| 204 | 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, |
| 205 | 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, |
| 206 | 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, |
| 207 | 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, |
| 208 | 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, |
| 209 | 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, |
| 210 | 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, |
| 211 | 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, |
| 212 | 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, |
| 213 | 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, |
| 214 | 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, |
| 215 | 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, |
| 216 | 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, |
| 217 | 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, |
| 218 | 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, |
| 219 | 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, |
| 220 | 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, |
| 221 | 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, |
| 222 | 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, |
| 223 | 8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, |
| 224 | 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, |
| 225 | 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, |
| 226 | 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, |
| 227 | 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, |
| 228 | 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, 9397, 9403, 9413, |
| 229 | 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, |
| 230 | 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, |
| 231 | 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, |
| 232 | 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, |
| 233 | 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, |
| 32874aea |
234 | 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973, 10007, |
| 22cc5f54 |
235 | 10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, |
| 236 | 10099, 10103, 10111, 10133, 10139, 10141, 10151, 10159, 10163, |
| 237 | 10169, 10177, 10181, 10193, 10211, 10223, 10243, 10247, 10253, |
| 238 | 10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321, |
| 239 | 10331, 10333, 10337, 10343, 10357, 10369, 10391, 10399, 10427, |
| 240 | 10429, 10433, 10453, 10457, 10459, 10463, 10477, 10487, 10499, |
| 241 | 10501, 10513, 10529, 10531, 10559, 10567, 10589, 10597, 10601, |
| 242 | 10607, 10613, 10627, 10631, 10639, 10651, 10657, 10663, 10667, |
| 243 | 10687, 10691, 10709, 10711, 10723, 10729, 10733, 10739, 10753, |
| 244 | 10771, 10781, 10789, 10799, 10831, 10837, 10847, 10853, 10859, |
| 245 | 10861, 10867, 10883, 10889, 10891, 10903, 10909, 10937, 10939, |
| 246 | 10949, 10957, 10973, 10979, 10987, 10993, 11003, 11027, 11047, |
| 247 | 11057, 11059, 11069, 11071, 11083, 11087, 11093, 11113, 11117, |
| 248 | 11119, 11131, 11149, 11159, 11161, 11171, 11173, 11177, 11197, |
| 249 | 11213, 11239, 11243, 11251, 11257, 11261, 11273, 11279, 11287, |
| 250 | 11299, 11311, 11317, 11321, 11329, 11351, 11353, 11369, 11383, |
| 251 | 11393, 11399, 11411, 11423, 11437, 11443, 11447, 11467, 11471, |
| 252 | 11483, 11489, 11491, 11497, 11503, 11519, 11527, 11549, 11551, |
| 253 | 11579, 11587, 11593, 11597, 11617, 11621, 11633, 11657, 11677, |
| 254 | 11681, 11689, 11699, 11701, 11717, 11719, 11731, 11743, 11777, |
| 255 | 11779, 11783, 11789, 11801, 11807, 11813, 11821, 11827, 11831, |
| 256 | 11833, 11839, 11863, 11867, 11887, 11897, 11903, 11909, 11923, |
| 257 | 11927, 11933, 11939, 11941, 11953, 11959, 11969, 11971, 11981, |
| 258 | 11987, 12007, 12011, 12037, 12041, 12043, 12049, 12071, 12073, |
| 259 | 12097, 12101, 12107, 12109, 12113, 12119, 12143, 12149, 12157, |
| 260 | 12161, 12163, 12197, 12203, 12211, 12227, 12239, 12241, 12251, |
| 261 | 12253, 12263, 12269, 12277, 12281, 12289, 12301, 12323, 12329, |
| 262 | 12343, 12347, 12373, 12377, 12379, 12391, 12401, 12409, 12413, |
| 263 | 12421, 12433, 12437, 12451, 12457, 12473, 12479, 12487, 12491, |
| 264 | 12497, 12503, 12511, 12517, 12527, 12539, 12541, 12547, 12553, |
| 265 | 12569, 12577, 12583, 12589, 12601, 12611, 12613, 12619, 12637, |
| 266 | 12641, 12647, 12653, 12659, 12671, 12689, 12697, 12703, 12713, |
| 267 | 12721, 12739, 12743, 12757, 12763, 12781, 12791, 12799, 12809, |
| 268 | 12821, 12823, 12829, 12841, 12853, 12889, 12893, 12899, 12907, |
| 269 | 12911, 12917, 12919, 12923, 12941, 12953, 12959, 12967, 12973, |
| 270 | 12979, 12983, 13001, 13003, 13007, 13009, 13033, 13037, 13043, |
| 271 | 13049, 13063, 13093, 13099, 13103, 13109, 13121, 13127, 13147, |
| 272 | 13151, 13159, 13163, 13171, 13177, 13183, 13187, 13217, 13219, |
| 273 | 13229, 13241, 13249, 13259, 13267, 13291, 13297, 13309, 13313, |
| 274 | 13327, 13331, 13337, 13339, 13367, 13381, 13397, 13399, 13411, |
| 275 | 13417, 13421, 13441, 13451, 13457, 13463, 13469, 13477, 13487, |
| 276 | 13499, 13513, 13523, 13537, 13553, 13567, 13577, 13591, 13597, |
| 277 | 13613, 13619, 13627, 13633, 13649, 13669, 13679, 13681, 13687, |
| 278 | 13691, 13693, 13697, 13709, 13711, 13721, 13723, 13729, 13751, |
| 279 | 13757, 13759, 13763, 13781, 13789, 13799, 13807, 13829, 13831, |
| 280 | 13841, 13859, 13873, 13877, 13879, 13883, 13901, 13903, 13907, |
| 281 | 13913, 13921, 13931, 13933, 13963, 13967, 13997, 13999, 14009, |
| 282 | 14011, 14029, 14033, 14051, 14057, 14071, 14081, 14083, 14087, |
| 283 | 14107, 14143, 14149, 14153, 14159, 14173, 14177, 14197, 14207, |
| 284 | 14221, 14243, 14249, 14251, 14281, 14293, 14303, 14321, 14323, |
| 285 | 14327, 14341, 14347, 14369, 14387, 14389, 14401, 14407, 14411, |
| 286 | 14419, 14423, 14431, 14437, 14447, 14449, 14461, 14479, 14489, |
| 287 | 14503, 14519, 14533, 14537, 14543, 14549, 14551, 14557, 14561, |
| 288 | 14563, 14591, 14593, 14621, 14627, 14629, 14633, 14639, 14653, |
| 289 | 14657, 14669, 14683, 14699, 14713, 14717, 14723, 14731, 14737, |
| 290 | 14741, 14747, 14753, 14759, 14767, 14771, 14779, 14783, 14797, |
| 291 | 14813, 14821, 14827, 14831, 14843, 14851, 14867, 14869, 14879, |
| 292 | 14887, 14891, 14897, 14923, 14929, 14939, 14947, 14951, 14957, |
| 293 | 14969, 14983, 15013, 15017, 15031, 15053, 15061, 15073, 15077, |
| 294 | 15083, 15091, 15101, 15107, 15121, 15131, 15137, 15139, 15149, |
| 295 | 15161, 15173, 15187, 15193, 15199, 15217, 15227, 15233, 15241, |
| 296 | 15259, 15263, 15269, 15271, 15277, 15287, 15289, 15299, 15307, |
| 297 | 15313, 15319, 15329, 15331, 15349, 15359, 15361, 15373, 15377, |
| 298 | 15383, 15391, 15401, 15413, 15427, 15439, 15443, 15451, 15461, |
| 299 | 15467, 15473, 15493, 15497, 15511, 15527, 15541, 15551, 15559, |
| 300 | 15569, 15581, 15583, 15601, 15607, 15619, 15629, 15641, 15643, |
| 301 | 15647, 15649, 15661, 15667, 15671, 15679, 15683, 15727, 15731, |
| 302 | 15733, 15737, 15739, 15749, 15761, 15767, 15773, 15787, 15791, |
| 303 | 15797, 15803, 15809, 15817, 15823, 15859, 15877, 15881, 15887, |
| 304 | 15889, 15901, 15907, 15913, 15919, 15923, 15937, 15959, 15971, |
| 305 | 15973, 15991, 16001, 16007, 16033, 16057, 16061, 16063, 16067, |
| 306 | 16069, 16073, 16087, 16091, 16097, 16103, 16111, 16127, 16139, |
| 307 | 16141, 16183, 16187, 16189, 16193, 16217, 16223, 16229, 16231, |
| 308 | 16249, 16253, 16267, 16273, 16301, 16319, 16333, 16339, 16349, |
| 309 | 16361, 16363, 16369, 16381, 16411, 16417, 16421, 16427, 16433, |
| 310 | 16447, 16451, 16453, 16477, 16481, 16487, 16493, 16519, 16529, |
| 311 | 16547, 16553, 16561, 16567, 16573, 16603, 16607, 16619, 16631, |
| 312 | 16633, 16649, 16651, 16657, 16661, 16673, 16691, 16693, 16699, |
| 313 | 16703, 16729, 16741, 16747, 16759, 16763, 16787, 16811, 16823, |
| 314 | 16829, 16831, 16843, 16871, 16879, 16883, 16889, 16901, 16903, |
| 315 | 16921, 16927, 16931, 16937, 16943, 16963, 16979, 16981, 16987, |
| 316 | 16993, 17011, 17021, 17027, 17029, 17033, 17041, 17047, 17053, |
| 317 | 17077, 17093, 17099, 17107, 17117, 17123, 17137, 17159, 17167, |
| 318 | 17183, 17189, 17191, 17203, 17207, 17209, 17231, 17239, 17257, |
| 319 | 17291, 17293, 17299, 17317, 17321, 17327, 17333, 17341, 17351, |
| 320 | 17359, 17377, 17383, 17387, 17389, 17393, 17401, 17417, 17419, |
| 321 | 17431, 17443, 17449, 17467, 17471, 17477, 17483, 17489, 17491, |
| 322 | 17497, 17509, 17519, 17539, 17551, 17569, 17573, 17579, 17581, |
| 323 | 17597, 17599, 17609, 17623, 17627, 17657, 17659, 17669, 17681, |
| 324 | 17683, 17707, 17713, 17729, 17737, 17747, 17749, 17761, 17783, |
| 325 | 17789, 17791, 17807, 17827, 17837, 17839, 17851, 17863, 17881, |
| 326 | 17891, 17903, 17909, 17911, 17921, 17923, 17929, 17939, 17957, |
| 327 | 17959, 17971, 17977, 17981, 17987, 17989, 18013, 18041, 18043, |
| 328 | 18047, 18049, 18059, 18061, 18077, 18089, 18097, 18119, 18121, |
| 329 | 18127, 18131, 18133, 18143, 18149, 18169, 18181, 18191, 18199, |
| 330 | 18211, 18217, 18223, 18229, 18233, 18251, 18253, 18257, 18269, |
| 331 | 18287, 18289, 18301, 18307, 18311, 18313, 18329, 18341, 18353, |
| 332 | 18367, 18371, 18379, 18397, 18401, 18413, 18427, 18433, 18439, |
| 333 | 18443, 18451, 18457, 18461, 18481, 18493, 18503, 18517, 18521, |
| 334 | 18523, 18539, 18541, 18553, 18583, 18587, 18593, 18617, 18637, |
| 335 | 18661, 18671, 18679, 18691, 18701, 18713, 18719, 18731, 18743, |
| 336 | 18749, 18757, 18773, 18787, 18793, 18797, 18803, 18839, 18859, |
| 337 | 18869, 18899, 18911, 18913, 18917, 18919, 18947, 18959, 18973, |
| 338 | 18979, 19001, 19009, 19013, 19031, 19037, 19051, 19069, 19073, |
| 339 | 19079, 19081, 19087, 19121, 19139, 19141, 19157, 19163, 19181, |
| 340 | 19183, 19207, 19211, 19213, 19219, 19231, 19237, 19249, 19259, |
| 341 | 19267, 19273, 19289, 19301, 19309, 19319, 19333, 19373, 19379, |
| 342 | 19381, 19387, 19391, 19403, 19417, 19421, 19423, 19427, 19429, |
| 343 | 19433, 19441, 19447, 19457, 19463, 19469, 19471, 19477, 19483, |
| 344 | 19489, 19501, 19507, 19531, 19541, 19543, 19553, 19559, 19571, |
| 345 | 19577, 19583, 19597, 19603, 19609, 19661, 19681, 19687, 19697, |
| 346 | 19699, 19709, 19717, 19727, 19739, 19751, 19753, 19759, 19763, |
| 347 | 19777, 19793, 19801, 19813, 19819, 19841, 19843, 19853, 19861, |
| 348 | 19867, 19889, 19891, 19913, 19919, 19927, 19937, 19949, 19961, |
| 349 | 19963, 19973, 19979, 19991, 19993, 19997, 20011, 20021, 20023, |
| 350 | 20029, 20047, 20051, 20063, 20071, 20089, 20101, 20107, 20113, |
| 351 | 20117, 20123, 20129, 20143, 20147, 20149, 20161, 20173, 20177, |
| 352 | 20183, 20201, 20219, 20231, 20233, 20249, 20261, 20269, 20287, |
| 353 | 20297, 20323, 20327, 20333, 20341, 20347, 20353, 20357, 20359, |
| 354 | 20369, 20389, 20393, 20399, 20407, 20411, 20431, 20441, 20443, |
| 355 | 20477, 20479, 20483, 20507, 20509, 20521, 20533, 20543, 20549, |
| 356 | 20551, 20563, 20593, 20599, 20611, 20627, 20639, 20641, 20663, |
| 357 | 20681, 20693, 20707, 20717, 20719, 20731, 20743, 20747, 20749, |
| 358 | 20753, 20759, 20771, 20773, 20789, 20807, 20809, 20849, 20857, |
| 359 | 20873, 20879, 20887, 20897, 20899, 20903, 20921, 20929, 20939, |
| 360 | 20947, 20959, 20963, 20981, 20983, 21001, 21011, 21013, 21017, |
| 361 | 21019, 21023, 21031, 21059, 21061, 21067, 21089, 21101, 21107, |
| 362 | 21121, 21139, 21143, 21149, 21157, 21163, 21169, 21179, 21187, |
| 363 | 21191, 21193, 21211, 21221, 21227, 21247, 21269, 21277, 21283, |
| 364 | 21313, 21317, 21319, 21323, 21341, 21347, 21377, 21379, 21383, |
| 365 | 21391, 21397, 21401, 21407, 21419, 21433, 21467, 21481, 21487, |
| 366 | 21491, 21493, 21499, 21503, 21517, 21521, 21523, 21529, 21557, |
| 367 | 21559, 21563, 21569, 21577, 21587, 21589, 21599, 21601, 21611, |
| 368 | 21613, 21617, 21647, 21649, 21661, 21673, 21683, 21701, 21713, |
| 369 | 21727, 21737, 21739, 21751, 21757, 21767, 21773, 21787, 21799, |
| 370 | 21803, 21817, 21821, 21839, 21841, 21851, 21859, 21863, 21871, |
| 371 | 21881, 21893, 21911, 21929, 21937, 21943, 21961, 21977, 21991, |
| 372 | 21997, 22003, 22013, 22027, 22031, 22037, 22039, 22051, 22063, |
| 373 | 22067, 22073, 22079, 22091, 22093, 22109, 22111, 22123, 22129, |
| 374 | 22133, 22147, 22153, 22157, 22159, 22171, 22189, 22193, 22229, |
| 375 | 22247, 22259, 22271, 22273, 22277, 22279, 22283, 22291, 22303, |
| 376 | 22307, 22343, 22349, 22367, 22369, 22381, 22391, 22397, 22409, |
| 377 | 22433, 22441, 22447, 22453, 22469, 22481, 22483, 22501, 22511, |
| 378 | 22531, 22541, 22543, 22549, 22567, 22571, 22573, 22613, 22619, |
| 379 | 22621, 22637, 22639, 22643, 22651, 22669, 22679, 22691, 22697, |
| 380 | 22699, 22709, 22717, 22721, 22727, 22739, 22741, 22751, 22769, |
| 381 | 22777, 22783, 22787, 22807, 22811, 22817, 22853, 22859, 22861, |
| 382 | 22871, 22877, 22901, 22907, 22921, 22937, 22943, 22961, 22963, |
| 383 | 22973, 22993, 23003, 23011, 23017, 23021, 23027, 23029, 23039, |
| 384 | 23041, 23053, 23057, 23059, 23063, 23071, 23081, 23087, 23099, |
| 385 | 23117, 23131, 23143, 23159, 23167, 23173, 23189, 23197, 23201, |
| 386 | 23203, 23209, 23227, 23251, 23269, 23279, 23291, 23293, 23297, |
| 387 | 23311, 23321, 23327, 23333, 23339, 23357, 23369, 23371, 23399, |
| 388 | 23417, 23431, 23447, 23459, 23473, 23497, 23509, 23531, 23537, |
| 389 | 23539, 23549, 23557, 23561, 23563, 23567, 23581, 23593, 23599, |
| 390 | 23603, 23609, 23623, 23627, 23629, 23633, 23663, 23669, 23671, |
| 391 | 23677, 23687, 23689, 23719, 23741, 23743, 23747, 23753, 23761, |
| 392 | 23767, 23773, 23789, 23801, 23813, 23819, 23827, 23831, 23833, |
| 393 | 23857, 23869, 23873, 23879, 23887, 23893, 23899, 23909, 23911, |
| 394 | 23917, 23929, 23957, 23971, 23977, 23981, 23993, 24001, 24007, |
| 395 | 24019, 24023, 24029, 24043, 24049, 24061, 24071, 24077, 24083, |
| 396 | 24091, 24097, 24103, 24107, 24109, 24113, 24121, 24133, 24137, |
| 397 | 24151, 24169, 24179, 24181, 24197, 24203, 24223, 24229, 24239, |
| 398 | 24247, 24251, 24281, 24317, 24329, 24337, 24359, 24371, 24373, |
| 399 | 24379, 24391, 24407, 24413, 24419, 24421, 24439, 24443, 24469, |
| 400 | 24473, 24481, 24499, 24509, 24517, 24527, 24533, 24547, 24551, |
| 401 | 24571, 24593, 24611, 24623, 24631, 24659, 24671, 24677, 24683, |
| 402 | 24691, 24697, 24709, 24733, 24749, 24763, 24767, 24781, 24793, |
| 403 | 24799, 24809, 24821, 24841, 24847, 24851, 24859, 24877, 24889, |
| 404 | 24907, 24917, 24919, 24923, 24943, 24953, 24967, 24971, 24977, |
| 405 | 24979, 24989, 25013, 25031, 25033, 25037, 25057, 25073, 25087, |
| 406 | 25097, 25111, 25117, 25121, 25127, 25147, 25153, 25163, 25169, |
| 407 | 25171, 25183, 25189, 25219, 25229, 25237, 25243, 25247, 25253, |
| 408 | 25261, 25301, 25303, 25307, 25309, 25321, 25339, 25343, 25349, |
| 409 | 25357, 25367, 25373, 25391, 25409, 25411, 25423, 25439, 25447, |
| 410 | 25453, 25457, 25463, 25469, 25471, 25523, 25537, 25541, 25561, |
| 411 | 25577, 25579, 25583, 25589, 25601, 25603, 25609, 25621, 25633, |
| 412 | 25639, 25643, 25657, 25667, 25673, 25679, 25693, 25703, 25717, |
| 413 | 25733, 25741, 25747, 25759, 25763, 25771, 25793, 25799, 25801, |
| 414 | 25819, 25841, 25847, 25849, 25867, 25873, 25889, 25903, 25913, |
| 415 | 25919, 25931, 25933, 25939, 25943, 25951, 25969, 25981, 25997, |
| 416 | 25999, 26003, 26017, 26021, 26029, 26041, 26053, 26083, 26099, |
| 417 | 26107, 26111, 26113, 26119, 26141, 26153, 26161, 26171, 26177, |
| 418 | 26183, 26189, 26203, 26209, 26227, 26237, 26249, 26251, 26261, |
| 419 | 26263, 26267, 26293, 26297, 26309, 26317, 26321, 26339, 26347, |
| 420 | 26357, 26371, 26387, 26393, 26399, 26407, 26417, 26423, 26431, |
| 421 | 26437, 26449, 26459, 26479, 26489, 26497, 26501, 26513, 26539, |
| 422 | 26557, 26561, 26573, 26591, 26597, 26627, 26633, 26641, 26647, |
| 423 | 26669, 26681, 26683, 26687, 26693, 26699, 26701, 26711, 26713, |
| 424 | 26717, 26723, 26729, 26731, 26737, 26759, 26777, 26783, 26801, |
| 425 | 26813, 26821, 26833, 26839, 26849, 26861, 26863, 26879, 26881, |
| 426 | 26891, 26893, 26903, 26921, 26927, 26947, 26951, 26953, 26959, |
| 427 | 26981, 26987, 26993, 27011, 27017, 27031, 27043, 27059, 27061, |
| 428 | 27067, 27073, 27077, 27091, 27103, 27107, 27109, 27127, 27143, |
| 429 | 27179, 27191, 27197, 27211, 27239, 27241, 27253, 27259, 27271, |
| 430 | 27277, 27281, 27283, 27299, 27329, 27337, 27361, 27367, 27397, |
| 431 | 27407, 27409, 27427, 27431, 27437, 27449, 27457, 27479, 27481, |
| 432 | 27487, 27509, 27527, 27529, 27539, 27541, 27551, 27581, 27583, |
| 433 | 27611, 27617, 27631, 27647, 27653, 27673, 27689, 27691, 27697, |
| 434 | 27701, 27733, 27737, 27739, 27743, 27749, 27751, 27763, 27767, |
| 435 | 27773, 27779, 27791, 27793, 27799, 27803, 27809, 27817, 27823, |
| 436 | 27827, 27847, 27851, 27883, 27893, 27901, 27917, 27919, 27941, |
| 437 | 27943, 27947, 27953, 27961, 27967, 27983, 27997, 28001, 28019, |
| 438 | 28027, 28031, 28051, 28057, 28069, 28081, 28087, 28097, 28099, |
| 439 | 28109, 28111, 28123, 28151, 28163, 28181, 28183, 28201, 28211, |
| 440 | 28219, 28229, 28277, 28279, 28283, 28289, 28297, 28307, 28309, |
| 441 | 28319, 28349, 28351, 28387, 28393, 28403, 28409, 28411, 28429, |
| 442 | 28433, 28439, 28447, 28463, 28477, 28493, 28499, 28513, 28517, |
| 443 | 28537, 28541, 28547, 28549, 28559, 28571, 28573, 28579, 28591, |
| 444 | 28597, 28603, 28607, 28619, 28621, 28627, 28631, 28643, 28649, |
| 445 | 28657, 28661, 28663, 28669, 28687, 28697, 28703, 28711, 28723, |
| 446 | 28729, 28751, 28753, 28759, 28771, 28789, 28793, 28807, 28813, |
| 447 | 28817, 28837, 28843, 28859, 28867, 28871, 28879, 28901, 28909, |
| 448 | 28921, 28927, 28933, 28949, 28961, 28979, 29009, 29017, 29021, |
| 449 | 29023, 29027, 29033, 29059, 29063, 29077, 29101, 29123, 29129, |
| 450 | 29131, 29137, 29147, 29153, 29167, 29173, 29179, 29191, 29201, |
| 451 | 29207, 29209, 29221, 29231, 29243, 29251, 29269, 29287, 29297, |
| 452 | 29303, 29311, 29327, 29333, 29339, 29347, 29363, 29383, 29387, |
| 453 | 29389, 29399, 29401, 29411, 29423, 29429, 29437, 29443, 29453, |
| 454 | 29473, 29483, 29501, 29527, 29531, 29537, 29567, 29569, 29573, |
| 455 | 29581, 29587, 29599, 29611, 29629, 29633, 29641, 29663, 29669, |
| 456 | 29671, 29683, 29717, 29723, 29741, 29753, 29759, 29761, 29789, |
| 457 | 29803, 29819, 29833, 29837, 29851, 29863, 29867, 29873, 29879, |
| 458 | 29881, 29917, 29921, 29927, 29947, 29959, 29983, 29989, 30011, |
| 459 | 30013, 30029, 30047, 30059, 30071, 30089, 30091, 30097, 30103, |
| 460 | 30109, 30113, 30119, 30133, 30137, 30139, 30161, 30169, 30181, |
| 461 | 30187, 30197, 30203, 30211, 30223, 30241, 30253, 30259, 30269, |
| 462 | 30271, 30293, 30307, 30313, 30319, 30323, 30341, 30347, 30367, |
| 463 | 30389, 30391, 30403, 30427, 30431, 30449, 30467, 30469, 30491, |
| 464 | 30493, 30497, 30509, 30517, 30529, 30539, 30553, 30557, 30559, |
| 465 | 30577, 30593, 30631, 30637, 30643, 30649, 30661, 30671, 30677, |
| 466 | 30689, 30697, 30703, 30707, 30713, 30727, 30757, 30763, 30773, |
| 467 | 30781, 30803, 30809, 30817, 30829, 30839, 30841, 30851, 30853, |
| 468 | 30859, 30869, 30871, 30881, 30893, 30911, 30931, 30937, 30941, |
| 469 | 30949, 30971, 30977, 30983, 31013, 31019, 31033, 31039, 31051, |
| 470 | 31063, 31069, 31079, 31081, 31091, 31121, 31123, 31139, 31147, |
| 471 | 31151, 31153, 31159, 31177, 31181, 31183, 31189, 31193, 31219, |
| 472 | 31223, 31231, 31237, 31247, 31249, 31253, 31259, 31267, 31271, |
| 473 | 31277, 31307, 31319, 31321, 31327, 31333, 31337, 31357, 31379, |
| 474 | 31387, 31391, 31393, 31397, 31469, 31477, 31481, 31489, 31511, |
| 475 | 31513, 31517, 31531, 31541, 31543, 31547, 31567, 31573, 31583, |
| 476 | 31601, 31607, 31627, 31643, 31649, 31657, 31663, 31667, 31687, |
| 477 | 31699, 31721, 31723, 31727, 31729, 31741, 31751, 31769, 31771, |
| 478 | 31793, 31799, 31817, 31847, 31849, 31859, 31873, 31883, 31891, |
| 479 | 31907, 31957, 31963, 31973, 31981, 31991, 32003, 32009, 32027, |
| 480 | 32029, 32051, 32057, 32059, 32063, 32069, 32077, 32083, 32089, |
| 481 | 32099, 32117, 32119, 32141, 32143, 32159, 32173, 32183, 32189, |
| 482 | 32191, 32203, 32213, 32233, 32237, 32251, 32257, 32261, 32297, |
| 483 | 32299, 32303, 32309, 32321, 32323, 32327, 32341, 32353, 32359, |
| 484 | 32363, 32369, 32371, 32377, 32381, 32401, 32411, 32413, 32423, |
| 485 | 32429, 32441, 32443, 32467, 32479, 32491, 32497, 32503, 32507, |
| 486 | 32531, 32533, 32537, 32561, 32563, 32569, 32573, 32579, 32587, |
| 487 | 32603, 32609, 32611, 32621, 32633, 32647, 32653, 32687, 32693, |
| 488 | 32707, 32713, 32717, 32719, 32749, 32771, 32779, 32783, 32789, |
| 489 | 32797, 32801, 32803, 32831, 32833, 32839, 32843, 32869, 32887, |
| 490 | 32909, 32911, 32917, 32933, 32939, 32941, 32957, 32969, 32971, |
| 491 | 32983, 32987, 32993, 32999, 33013, 33023, 33029, 33037, 33049, |
| 492 | 33053, 33071, 33073, 33083, 33091, 33107, 33113, 33119, 33149, |
| 493 | 33151, 33161, 33179, 33181, 33191, 33199, 33203, 33211, 33223, |
| 494 | 33247, 33287, 33289, 33301, 33311, 33317, 33329, 33331, 33343, |
| 495 | 33347, 33349, 33353, 33359, 33377, 33391, 33403, 33409, 33413, |
| 496 | 33427, 33457, 33461, 33469, 33479, 33487, 33493, 33503, 33521, |
| 497 | 33529, 33533, 33547, 33563, 33569, 33577, 33581, 33587, 33589, |
| 498 | 33599, 33601, 33613, 33617, 33619, 33623, 33629, 33637, 33641, |
| 499 | 33647, 33679, 33703, 33713, 33721, 33739, 33749, 33751, 33757, |
| 500 | 33767, 33769, 33773, 33791, 33797, 33809, 33811, 33827, 33829, |
| 501 | 33851, 33857, 33863, 33871, 33889, 33893, 33911, 33923, 33931, |
| 502 | 33937, 33941, 33961, 33967, 33997, 34019, 34031, 34033, 34039, |
| 503 | 34057, 34061, 34123, 34127, 34129, 34141, 34147, 34157, 34159, |
| 504 | 34171, 34183, 34211, 34213, 34217, 34231, 34253, 34259, 34261, |
| 505 | 34267, 34273, 34283, 34297, 34301, 34303, 34313, 34319, 34327, |
| 506 | 34337, 34351, 34361, 34367, 34369, 34381, 34403, 34421, 34429, |
| 507 | 34439, 34457, 34469, 34471, 34483, 34487, 34499, 34501, 34511, |
| 508 | 34513, 34519, 34537, 34543, 34549, 34583, 34589, 34591, 34603, |
| 509 | 34607, 34613, 34631, 34649, 34651, 34667, 34673, 34679, 34687, |
| 510 | 34693, 34703, 34721, 34729, 34739, 34747, 34757, 34759, 34763, |
| 511 | 34781, 34807, 34819, 34841, 34843, 34847, 34849, 34871, 34877, |
| 512 | 34883, 34897, 34913, 34919, 34939, 34949, 34961, 34963, 34981, |
| 513 | 35023, 35027, 35051, 35053, 35059, 35069, 35081, 35083, 35089, |
| 514 | 35099, 35107, 35111, 35117, 35129, 35141, 35149, 35153, 35159, |
| 515 | 35171, 35201, 35221, 35227, 35251, 35257, 35267, 35279, 35281, |
| 516 | 35291, 35311, 35317, 35323, 35327, 35339, 35353, 35363, 35381, |
| 517 | 35393, 35401, 35407, 35419, 35423, 35437, 35447, 35449, 35461, |
| 518 | 35491, 35507, 35509, 35521, 35527, 35531, 35533, 35537, 35543, |
| 519 | 35569, 35573, 35591, 35593, 35597, 35603, 35617, 35671, 35677, |
| 520 | 35729, 35731, 35747, 35753, 35759, 35771, 35797, 35801, 35803, |
| 521 | 35809, 35831, 35837, 35839, 35851, 35863, 35869, 35879, 35897, |
| 522 | 35899, 35911, 35923, 35933, 35951, 35963, 35969, 35977, 35983, |
| 523 | 35993, 35999, 36007, 36011, 36013, 36017, 36037, 36061, 36067, |
| 524 | 36073, 36083, 36097, 36107, 36109, 36131, 36137, 36151, 36161, |
| 525 | 36187, 36191, 36209, 36217, 36229, 36241, 36251, 36263, 36269, |
| 526 | 36277, 36293, 36299, 36307, 36313, 36319, 36341, 36343, 36353, |
| 527 | 36373, 36383, 36389, 36433, 36451, 36457, 36467, 36469, 36473, |
| 528 | 36479, 36493, 36497, 36523, 36527, 36529, 36541, 36551, 36559, |
| 529 | 36563, 36571, 36583, 36587, 36599, 36607, 36629, 36637, 36643, |
| 530 | 36653, 36671, 36677, 36683, 36691, 36697, 36709, 36713, 36721, |
| 531 | 36739, 36749, 36761, 36767, 36779, 36781, 36787, 36791, 36793, |
| 532 | 36809, 36821, 36833, 36847, 36857, 36871, 36877, 36887, 36899, |
| 533 | 36901, 36913, 36919, 36923, 36929, 36931, 36943, 36947, 36973, |
| 534 | 36979, 36997, 37003, 37013, 37019, 37021, 37039, 37049, 37057, |
| 535 | 37061, 37087, 37097, 37117, 37123, 37139, 37159, 37171, 37181, |
| 536 | 37189, 37199, 37201, 37217, 37223, 37243, 37253, 37273, 37277, |
| 537 | 37307, 37309, 37313, 37321, 37337, 37339, 37357, 37361, 37363, |
| 538 | 37369, 37379, 37397, 37409, 37423, 37441, 37447, 37463, 37483, |
| 539 | 37489, 37493, 37501, 37507, 37511, 37517, 37529, 37537, 37547, |
| 540 | 37549, 37561, 37567, 37571, 37573, 37579, 37589, 37591, 37607, |
| 541 | 37619, 37633, 37643, 37649, 37657, 37663, 37691, 37693, 37699, |
| 542 | 37717, 37747, 37781, 37783, 37799, 37811, 37813, 37831, 37847, |
| 543 | 37853, 37861, 37871, 37879, 37889, 37897, 37907, 37951, 37957, |
| 544 | 37963, 37967, 37987, 37991, 37993, 37997, 38011, 38039, 38047, |
| 545 | 38053, 38069, 38083, 38113, 38119, 38149, 38153, 38167, 38177, |
| 546 | 38183, 38189, 38197, 38201, 38219, 38231, 38237, 38239, 38261, |
| 547 | 38273, 38281, 38287, 38299, 38303, 38317, 38321, 38327, 38329, |
| 548 | 38333, 38351, 38371, 38377, 38393, 38431, 38447, 38449, 38453, |
| 549 | 38459, 38461, 38501, 38543, 38557, 38561, 38567, 38569, 38593, |
| 550 | 38603, 38609, 38611, 38629, 38639, 38651, 38653, 38669, 38671, |
| 551 | 38677, 38693, 38699, 38707, 38711, 38713, 38723, 38729, 38737, |
| 552 | 38747, 38749, 38767, 38783, 38791, 38803, 38821, 38833, 38839, |
| 553 | 38851, 38861, 38867, 38873, 38891, 38903, 38917, 38921, 38923, |
| 554 | 38933, 38953, 38959, 38971, 38977, 38993, 39019, 39023, 39041, |
| 555 | 39043, 39047, 39079, 39089, 39097, 39103, 39107, 39113, 39119, |
| 556 | 39133, 39139, 39157, 39161, 39163, 39181, 39191, 39199, 39209, |
| 557 | 39217, 39227, 39229, 39233, 39239, 39241, 39251, 39293, 39301, |
| 558 | 39313, 39317, 39323, 39341, 39343, 39359, 39367, 39371, 39373, |
| 559 | 39383, 39397, 39409, 39419, 39439, 39443, 39451, 39461, 39499, |
| 560 | 39503, 39509, 39511, 39521, 39541, 39551, 39563, 39569, 39581, |
| 561 | 39607, 39619, 39623, 39631, 39659, 39667, 39671, 39679, 39703, |
| 562 | 39709, 39719, 39727, 39733, 39749, 39761, 39769, 39779, 39791, |
| 563 | 39799, 39821, 39827, 39829, 39839, 39841, 39847, 39857, 39863, |
| 564 | 39869, 39877, 39883, 39887, 39901, 39929, 39937, 39953, 39971, |
| 565 | 39979, 39983, 39989, 40009, 40013, 40031, 40037, 40039, 40063, |
| 566 | 40087, 40093, 40099, 40111, 40123, 40127, 40129, 40151, 40153, |
| 567 | 40163, 40169, 40177, 40189, 40193, 40213, 40231, 40237, 40241, |
| 568 | 40253, 40277, 40283, 40289, 40343, 40351, 40357, 40361, 40387, |
| 569 | 40423, 40427, 40429, 40433, 40459, 40471, 40483, 40487, 40493, |
| 570 | 40499, 40507, 40519, 40529, 40531, 40543, 40559, 40577, 40583, |
| 571 | 40591, 40597, 40609, 40627, 40637, 40639, 40693, 40697, 40699, |
| 572 | 40709, 40739, 40751, 40759, 40763, 40771, 40787, 40801, 40813, |
| 573 | 40819, 40823, 40829, 40841, 40847, 40849, 40853, 40867, 40879, |
| 574 | 40883, 40897, 40903, 40927, 40933, 40939, 40949, 40961, 40973, |
| 575 | 40993, 41011, 41017, 41023, 41039, 41047, 41051, 41057, 41077, |
| 576 | 41081, 41113, 41117, 41131, 41141, 41143, 41149, 41161, 41177, |
| 577 | 41179, 41183, 41189, 41201, 41203, 41213, 41221, 41227, 41231, |
| 578 | 41233, 41243, 41257, 41263, 41269, 41281, 41299, 41333, 41341, |
| 579 | 41351, 41357, 41381, 41387, 41389, 41399, 41411, 41413, 41443, |
| 580 | 41453, 41467, 41479, 41491, 41507, 41513, 41519, 41521, 41539, |
| 581 | 41543, 41549, 41579, 41593, 41597, 41603, 41609, 41611, 41617, |
| 582 | 41621, 41627, 41641, 41647, 41651, 41659, 41669, 41681, 41687, |
| 583 | 41719, 41729, 41737, 41759, 41761, 41771, 41777, 41801, 41809, |
| 584 | 41813, 41843, 41849, 41851, 41863, 41879, 41887, 41893, 41897, |
| 585 | 41903, 41911, 41927, 41941, 41947, 41953, 41957, 41959, 41969, |
| 586 | 41981, 41983, 41999, 42013, 42017, 42019, 42023, 42043, 42061, |
| 587 | 42071, 42073, 42083, 42089, 42101, 42131, 42139, 42157, 42169, |
| 588 | 42179, 42181, 42187, 42193, 42197, 42209, 42221, 42223, 42227, |
| 589 | 42239, 42257, 42281, 42283, 42293, 42299, 42307, 42323, 42331, |
| 590 | 42337, 42349, 42359, 42373, 42379, 42391, 42397, 42403, 42407, |
| 591 | 42409, 42433, 42437, 42443, 42451, 42457, 42461, 42463, 42467, |
| 592 | 42473, 42487, 42491, 42499, 42509, 42533, 42557, 42569, 42571, |
| 593 | 42577, 42589, 42611, 42641, 42643, 42649, 42667, 42677, 42683, |
| 594 | 42689, 42697, 42701, 42703, 42709, 42719, 42727, 42737, 42743, |
| 595 | 42751, 42767, 42773, 42787, 42793, 42797, 42821, 42829, 42839, |
| 596 | 42841, 42853, 42859, 42863, 42899, 42901, 42923, 42929, 42937, |
| 597 | 42943, 42953, 42961, 42967, 42979, 42989, 43003, 43013, 43019, |
| 598 | 43037, 43049, 43051, 43063, 43067, 43093, 43103, 43117, 43133, |
| 599 | 43151, 43159, 43177, 43189, 43201, 43207, 43223, 43237, 43261, |
| 600 | 43271, 43283, 43291, 43313, 43319, 43321, 43331, 43391, 43397, |
| 601 | 43399, 43403, 43411, 43427, 43441, 43451, 43457, 43481, 43487, |
| 602 | 43499, 43517, 43541, 43543, 43573, 43577, 43579, 43591, 43597, |
| 603 | 43607, 43609, 43613, 43627, 43633, 43649, 43651, 43661, 43669, |
| 604 | 43691, 43711, 43717, 43721, 43753, 43759, 43777, 43781, 43783, |
| 605 | 43787, 43789, 43793, 43801, 43853, 43867, 43889, 43891, 43913, |
| 606 | 43933, 43943, 43951, 43961, 43963, 43969, 43973, 43987, 43991, |
| 607 | 43997, 44017, 44021, 44027, 44029, 44041, 44053, 44059, 44071, |
| 608 | 44087, 44089, 44101, 44111, 44119, 44123, 44129, 44131, 44159, |
| 609 | 44171, 44179, 44189, 44201, 44203, 44207, 44221, 44249, 44257, |
| 610 | 44263, 44267, 44269, 44273, 44279, 44281, 44293, 44351, 44357, |
| 611 | 44371, 44381, 44383, 44389, 44417, 44449, 44453, 44483, 44491, |
| 612 | 44497, 44501, 44507, 44519, 44531, 44533, 44537, 44543, 44549, |
| 613 | 44563, 44579, 44587, 44617, 44621, 44623, 44633, 44641, 44647, |
| 614 | 44651, 44657, 44683, 44687, 44699, 44701, 44711, 44729, 44741, |
| 615 | 44753, 44771, 44773, 44777, 44789, 44797, 44809, 44819, 44839, |
| 616 | 44843, 44851, 44867, 44879, 44887, 44893, 44909, 44917, 44927, |
| 617 | 44939, 44953, 44959, 44963, 44971, 44983, 44987, 45007, 45013, |
| 618 | 45053, 45061, 45077, 45083, 45119, 45121, 45127, 45131, 45137, |
| 619 | 45139, 45161, 45179, 45181, 45191, 45197, 45233, 45247, 45259, |
| 620 | 45263, 45281, 45289, 45293, 45307, 45317, 45319, 45329, 45337, |
| 621 | 45341, 45343, 45361, 45377, 45389, 45403, 45413, 45427, 45433, |
| 622 | 45439, 45481, 45491, 45497, 45503, 45523, 45533, 45541, 45553, |
| 623 | 45557, 45569, 45587, 45589, 45599, 45613, 45631, 45641, 45659, |
| 624 | 45667, 45673, 45677, 45691, 45697, 45707, 45737, 45751, 45757, |
| 625 | 45763, 45767, 45779, 45817, 45821, 45823, 45827, 45833, 45841, |
| 626 | 45853, 45863, 45869, 45887, 45893, 45943, 45949, 45953, 45959, |
| 627 | 45971, 45979, 45989, 46021, 46027, 46049, 46051, 46061, 46073, |
| 628 | 46091, 46093, 46099, 46103, 46133, 46141, 46147, 46153, 46171, |
| 629 | 46181, 46183, 46187, 46199, 46219, 46229, 46237, 46261, 46271, |
| 630 | 46273, 46279, 46301, 46307, 46309, 46327, 46337, 46349, 46351, |
| 631 | 46381, 46399, 46411, 46439, 46441, 46447, 46451, 46457, 46471, |
| 632 | 46477, 46489, 46499, 46507, 46511, 46523, 46549, 46559, 46567, |
| 633 | 46573, 46589, 46591, 46601, 46619, 46633, 46639, 46643, 46649, |
| 634 | 46663, 46679, 46681, 46687, 46691, 46703, 46723, 46727, 46747, |
| 635 | 46751, 46757, 46769, 46771, 46807, 46811, 46817, 46819, 46829, |
| 636 | 46831, 46853, 46861, 46867, 46877, 46889, 46901, 46919, 46933, |
| 637 | 46957, 46993, 46997, 47017, 47041, 47051, 47057, 47059, 47087, |
| 638 | 47093, 47111, 47119, 47123, 47129, 47137, 47143, 47147, 47149, |
| 639 | 47161, 47189, 47207, 47221, 47237, 47251, 47269, 47279, 47287, |
| 640 | 47293, 47297, 47303, 47309, 47317, 47339, 47351, 47353, 47363, |
| 641 | 47381, 47387, 47389, 47407, 47417, 47419, 47431, 47441, 47459, |
| 642 | 47491, 47497, 47501, 47507, 47513, 47521, 47527, 47533, 47543, |
| 643 | 47563, 47569, 47581, 47591, 47599, 47609, 47623, 47629, 47639, |
| 644 | 47653, 47657, 47659, 47681, 47699, 47701, 47711, 47713, 47717, |
| 645 | 47737, 47741, 47743, 47777, 47779, 47791, 47797, 47807, 47809, |
| 646 | 47819, 47837, 47843, 47857, 47869, 47881, 47903, 47911, 47917, |
| 647 | 47933, 47939, 47947, 47951, 47963, 47969, 47977, 47981, 48017, |
| 648 | 48023, 48029, 48049, 48073, 48079, 48091, 48109, 48119, 48121, |
| 649 | 48131, 48157, 48163, 48179, 48187, 48193, 48197, 48221, 48239, |
| 650 | 48247, 48259, 48271, 48281, 48299, 48311, 48313, 48337, 48341, |
| 651 | 48353, 48371, 48383, 48397, 48407, 48409, 48413, 48437, 48449, |
| 652 | 48463, 48473, 48479, 48481, 48487, 48491, 48497, 48523, 48527, |
| 653 | 48533, 48539, 48541, 48563, 48571, 48589, 48593, 48611, 48619, |
| 654 | 48623, 48647, 48649, 48661, 48673, 48677, 48679, 48731, 48733, |
| 655 | 48751, 48757, 48761, 48767, 48779, 48781, 48787, 48799, 48809, |
| 656 | 48817, 48821, 48823, 48847, 48857, 48859, 48869, 48871, 48883, |
| 657 | 48889, 48907, 48947, 48953, 48973, 48989, 48991, 49003, 49009, |
| 658 | 49019, 49031, 49033, 49037, 49043, 49057, 49069, 49081, 49103, |
| 659 | 49109, 49117, 49121, 49123, 49139, 49157, 49169, 49171, 49177, |
| 660 | 49193, 49199, 49201, 49207, 49211, 49223, 49253, 49261, 49277, |
| 661 | 49279, 49297, 49307, 49331, 49333, 49339, 49363, 49367, 49369, |
| 662 | 49391, 49393, 49409, 49411, 49417, 49429, 49433, 49451, 49459, |
| 663 | 49463, 49477, 49481, 49499, 49523, 49529, 49531, 49537, 49547, |
| 664 | 49549, 49559, 49597, 49603, 49613, 49627, 49633, 49639, 49663, |
| 665 | 49667, 49669, 49681, 49697, 49711, 49727, 49739, 49741, 49747, |
| 666 | 49757, 49783, 49787, 49789, 49801, 49807, 49811, 49823, 49831, |
| 667 | 49843, 49853, 49871, 49877, 49891, 49919, 49921, 49927, 49937, |
| 668 | 49939, 49943, 49957, 49991, 49993, 49999, 50021, 50023, 50033, |
| 669 | 50047, 50051, 50053, 50069, 50077, 50087, 50093, 50101, 50111, |
| 670 | 50119, 50123, 50129, 50131, 50147, 50153, 50159, 50177, 50207, |
| 671 | 50221, 50227, 50231, 50261, 50263, 50273, 50287, 50291, 50311, |
| 672 | 50321, 50329, 50333, 50341, 50359, 50363, 50377, 50383, 50387, |
| 673 | 50411, 50417, 50423, 50441, 50459, 50461, 50497, 50503, 50513, |
| 674 | 50527, 50539, 50543, 50549, 50551, 50581, 50587, 50591, 50593, |
| 675 | 50599, 50627, 50647, 50651, 50671, 50683, 50707, 50723, 50741, |
| 676 | 50753, 50767, 50773, 50777, 50789, 50821, 50833, 50839, 50849, |
| 677 | 50857, 50867, 50873, 50891, 50893, 50909, 50923, 50929, 50951, |
| 678 | 50957, 50969, 50971, 50989, 50993, 51001, 51031, 51043, 51047, |
| 679 | 51059, 51061, 51071, 51109, 51131, 51133, 51137, 51151, 51157, |
| 680 | 51169, 51193, 51197, 51199, 51203, 51217, 51229, 51239, 51241, |
| 681 | 51257, 51263, 51283, 51287, 51307, 51329, 51341, 51343, 51347, |
| 682 | 51349, 51361, 51383, 51407, 51413, 51419, 51421, 51427, 51431, |
| 683 | 51437, 51439, 51449, 51461, 51473, 51479, 51481, 51487, 51503, |
| 684 | 51511, 51517, 51521, 51539, 51551, 51563, 51577, 51581, 51593, |
| 685 | 51599, 51607, 51613, 51631, 51637, 51647, 51659, 51673, 51679, |
| 686 | 51683, 51691, 51713, 51719, 51721, 51749, 51767, 51769, 51787, |
| 687 | 51797, 51803, 51817, 51827, 51829, 51839, 51853, 51859, 51869, |
| 688 | 51871, 51893, 51899, 51907, 51913, 51929, 51941, 51949, 51971, |
| 689 | 51973, 51977, 51991, 52009, 52021, 52027, 52051, 52057, 52067, |
| 690 | 52069, 52081, 52103, 52121, 52127, 52147, 52153, 52163, 52177, |
| 691 | 52181, 52183, 52189, 52201, 52223, 52237, 52249, 52253, 52259, |
| 692 | 52267, 52289, 52291, 52301, 52313, 52321, 52361, 52363, 52369, |
| 693 | 52379, 52387, 52391, 52433, 52453, 52457, 52489, 52501, 52511, |
| 694 | 52517, 52529, 52541, 52543, 52553, 52561, 52567, 52571, 52579, |
| 695 | 52583, 52609, 52627, 52631, 52639, 52667, 52673, 52691, 52697, |
| 696 | 52709, 52711, 52721, 52727, 52733, 52747, 52757, 52769, 52783, |
| 697 | 52807, 52813, 52817, 52837, 52859, 52861, 52879, 52883, 52889, |
| 698 | 52901, 52903, 52919, 52937, 52951, 52957, 52963, 52967, 52973, |
| 699 | 52981, 52999, 53003, 53017, 53047, 53051, 53069, 53077, 53087, |
| 700 | 53089, 53093, 53101, 53113, 53117, 53129, 53147, 53149, 53161, |
| 701 | 53171, 53173, 53189, 53197, 53201, 53231, 53233, 53239, 53267, |
| 702 | 53269, 53279, 53281, 53299, 53309, 53323, 53327, 53353, 53359, |
| 703 | 53377, 53381, 53401, 53407, 53411, 53419, 53437, 53441, 53453, |
| 704 | 53479, 53503, 53507, 53527, 53549, 53551, 53569, 53591, 53593, |
| 705 | 53597, 53609, 53611, 53617, 53623, 53629, 53633, 53639, 53653, |
| 706 | 53657, 53681, 53693, 53699, 53717, 53719, 53731, 53759, 53773, |
| 707 | 53777, 53783, 53791, 53813, 53819, 53831, 53849, 53857, 53861, |
| 708 | 53881, 53887, 53891, 53897, 53899, 53917, 53923, 53927, 53939, |
| 709 | 53951, 53959, 53987, 53993, 54001, 54011, 54013, 54037, 54049, |
| 710 | 54059, 54083, 54091, 54101, 54121, 54133, 54139, 54151, 54163, |
| 711 | 54167, 54181, 54193, 54217, 54251, 54269, 54277, 54287, 54293, |
| 712 | 54311, 54319, 54323, 54331, 54347, 54361, 54367, 54371, 54377, |
| 713 | 54401, 54403, 54409, 54413, 54419, 54421, 54437, 54443, 54449, |
| 714 | 54469, 54493, 54497, 54499, 54503, 54517, 54521, 54539, 54541, |
| 715 | 54547, 54559, 54563, 54577, 54581, 54583, 54601, 54617, 54623, |
| 716 | 54629, 54631, 54647, 54667, 54673, 54679, 54709, 54713, 54721, |
| 717 | 54727, 54751, 54767, 54773, 54779, 54787, 54799, 54829, 54833, |
| 718 | 54851, 54869, 54877, 54881, 54907, 54917, 54919, 54941, 54949, |
| 719 | 54959, 54973, 54979, 54983, 55001, 55009, 55021, 55049, 55051, |
| 720 | 55057, 55061, 55073, 55079, 55103, 55109, 55117, 55127, 55147, |
| 721 | 55163, 55171, 55201, 55207, 55213, 55217, 55219, 55229, 55243, |
| 722 | 55249, 55259, 55291, 55313, 55331, 55333, 55337, 55339, 55343, |
| 723 | 55351, 55373, 55381, 55399, 55411, 55439, 55441, 55457, 55469, |
| 724 | 55487, 55501, 55511, 55529, 55541, 55547, 55579, 55589, 55603, |
| 725 | 55609, 55619, 55621, 55631, 55633, 55639, 55661, 55663, 55667, |
| 726 | 55673, 55681, 55691, 55697, 55711, 55717, 55721, 55733, 55763, |
| 727 | 55787, 55793, 55799, 55807, 55813, 55817, 55819, 55823, 55829, |
| 728 | 55837, 55843, 55849, 55871, 55889, 55897, 55901, 55903, 55921, |
| 729 | 55927, 55931, 55933, 55949, 55967, 55987, 55997, 56003, 56009, |
| 730 | 56039, 56041, 56053, 56081, 56087, 56093, 56099, 56101, 56113, |
| 731 | 56123, 56131, 56149, 56167, 56171, 56179, 56197, 56207, 56209, |
| 732 | 56237, 56239, 56249, 56263, 56267, 56269, 56299, 56311, 56333, |
| 733 | 56359, 56369, 56377, 56383, 56393, 56401, 56417, 56431, 56437, |
| 734 | 56443, 56453, 56467, 56473, 56477, 56479, 56489, 56501, 56503, |
| 735 | 56509, 56519, 56527, 56531, 56533, 56543, 56569, 56591, 56597, |
| 736 | 56599, 56611, 56629, 56633, 56659, 56663, 56671, 56681, 56687, |
| 737 | 56701, 56711, 56713, 56731, 56737, 56747, 56767, 56773, 56779, |
| 738 | 56783, 56807, 56809, 56813, 56821, 56827, 56843, 56857, 56873, |
| 739 | 56891, 56893, 56897, 56909, 56911, 56921, 56923, 56929, 56941, |
| 740 | 56951, 56957, 56963, 56983, 56989, 56993, 56999, 57037, 57041, |
| 741 | 57047, 57059, 57073, 57077, 57089, 57097, 57107, 57119, 57131, |
| 742 | 57139, 57143, 57149, 57163, 57173, 57179, 57191, 57193, 57203, |
| 743 | 57221, 57223, 57241, 57251, 57259, 57269, 57271, 57283, 57287, |
| 744 | 57301, 57329, 57331, 57347, 57349, 57367, 57373, 57383, 57389, |
| 745 | 57397, 57413, 57427, 57457, 57467, 57487, 57493, 57503, 57527, |
| 746 | 57529, 57557, 57559, 57571, 57587, 57593, 57601, 57637, 57641, |
| 747 | 57649, 57653, 57667, 57679, 57689, 57697, 57709, 57713, 57719, |
| 748 | 57727, 57731, 57737, 57751, 57773, 57781, 57787, 57791, 57793, |
| 749 | 57803, 57809, 57829, 57839, 57847, 57853, 57859, 57881, 57899, |
| 750 | 57901, 57917, 57923, 57943, 57947, 57973, 57977, 57991, 58013, |
| 751 | 58027, 58031, 58043, 58049, 58057, 58061, 58067, 58073, 58099, |
| 752 | 58109, 58111, 58129, 58147, 58151, 58153, 58169, 58171, 58189, |
| 753 | 58193, 58199, 58207, 58211, 58217, 58229, 58231, 58237, 58243, |
| 754 | 58271, 58309, 58313, 58321, 58337, 58363, 58367, 58369, 58379, |
| 755 | 58391, 58393, 58403, 58411, 58417, 58427, 58439, 58441, 58451, |
| 756 | 58453, 58477, 58481, 58511, 58537, 58543, 58549, 58567, 58573, |
| 757 | 58579, 58601, 58603, 58613, 58631, 58657, 58661, 58679, 58687, |
| 758 | 58693, 58699, 58711, 58727, 58733, 58741, 58757, 58763, 58771, |
| 759 | 58787, 58789, 58831, 58889, 58897, 58901, 58907, 58909, 58913, |
| 760 | 58921, 58937, 58943, 58963, 58967, 58979, 58991, 58997, 59009, |
| 761 | 59011, 59021, 59023, 59029, 59051, 59053, 59063, 59069, 59077, |
| 762 | 59083, 59093, 59107, 59113, 59119, 59123, 59141, 59149, 59159, |
| 763 | 59167, 59183, 59197, 59207, 59209, 59219, 59221, 59233, 59239, |
| 764 | 59243, 59263, 59273, 59281, 59333, 59341, 59351, 59357, 59359, |
| 765 | 59369, 59377, 59387, 59393, 59399, 59407, 59417, 59419, 59441, |
| 766 | 59443, 59447, 59453, 59467, 59471, 59473, 59497, 59509, 59513, |
| 767 | 59539, 59557, 59561, 59567, 59581, 59611, 59617, 59621, 59627, |
| 768 | 59629, 59651, 59659, 59663, 59669, 59671, 59693, 59699, 59707, |
| 769 | 59723, 59729, 59743, 59747, 59753, 59771, 59779, 59791, 59797, |
| 770 | 59809, 59833, 59863, 59879, 59887, 59921, 59929, 59951, 59957, |
| 771 | 59971, 59981, 59999, 60013, 60017, 60029, 60037, 60041, 60077, |
| 772 | 60083, 60089, 60091, 60101, 60103, 60107, 60127, 60133, 60139, |
| 773 | 60149, 60161, 60167, 60169, 60209, 60217, 60223, 60251, 60257, |
| 774 | 60259, 60271, 60289, 60293, 60317, 60331, 60337, 60343, 60353, |
| 775 | 60373, 60383, 60397, 60413, 60427, 60443, 60449, 60457, 60493, |
| 776 | 60497, 60509, 60521, 60527, 60539, 60589, 60601, 60607, 60611, |
| 777 | 60617, 60623, 60631, 60637, 60647, 60649, 60659, 60661, 60679, |
| 778 | 60689, 60703, 60719, 60727, 60733, 60737, 60757, 60761, 60763, |
| 779 | 60773, 60779, 60793, 60811, 60821, 60859, 60869, 60887, 60889, |
| 780 | 60899, 60901, 60913, 60917, 60919, 60923, 60937, 60943, 60953, |
| 781 | 60961, 61001, 61007, 61027, 61031, 61043, 61051, 61057, 61091, |
| 782 | 61099, 61121, 61129, 61141, 61151, 61153, 61169, 61211, 61223, |
| 783 | 61231, 61253, 61261, 61283, 61291, 61297, 61331, 61333, 61339, |
| 784 | 61343, 61357, 61363, 61379, 61381, 61403, 61409, 61417, 61441, |
| 785 | 61463, 61469, 61471, 61483, 61487, 61493, 61507, 61511, 61519, |
| 786 | 61543, 61547, 61553, 61559, 61561, 61583, 61603, 61609, 61613, |
| 787 | 61627, 61631, 61637, 61643, 61651, 61657, 61667, 61673, 61681, |
| 788 | 61687, 61703, 61717, 61723, 61729, 61751, 61757, 61781, 61813, |
| 789 | 61819, 61837, 61843, 61861, 61871, 61879, 61909, 61927, 61933, |
| 790 | 61949, 61961, 61967, 61979, 61981, 61987, 61991, 62003, 62011, |
| 791 | 62017, 62039, 62047, 62053, 62057, 62071, 62081, 62099, 62119, |
| 792 | 62129, 62131, 62137, 62141, 62143, 62171, 62189, 62191, 62201, |
| 793 | 62207, 62213, 62219, 62233, 62273, 62297, 62299, 62303, 62311, |
| 794 | 62323, 62327, 62347, 62351, 62383, 62401, 62417, 62423, 62459, |
| 795 | 62467, 62473, 62477, 62483, 62497, 62501, 62507, 62533, 62539, |
| 796 | 62549, 62563, 62581, 62591, 62597, 62603, 62617, 62627, 62633, |
| 797 | 62639, 62653, 62659, 62683, 62687, 62701, 62723, 62731, 62743, |
| 798 | 62753, 62761, 62773, 62791, 62801, 62819, 62827, 62851, 62861, |
| 799 | 62869, 62873, 62897, 62903, 62921, 62927, 62929, 62939, 62969, |
| 800 | 62971, 62981, 62983, 62987, 62989, 63029, 63031, 63059, 63067, |
| 801 | 63073, 63079, 63097, 63103, 63113, 63127, 63131, 63149, 63179, |
| 802 | 63197, 63199, 63211, 63241, 63247, 63277, 63281, 63299, 63311, |
| 803 | 63313, 63317, 63331, 63337, 63347, 63353, 63361, 63367, 63377, |
| 804 | 63389, 63391, 63397, 63409, 63419, 63421, 63439, 63443, 63463, |
| 805 | 63467, 63473, 63487, 63493, 63499, 63521, 63527, 63533, 63541, |
| 806 | 63559, 63577, 63587, 63589, 63599, 63601, 63607, 63611, 63617, |
| 807 | 63629, 63647, 63649, 63659, 63667, 63671, 63689, 63691, 63697, |
| 808 | 63703, 63709, 63719, 63727, 63737, 63743, 63761, 63773, 63781, |
| 809 | 63793, 63799, 63803, 63809, 63823, 63839, 63841, 63853, 63857, |
| 810 | 63863, 63901, 63907, 63913, 63929, 63949, 63977, 63997, 64007, |
| 811 | 64013, 64019, 64033, 64037, 64063, 64067, 64081, 64091, 64109, |
| 812 | 64123, 64151, 64153, 64157, 64171, 64187, 64189, 64217, 64223, |
| 813 | 64231, 64237, 64271, 64279, 64283, 64301, 64303, 64319, 64327, |
| 814 | 64333, 64373, 64381, 64399, 64403, 64433, 64439, 64451, 64453, |
| 815 | 64483, 64489, 64499, 64513, 64553, 64567, 64577, 64579, 64591, |
| 816 | 64601, 64609, 64613, 64621, 64627, 64633, 64661, 64663, 64667, |
| 817 | 64679, 64693, 64709, 64717, 64747, 64763, 64781, 64783, 64793, |
| 818 | 64811, 64817, 64849, 64853, 64871, 64877, 64879, 64891, 64901, |
| 819 | 64919, 64921, 64927, 64937, 64951, 64969, 64997, 65003, 65011, |
| 820 | 65027, 65029, 65033, 65053, 65063, 65071, 65089, 65099, 65101, |
| 821 | 65111, 65119, 65123, 65129, 65141, 65147, 65167, 65171, 65173, |
| 822 | 65179, 65183, 65203, 65213, 65239, 65257, 65267, 65269, 65287, |
| 823 | 65293, 65309, 65323, 65327, 65353, 65357, 65371, 65381, 65393, |
| 824 | 65407, 65413, 65419, 65423, 65437, 65447, 65449, 65479, 65497, |
| 825 | 65519, 65521, |
| 9400cf6f |
826 | }; |
| 827 | |
| 828 | #define NPRIMES (sizeof(primes) / sizeof(*primes)) |
| 829 | |
| 830 | /* |
| 5c72ca61 |
831 | * Generate a prime. We can deal with various extra properties of |
| 832 | * the prime: |
| 833 | * |
| 834 | * - to speed up use in RSA, we can arrange to select a prime with |
| 835 | * the property (prime % modulus) != residue. |
| 836 | * |
| 837 | * - for use in DSA, we can arrange to select a prime which is one |
| 838 | * more than a multiple of a dirty great bignum. In this case |
| 839 | * `bits' gives the size of the factor by which we _multiply_ |
| 840 | * that bignum, rather than the size of the whole number. |
| 79dae043 |
841 | * |
| 842 | * - for the basically cosmetic purposes of generating keys of the |
| 843 | * length actually specified rather than off by one bit, we permit |
| 844 | * the caller to provide an unsigned integer 'firstbits' which will |
| 845 | * match the top few bits of the returned prime. (That is, there |
| 846 | * will exist some n such that (returnvalue >> n) == firstbits.) If |
| 847 | * 'firstbits' is not needed, specifying it to either 0 or 1 is |
| 848 | * an adequate no-op. |
| 9400cf6f |
849 | */ |
| 5c72ca61 |
850 | Bignum primegen(int bits, int modulus, int residue, Bignum factor, |
| 79dae043 |
851 | int phase, progfn_t pfn, void *pfnparam, unsigned firstbits) |
| 32874aea |
852 | { |
| 79dae043 |
853 | int i, k, v, byte, bitsleft, check, checks, fbsize; |
| 5c72ca61 |
854 | unsigned long delta; |
| 855 | unsigned long moduli[NPRIMES + 1]; |
| 856 | unsigned long residues[NPRIMES + 1]; |
| 857 | unsigned long multipliers[NPRIMES + 1]; |
| b73c0b37 |
858 | Bignum p, pm1, q, wqp, wqp2; |
| 9400cf6f |
859 | int progress = 0; |
| 860 | |
| 32874aea |
861 | byte = 0; |
| 862 | bitsleft = 0; |
| 9400cf6f |
863 | |
| 79dae043 |
864 | fbsize = 0; |
| 865 | while (firstbits >> fbsize) /* work out how to align this */ |
| 866 | fbsize++; |
| 867 | |
| 32874aea |
868 | STARTOVER: |
| 9400cf6f |
869 | |
| 5c72ca61 |
870 | pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress); |
| 9400cf6f |
871 | |
| 872 | /* |
| 873 | * Generate a k-bit random number with top and bottom bits set. |
| 5c72ca61 |
874 | * Alternatively, if `factor' is nonzero, generate a k-bit |
| 875 | * random number with the top bit set and the bottom bit clear, |
| 876 | * multiply it by `factor', and add one. |
| 9400cf6f |
877 | */ |
| 32874aea |
878 | p = bn_power_2(bits - 1); |
| 9400cf6f |
879 | for (i = 0; i < bits; i++) { |
| 79dae043 |
880 | if (i == 0 || i == bits - 1) { |
| 5c72ca61 |
881 | v = (i != 0 || !factor) ? 1 : 0; |
| 79dae043 |
882 | } else if (i >= bits - fbsize) { |
| 883 | v = (firstbits >> (i - (bits - fbsize))) & 1; |
| 884 | } else { |
| 32874aea |
885 | if (bitsleft <= 0) |
| 886 | bitsleft = 8, byte = random_byte(); |
| 887 | v = byte & 1; |
| 888 | byte >>= 1; |
| 889 | bitsleft--; |
| 890 | } |
| 891 | bignum_set_bit(p, i, v); |
| 9400cf6f |
892 | } |
| 5c72ca61 |
893 | if (factor) { |
| 894 | Bignum tmp = p; |
| 895 | p = bigmul(tmp, factor); |
| 896 | freebn(tmp); |
| 897 | assert(bignum_bit(p, 0) == 0); |
| 898 | bignum_set_bit(p, 0, 1); |
| 899 | } |
| 9400cf6f |
900 | |
| 901 | /* |
| 902 | * Ensure this random number is coprime to the first few |
| 5c72ca61 |
903 | * primes, by repeatedly adding either 2 or 2*factor to it |
| 904 | * until it is. |
| 9400cf6f |
905 | */ |
| 906 | for (i = 0; i < NPRIMES; i++) { |
| 32874aea |
907 | moduli[i] = primes[i]; |
| 908 | residues[i] = bignum_mod_short(p, primes[i]); |
| 5c72ca61 |
909 | if (factor) |
| 910 | multipliers[i] = bignum_mod_short(factor, primes[i]); |
| 911 | else |
| 912 | multipliers[i] = 1; |
| 9400cf6f |
913 | } |
| 914 | moduli[NPRIMES] = modulus; |
| 32874aea |
915 | residues[NPRIMES] = (bignum_mod_short(p, (unsigned short) modulus) |
| 916 | + modulus - residue); |
| f26b3d77 |
917 | if (factor) |
| 918 | multipliers[NPRIMES] = bignum_mod_short(factor, modulus); |
| 919 | else |
| 920 | multipliers[NPRIMES] = 1; |
| 9400cf6f |
921 | delta = 0; |
| 922 | while (1) { |
| 32874aea |
923 | for (i = 0; i < (sizeof(moduli) / sizeof(*moduli)); i++) |
| 5c72ca61 |
924 | if (!((residues[i] + delta * multipliers[i]) % moduli[i])) |
| 32874aea |
925 | break; |
| 926 | if (i < (sizeof(moduli) / sizeof(*moduli))) { /* we broke */ |
| 927 | delta += 2; |
| 5c72ca61 |
928 | if (delta > 65536) { |
| 32874aea |
929 | freebn(p); |
| 930 | goto STARTOVER; |
| 931 | } |
| 932 | continue; |
| 933 | } |
| 934 | break; |
| 9400cf6f |
935 | } |
| 936 | q = p; |
| 5c72ca61 |
937 | if (factor) { |
| 938 | Bignum tmp; |
| 939 | tmp = bignum_from_long(delta); |
| 940 | p = bigmuladd(tmp, factor, q); |
| 941 | freebn(tmp); |
| 942 | } else { |
| 943 | p = bignum_add_long(q, delta); |
| 944 | } |
| 9400cf6f |
945 | freebn(q); |
| 946 | |
| 947 | /* |
| b73c0b37 |
948 | * Now apply the Miller-Rabin primality test a few times. First |
| 949 | * work out how many checks are needed. |
| 9400cf6f |
950 | */ |
| 951 | checks = 27; |
| 32874aea |
952 | if (bits >= 150) |
| 953 | checks = 18; |
| 954 | if (bits >= 200) |
| 955 | checks = 15; |
| 956 | if (bits >= 250) |
| 957 | checks = 12; |
| 958 | if (bits >= 300) |
| 959 | checks = 9; |
| 960 | if (bits >= 350) |
| 961 | checks = 8; |
| 962 | if (bits >= 400) |
| 963 | checks = 7; |
| 964 | if (bits >= 450) |
| 965 | checks = 6; |
| 966 | if (bits >= 550) |
| 967 | checks = 5; |
| 968 | if (bits >= 650) |
| 969 | checks = 4; |
| 970 | if (bits >= 850) |
| 971 | checks = 3; |
| 972 | if (bits >= 1300) |
| 973 | checks = 2; |
| 9400cf6f |
974 | |
| 975 | /* |
| 976 | * Next, write p-1 as q*2^k. |
| 977 | */ |
| e4b4f5a1 |
978 | for (k = 0; bignum_bit(p, k) == !k; k++) |
| 979 | continue; /* find first 1 bit in p-1 */ |
| 9400cf6f |
980 | q = bignum_rshift(p, k); |
| b73c0b37 |
981 | /* And store p-1 itself, which we'll need. */ |
| 982 | pm1 = copybn(p); |
| 983 | decbn(pm1); |
| 9400cf6f |
984 | |
| 985 | /* |
| 986 | * Now, for each check ... |
| 987 | */ |
| 988 | for (check = 0; check < checks; check++) { |
| 32874aea |
989 | Bignum w; |
| 9400cf6f |
990 | |
| 32874aea |
991 | /* |
| 992 | * Invent a random number between 1 and p-1 inclusive. |
| 993 | */ |
| 994 | while (1) { |
| 995 | w = bn_power_2(bits - 1); |
| 996 | for (i = 0; i < bits; i++) { |
| 997 | if (bitsleft <= 0) |
| 998 | bitsleft = 8, byte = random_byte(); |
| 999 | v = byte & 1; |
| 1000 | byte >>= 1; |
| 1001 | bitsleft--; |
| 1002 | bignum_set_bit(w, i, v); |
| 1003 | } |
| 1004 | bn_restore_invariant(w); |
| 1005 | if (bignum_cmp(w, p) >= 0 || bignum_cmp(w, Zero) == 0) { |
| 1006 | freebn(w); |
| 1007 | continue; |
| 1008 | } |
| 1009 | break; |
| 1010 | } |
| 9400cf6f |
1011 | |
| 5c72ca61 |
1012 | pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress); |
| 9400cf6f |
1013 | |
| 32874aea |
1014 | /* |
| 1015 | * Compute w^q mod p. |
| 1016 | */ |
| 1017 | wqp = modpow(w, q, p); |
| 1018 | freebn(w); |
| 9400cf6f |
1019 | |
| 32874aea |
1020 | /* |
| 1021 | * See if this is 1, or if it is -1, or if it becomes -1 |
| 1022 | * when squared at most k-1 times. |
| 1023 | */ |
| 1024 | if (bignum_cmp(wqp, One) == 0 || bignum_cmp(wqp, pm1) == 0) { |
| 1025 | freebn(wqp); |
| 1026 | continue; |
| 1027 | } |
| 1028 | for (i = 0; i < k - 1; i++) { |
| 1029 | wqp2 = modmul(wqp, wqp, p); |
| 1030 | freebn(wqp); |
| 1031 | wqp = wqp2; |
| 1032 | if (bignum_cmp(wqp, pm1) == 0) |
| 1033 | break; |
| 1034 | } |
| 1035 | if (i < k - 1) { |
| 1036 | freebn(wqp); |
| 1037 | continue; |
| 1038 | } |
| 9400cf6f |
1039 | |
| 32874aea |
1040 | /* |
| 1041 | * It didn't. Therefore, w is a witness for the |
| 1042 | * compositeness of p. |
| 1043 | */ |
| 32a94223 |
1044 | freebn(wqp); |
| 32874aea |
1045 | freebn(p); |
| 1046 | freebn(pm1); |
| 1047 | freebn(q); |
| 1048 | goto STARTOVER; |
| 9400cf6f |
1049 | } |
| 1050 | |
| 1051 | /* |
| 1052 | * We have a prime! |
| 1053 | */ |
| 1054 | freebn(q); |
| b73c0b37 |
1055 | freebn(pm1); |
| 9400cf6f |
1056 | return p; |
| 1057 | } |
| 79dae043 |
1058 | |
| 1059 | /* |
| 1060 | * Invent a pair of values suitable for use as 'firstbits' in the |
| 1061 | * above function, such that their product is at least 2. |
| 1062 | * |
| 1063 | * This is used for generating both RSA and DSA keys which have |
| 1064 | * exactly the specified number of bits rather than one fewer - if you |
| 1065 | * generate an a-bit and a b-bit number completely at random and |
| 1066 | * multiply them together, you could end up with either an (ab-1)-bit |
| 1067 | * number or an (ab)-bit number. The former happens log(2)*2-1 of the |
| 1068 | * time (about 39%) and, though actually harmless, every time it |
| 1069 | * occurs it has a non-zero probability of sparking a user email along |
| 1070 | * the lines of 'Hey, I asked PuTTYgen for a 2048-bit key and I only |
| 1071 | * got 2047 bits! Bug!' |
| 1072 | */ |
| 1073 | void invent_firstbits(unsigned *one, unsigned *two) |
| 1074 | { |
| 1075 | /* |
| 1076 | * Our criterion is that any number in the range [one,one+1) |
| 1077 | * multiplied by any number in the range [two,two+1) should have |
| 1078 | * the highest bit set. It should be clear that we can trivially |
| 1079 | * test this by multiplying the smallest values in each interval, |
| 1080 | * i.e. the ones we actually invented. |
| 1081 | */ |
| 1082 | do { |
| 1083 | *one = 0x100 | random_byte(); |
| 1084 | *two = 0x100 | random_byte(); |
| 1085 | } while (*one * *two < 0x20000); |
| 1086 | } |
projects / u / mdw / putty / blame