+++ /dev/null
-/*\r
- * Prime generation.\r
- */\r
-\r
-#include <assert.h>\r
-#include "ssh.h"\r
-\r
-/*\r
- * This prime generation algorithm is pretty much cribbed from\r
- * OpenSSL. The algorithm is:\r
- * \r
- * - invent a B-bit random number and ensure the top and bottom\r
- * bits are set (so it's definitely B-bit, and it's definitely\r
- * odd)\r
- * \r
- * - see if it's coprime to all primes below 2^16; increment it by\r
- * two until it is (this shouldn't take long in general)\r
- * \r
- * - perform the Miller-Rabin primality test enough times to\r
- * ensure the probability of it being composite is 2^-80 or\r
- * less\r
- * \r
- * - go back to square one if any M-R test fails.\r
- */\r
-\r
-/*\r
- * The Miller-Rabin primality test is an extension to the Fermat\r
- * test. The Fermat test just checks that a^(p-1) == 1 mod p; this\r
- * is vulnerable to Carmichael numbers. Miller-Rabin considers how\r
- * that 1 is derived as well.\r
- * \r
- * Lemma: if a^2 == 1 (mod p), and p is prime, then either a == 1\r
- * or a == -1 (mod p).\r
- * \r
- * Proof: p divides a^2-1, i.e. p divides (a+1)(a-1). Hence,\r
- * since p is prime, either p divides (a+1) or p divides (a-1).\r
- * But this is the same as saying that either a is congruent to\r
- * -1 mod p or a is congruent to +1 mod p. []\r
- * \r
- * Comment: This fails when p is not prime. Consider p=mn, so\r
- * that mn divides (a+1)(a-1). Now we could have m dividing (a+1)\r
- * and n dividing (a-1), without the whole of mn dividing either.\r
- * For example, consider a=10 and p=99. 99 = 9 * 11; 9 divides\r
- * 10-1 and 11 divides 10+1, so a^2 is congruent to 1 mod p\r
- * without a having to be congruent to either 1 or -1.\r
- * \r
- * So the Miller-Rabin test, as well as considering a^(p-1),\r
- * considers a^((p-1)/2), a^((p-1)/4), and so on as far as it can\r
- * go. In other words. we write p-1 as q * 2^k, with k as large as\r
- * possible (i.e. q must be odd), and we consider the powers\r
- * \r
- * a^(q*2^0) a^(q*2^1) ... a^(q*2^(k-1)) a^(q*2^k)\r
- * i.e. a^((n-1)/2^k) a^((n-1)/2^(k-1)) ... a^((n-1)/2) a^(n-1)\r
- * \r
- * If p is to be prime, the last of these must be 1. Therefore, by\r
- * the above lemma, the one before it must be either 1 or -1. And\r
- * _if_ it's 1, then the one before that must be either 1 or -1,\r
- * and so on ... In other words, we expect to see a trailing chain\r
- * of 1s preceded by a -1. (If we're unlucky, our trailing chain of\r
- * 1s will be as long as the list so we'll never get to see what\r
- * lies before it. This doesn't count as a test failure because it\r
- * hasn't _proved_ that p is not prime.)\r
- * \r
- * For example, consider a=2 and p=1729. 1729 is a Carmichael\r
- * number: although it's not prime, it satisfies a^(p-1) == 1 mod p\r
- * for any a coprime to it. So the Fermat test wouldn't have a\r
- * problem with it at all, unless we happened to stumble on an a\r
- * which had a common factor.\r
- * \r
- * So. 1729 - 1 equals 27 * 2^6. So we look at\r
- * \r
- * 2^27 mod 1729 == 645\r
- * 2^108 mod 1729 == 1065\r
- * 2^216 mod 1729 == 1\r
- * 2^432 mod 1729 == 1\r
- * 2^864 mod 1729 == 1\r
- * 2^1728 mod 1729 == 1\r
- * \r
- * We do have a trailing string of 1s, so the Fermat test would\r
- * have been happy. But this trailing string of 1s is preceded by\r
- * 1065; whereas if 1729 were prime, we'd expect to see it preceded\r
- * by -1 (i.e. 1728.). Guards! Seize this impostor.\r
- * \r
- * (If we were unlucky, we might have tried a=16 instead of a=2;\r
- * now 16^27 mod 1729 == 1, so we would have seen a long string of\r
- * 1s and wouldn't have seen the thing _before_ the 1s. So, just\r
- * like the Fermat test, for a given p there may well exist values\r
- * of a which fail to show up its compositeness. So we try several,\r
- * just like the Fermat test. The difference is that Miller-Rabin\r
- * is not _in general_ fooled by Carmichael numbers.)\r
- * \r
- * Put simply, then, the Miller-Rabin test requires us to:\r
- * \r
- * 1. write p-1 as q * 2^k, with q odd\r
- * 2. compute z = (a^q) mod p.\r
- * 3. report success if z == 1 or z == -1.\r
- * 4. square z at most k-1 times, and report success if it becomes\r
- * -1 at any point.\r
- * 5. report failure otherwise.\r
- * \r
- * (We expect z to become -1 after at most k-1 squarings, because\r
- * if it became -1 after k squarings then a^(p-1) would fail to be\r
- * 1. And we don't need to investigate what happens after we see a\r
- * -1, because we _know_ that -1 squared is 1 modulo anything at\r
- * all, so after we've seen a -1 we can be sure of seeing nothing\r
- * but 1s.)\r
- */\r
-\r
-/*\r
- * The first few odd primes.\r
- *\r
- * import sys\r
- * def sieve(n):\r
- * z = []\r
- * list = []\r
- * for i in range(n): z.append(1)\r
- * for i in range(2,n):\r
- * if z[i]:\r
- * list.append(i)\r
- * for j in range(i,n,i): z[j] = 0\r
- * return list\r
- * list = sieve(65535)\r
- * for i in list[1:]: sys.stdout.write("%d," % i)\r
- */\r
-static const unsigned short primes[] = {\r
- 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,\r
- 71, 73, 79, 83, 89, 97, 101,\r
- 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173,\r
- 179, 181, 191, 193,\r
- 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271,\r
- 277, 281, 283, 293,\r
- 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383,\r
- 389, 397, 401, 409,\r
- 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491,\r
- 499, 503, 509, 521,\r
- 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613,\r
- 617, 619, 631, 641,\r
- 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733,\r
- 739, 743, 751, 757,\r
- 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857,\r
- 859, 863, 877, 881,\r
- 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983,\r
- 991, 997, 1009,\r
- 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087,\r
- 1091, 1093,\r
- 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187,\r
- 1193, 1201,\r
- 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289,\r
- 1291, 1297,\r
- 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409,\r
- 1423, 1427,\r
- 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489,\r
- 1493, 1499,\r
- 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597,\r
- 1601, 1607,\r
- 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697,\r
- 1699, 1709,\r
- 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801,\r
- 1811, 1823,\r
- 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913,\r
- 1931, 1933,\r
- 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027,\r
- 2029, 2039,\r
- 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131,\r
- 2137, 2141,\r
- 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251,\r
- 2267, 2269,\r
- 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351,\r
- 2357, 2371,\r
- 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447,\r
- 2459, 2467,\r
- 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591,\r
- 2593, 2609,\r
- 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689,\r
- 2693, 2699,\r
- 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789,\r
- 2791, 2797,\r
- 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897,\r
- 2903, 2909,\r
- 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019,\r
- 3023, 3037,\r
- 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163,\r
- 3167, 3169,\r
- 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259,\r
- 3271, 3299,\r
- 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371,\r
- 3373, 3389,\r
- 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499,\r
- 3511, 3517,\r
- 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593,\r
- 3607, 3613,\r
- 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701,\r
- 3709, 3719,\r
- 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823,\r
- 3833, 3847,\r
- 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929,\r
- 3931, 3943,\r
- 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051,\r
- 4057, 4073,\r
- 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159,\r
- 4177, 4201,\r
- 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273,\r
- 4283, 4289,\r
- 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421,\r
- 4423, 4441,\r
- 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523,\r
- 4547, 4549,\r
- 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651,\r
- 4657, 4663,\r
- 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787,\r
- 4789, 4793,\r
- 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919,\r
- 4931, 4933,\r
- 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009,\r
- 5011, 5021,\r
- 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119,\r
- 5147, 5153,\r
- 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273,\r
- 5279, 5281,\r
- 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407,\r
- 5413, 5417,\r
- 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503,\r
- 5507, 5519,\r
- 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641,\r
- 5647, 5651,\r
- 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741,\r
- 5743, 5749,\r
- 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851,\r
- 5857, 5861,\r
- 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987,\r
- 6007, 6011,\r
- 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113,\r
- 6121, 6131,\r
- 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229,\r
- 6247, 6257,\r
- 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337,\r
- 6343, 6353,\r
- 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469,\r
- 6473, 6481,\r
- 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599,\r
- 6607, 6619,\r
- 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719,\r
- 6733, 6737,\r
- 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841,\r
- 6857, 6863,\r
- 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967,\r
- 6971, 6977,\r
- 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079,\r
- 7103, 7109,\r
- 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219,\r
- 7229, 7237,\r
- 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351,\r
- 7369, 7393,\r
- 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507,\r
- 7517, 7523,\r
- 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591,\r
- 7603, 7607,\r
- 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717,\r
- 7723, 7727,\r
- 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867,\r
- 7873, 7877,\r
- 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993,\r
- 8009, 8011,\r
- 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117,\r
- 8123, 8147,\r
- 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243,\r
- 8263, 8269,\r
- 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377,\r
- 8387, 8389,\r
- 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527,\r
- 8537, 8539,\r
- 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647,\r
- 8663, 8669,\r
- 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747,\r
- 8753, 8761,\r
- 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863,\r
- 8867, 8887,\r
- 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007,\r
- 9011, 9013,\r
- 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137,\r
- 9151, 9157,\r
- 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257,\r
- 9277, 9281,\r
- 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391,\r
- 9397, 9403,\r
- 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479,\r
- 9491, 9497,\r
- 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629,\r
- 9631, 9643,\r
- 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749,\r
- 9767, 9769,\r
- 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859,\r
- 9871, 9883,\r
- 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973, 10007,\r
- 10009, 10037,\r
- 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099, 10103, 10111,\r
- 10133, 10139,\r
- 10141, 10151, 10159, 10163, 10169, 10177, 10181, 10193, 10211, 10223,\r
- 10243, 10247,\r
- 10253, 10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321,\r
- 10331, 10333,\r
- 10337, 10343, 10357, 10369, 10391, 10399, 10427, 10429, 10433, 10453,\r
- 10457, 10459,\r
- 10463, 10477, 10487, 10499, 10501, 10513, 10529, 10531, 10559, 10567,\r
- 10589, 10597,\r
- 10601, 10607, 10613, 10627, 10631, 10639, 10651, 10657, 10663, 10667,\r
- 10687, 10691,\r
- 10709, 10711, 10723, 10729, 10733, 10739, 10753, 10771, 10781, 10789,\r
- 10799, 10831,\r
- 10837, 10847, 10853, 10859, 10861, 10867, 10883, 10889, 10891, 10903,\r
- 10909, 10937,\r
- 10939, 10949, 10957, 10973, 10979, 10987, 10993, 11003, 11027, 11047,\r
- 11057, 11059,\r
- 11069, 11071, 11083, 11087, 11093, 11113, 11117, 11119, 11131, 11149,\r
- 11159, 11161,\r
- 11171, 11173, 11177, 11197, 11213, 11239, 11243, 11251, 11257, 11261,\r
- 11273, 11279,\r
- 11287, 11299, 11311, 11317, 11321, 11329, 11351, 11353, 11369, 11383,\r
- 11393, 11399,\r
- 11411, 11423, 11437, 11443, 11447, 11467, 11471, 11483, 11489, 11491,\r
- 11497, 11503,\r
- 11519, 11527, 11549, 11551, 11579, 11587, 11593, 11597, 11617, 11621,\r
- 11633, 11657,\r
- 11677, 11681, 11689, 11699, 11701, 11717, 11719, 11731, 11743, 11777,\r
- 11779, 11783,\r
- 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833, 11839, 11863,\r
- 11867, 11887,\r
- 11897, 11903, 11909, 11923, 11927, 11933, 11939, 11941, 11953, 11959,\r
- 11969, 11971,\r
- 11981, 11987, 12007, 12011, 12037, 12041, 12043, 12049, 12071, 12073,\r
- 12097, 12101,\r
- 12107, 12109, 12113, 12119, 12143, 12149, 12157, 12161, 12163, 12197,\r
- 12203, 12211,\r
- 12227, 12239, 12241, 12251, 12253, 12263, 12269, 12277, 12281, 12289,\r
- 12301, 12323,\r
- 12329, 12343, 12347, 12373, 12377, 12379, 12391, 12401, 12409, 12413,\r
- 12421, 12433,\r
- 12437, 12451, 12457, 12473, 12479, 12487, 12491, 12497, 12503, 12511,\r
- 12517, 12527,\r
- 12539, 12541, 12547, 12553, 12569, 12577, 12583, 12589, 12601, 12611,\r
- 12613, 12619,\r
- 12637, 12641, 12647, 12653, 12659, 12671, 12689, 12697, 12703, 12713,\r
- 12721, 12739,\r
- 12743, 12757, 12763, 12781, 12791, 12799, 12809, 12821, 12823, 12829,\r
- 12841, 12853,\r
- 12889, 12893, 12899, 12907, 12911, 12917, 12919, 12923, 12941, 12953,\r
- 12959, 12967,\r
- 12973, 12979, 12983, 13001, 13003, 13007, 13009, 13033, 13037, 13043,\r
- 13049, 13063,\r
- 13093, 13099, 13103, 13109, 13121, 13127, 13147, 13151, 13159, 13163,\r
- 13171, 13177,\r
- 13183, 13187, 13217, 13219, 13229, 13241, 13249, 13259, 13267, 13291,\r
- 13297, 13309,\r
- 13313, 13327, 13331, 13337, 13339, 13367, 13381, 13397, 13399, 13411,\r
- 13417, 13421,\r
- 13441, 13451, 13457, 13463, 13469, 13477, 13487, 13499, 13513, 13523,\r
- 13537, 13553,\r
- 13567, 13577, 13591, 13597, 13613, 13619, 13627, 13633, 13649, 13669,\r
- 13679, 13681,\r
- 13687, 13691, 13693, 13697, 13709, 13711, 13721, 13723, 13729, 13751,\r
- 13757, 13759,\r
- 13763, 13781, 13789, 13799, 13807, 13829, 13831, 13841, 13859, 13873,\r
- 13877, 13879,\r
- 13883, 13901, 13903, 13907, 13913, 13921, 13931, 13933, 13963, 13967,\r
- 13997, 13999,\r
- 14009, 14011, 14029, 14033, 14051, 14057, 14071, 14081, 14083, 14087,\r
- 14107, 14143,\r
- 14149, 14153, 14159, 14173, 14177, 14197, 14207, 14221, 14243, 14249,\r
- 14251, 14281,\r
- 14293, 14303, 14321, 14323, 14327, 14341, 14347, 14369, 14387, 14389,\r
- 14401, 14407,\r
- 14411, 14419, 14423, 14431, 14437, 14447, 14449, 14461, 14479, 14489,\r
- 14503, 14519,\r
- 14533, 14537, 14543, 14549, 14551, 14557, 14561, 14563, 14591, 14593,\r
- 14621, 14627,\r
- 14629, 14633, 14639, 14653, 14657, 14669, 14683, 14699, 14713, 14717,\r
- 14723, 14731,\r
- 14737, 14741, 14747, 14753, 14759, 14767, 14771, 14779, 14783, 14797,\r
- 14813, 14821,\r
- 14827, 14831, 14843, 14851, 14867, 14869, 14879, 14887, 14891, 14897,\r
- 14923, 14929,\r
- 14939, 14947, 14951, 14957, 14969, 14983, 15013, 15017, 15031, 15053,\r
- 15061, 15073,\r
- 15077, 15083, 15091, 15101, 15107, 15121, 15131, 15137, 15139, 15149,\r
- 15161, 15173,\r
- 15187, 15193, 15199, 15217, 15227, 15233, 15241, 15259, 15263, 15269,\r
- 15271, 15277,\r
- 15287, 15289, 15299, 15307, 15313, 15319, 15329, 15331, 15349, 15359,\r
- 15361, 15373,\r
- 15377, 15383, 15391, 15401, 15413, 15427, 15439, 15443, 15451, 15461,\r
- 15467, 15473,\r
- 15493, 15497, 15511, 15527, 15541, 15551, 15559, 15569, 15581, 15583,\r
- 15601, 15607,\r
- 15619, 15629, 15641, 15643, 15647, 15649, 15661, 15667, 15671, 15679,\r
- 15683, 15727,\r
- 15731, 15733, 15737, 15739, 15749, 15761, 15767, 15773, 15787, 15791,\r
- 15797, 15803,\r
- 15809, 15817, 15823, 15859, 15877, 15881, 15887, 15889, 15901, 15907,\r
- 15913, 15919,\r
- 15923, 15937, 15959, 15971, 15973, 15991, 16001, 16007, 16033, 16057,\r
- 16061, 16063,\r
- 16067, 16069, 16073, 16087, 16091, 16097, 16103, 16111, 16127, 16139,\r
- 16141, 16183,\r
- 16187, 16189, 16193, 16217, 16223, 16229, 16231, 16249, 16253, 16267,\r
- 16273, 16301,\r
- 16319, 16333, 16339, 16349, 16361, 16363, 16369, 16381, 16411, 16417,\r
- 16421, 16427,\r
- 16433, 16447, 16451, 16453, 16477, 16481, 16487, 16493, 16519, 16529,\r
- 16547, 16553,\r
- 16561, 16567, 16573, 16603, 16607, 16619, 16631, 16633, 16649, 16651,\r
- 16657, 16661,\r
- 16673, 16691, 16693, 16699, 16703, 16729, 16741, 16747, 16759, 16763,\r
- 16787, 16811,\r
- 16823, 16829, 16831, 16843, 16871, 16879, 16883, 16889, 16901, 16903,\r
- 16921, 16927,\r
- 16931, 16937, 16943, 16963, 16979, 16981, 16987, 16993, 17011, 17021,\r
- 17027, 17029,\r
- 17033, 17041, 17047, 17053, 17077, 17093, 17099, 17107, 17117, 17123,\r
- 17137, 17159,\r
- 17167, 17183, 17189, 17191, 17203, 17207, 17209, 17231, 17239, 17257,\r
- 17291, 17293,\r
- 17299, 17317, 17321, 17327, 17333, 17341, 17351, 17359, 17377, 17383,\r
- 17387, 17389,\r
- 17393, 17401, 17417, 17419, 17431, 17443, 17449, 17467, 17471, 17477,\r
- 17483, 17489,\r
- 17491, 17497, 17509, 17519, 17539, 17551, 17569, 17573, 17579, 17581,\r
- 17597, 17599,\r
- 17609, 17623, 17627, 17657, 17659, 17669, 17681, 17683, 17707, 17713,\r
- 17729, 17737,\r
- 17747, 17749, 17761, 17783, 17789, 17791, 17807, 17827, 17837, 17839,\r
- 17851, 17863,\r
- 17881, 17891, 17903, 17909, 17911, 17921, 17923, 17929, 17939, 17957,\r
- 17959, 17971,\r
- 17977, 17981, 17987, 17989, 18013, 18041, 18043, 18047, 18049, 18059,\r
- 18061, 18077,\r
- 18089, 18097, 18119, 18121, 18127, 18131, 18133, 18143, 18149, 18169,\r
- 18181, 18191,\r
- 18199, 18211, 18217, 18223, 18229, 18233, 18251, 18253, 18257, 18269,\r
- 18287, 18289,\r
- 18301, 18307, 18311, 18313, 18329, 18341, 18353, 18367, 18371, 18379,\r
- 18397, 18401,\r
- 18413, 18427, 18433, 18439, 18443, 18451, 18457, 18461, 18481, 18493,\r
- 18503, 18517,\r
- 18521, 18523, 18539, 18541, 18553, 18583, 18587, 18593, 18617, 18637,\r
- 18661, 18671,\r
- 18679, 18691, 18701, 18713, 18719, 18731, 18743, 18749, 18757, 18773,\r
- 18787, 18793,\r
- 18797, 18803, 18839, 18859, 18869, 18899, 18911, 18913, 18917, 18919,\r
- 18947, 18959,\r
- 18973, 18979, 19001, 19009, 19013, 19031, 19037, 19051, 19069, 19073,\r
- 19079, 19081,\r
- 19087, 19121, 19139, 19141, 19157, 19163, 19181, 19183, 19207, 19211,\r
- 19213, 19219,\r
- 19231, 19237, 19249, 19259, 19267, 19273, 19289, 19301, 19309, 19319,\r
- 19333, 19373,\r
- 19379, 19381, 19387, 19391, 19403, 19417, 19421, 19423, 19427, 19429,\r
- 19433, 19441,\r
- 19447, 19457, 19463, 19469, 19471, 19477, 19483, 19489, 19501, 19507,\r
- 19531, 19541,\r
- 19543, 19553, 19559, 19571, 19577, 19583, 19597, 19603, 19609, 19661,\r
- 19681, 19687,\r
- 19697, 19699, 19709, 19717, 19727, 19739, 19751, 19753, 19759, 19763,\r
- 19777, 19793,\r
- 19801, 19813, 19819, 19841, 19843, 19853, 19861, 19867, 19889, 19891,\r
- 19913, 19919,\r
- 19927, 19937, 19949, 19961, 19963, 19973, 19979, 19991, 19993, 19997,\r
- 20011, 20021,\r
- 20023, 20029, 20047, 20051, 20063, 20071, 20089, 20101, 20107, 20113,\r
- 20117, 20123,\r
- 20129, 20143, 20147, 20149, 20161, 20173, 20177, 20183, 20201, 20219,\r
- 20231, 20233,\r
- 20249, 20261, 20269, 20287, 20297, 20323, 20327, 20333, 20341, 20347,\r
- 20353, 20357,\r
- 20359, 20369, 20389, 20393, 20399, 20407, 20411, 20431, 20441, 20443,\r
- 20477, 20479,\r
- 20483, 20507, 20509, 20521, 20533, 20543, 20549, 20551, 20563, 20593,\r
- 20599, 20611,\r
- 20627, 20639, 20641, 20663, 20681, 20693, 20707, 20717, 20719, 20731,\r
- 20743, 20747,\r
- 20749, 20753, 20759, 20771, 20773, 20789, 20807, 20809, 20849, 20857,\r
- 20873, 20879,\r
- 20887, 20897, 20899, 20903, 20921, 20929, 20939, 20947, 20959, 20963,\r
- 20981, 20983,\r
- 21001, 21011, 21013, 21017, 21019, 21023, 21031, 21059, 21061, 21067,\r
- 21089, 21101,\r
- 21107, 21121, 21139, 21143, 21149, 21157, 21163, 21169, 21179, 21187,\r
- 21191, 21193,\r
- 21211, 21221, 21227, 21247, 21269, 21277, 21283, 21313, 21317, 21319,\r
- 21323, 21341,\r
- 21347, 21377, 21379, 21383, 21391, 21397, 21401, 21407, 21419, 21433,\r
- 21467, 21481,\r
- 21487, 21491, 21493, 21499, 21503, 21517, 21521, 21523, 21529, 21557,\r
- 21559, 21563,\r
- 21569, 21577, 21587, 21589, 21599, 21601, 21611, 21613, 21617, 21647,\r
- 21649, 21661,\r
- 21673, 21683, 21701, 21713, 21727, 21737, 21739, 21751, 21757, 21767,\r
- 21773, 21787,\r
- 21799, 21803, 21817, 21821, 21839, 21841, 21851, 21859, 21863, 21871,\r
- 21881, 21893,\r
- 21911, 21929, 21937, 21943, 21961, 21977, 21991, 21997, 22003, 22013,\r
- 22027, 22031,\r
- 22037, 22039, 22051, 22063, 22067, 22073, 22079, 22091, 22093, 22109,\r
- 22111, 22123,\r
- 22129, 22133, 22147, 22153, 22157, 22159, 22171, 22189, 22193, 22229,\r
- 22247, 22259,\r
- 22271, 22273, 22277, 22279, 22283, 22291, 22303, 22307, 22343, 22349,\r
- 22367, 22369,\r
- 22381, 22391, 22397, 22409, 22433, 22441, 22447, 22453, 22469, 22481,\r
- 22483, 22501,\r
- 22511, 22531, 22541, 22543, 22549, 22567, 22571, 22573, 22613, 22619,\r
- 22621, 22637,\r
- 22639, 22643, 22651, 22669, 22679, 22691, 22697, 22699, 22709, 22717,\r
- 22721, 22727,\r
- 22739, 22741, 22751, 22769, 22777, 22783, 22787, 22807, 22811, 22817,\r
- 22853, 22859,\r
- 22861, 22871, 22877, 22901, 22907, 22921, 22937, 22943, 22961, 22963,\r
- 22973, 22993,\r
- 23003, 23011, 23017, 23021, 23027, 23029, 23039, 23041, 23053, 23057,\r
- 23059, 23063,\r
- 23071, 23081, 23087, 23099, 23117, 23131, 23143, 23159, 23167, 23173,\r
- 23189, 23197,\r
- 23201, 23203, 23209, 23227, 23251, 23269, 23279, 23291, 23293, 23297,\r
- 23311, 23321,\r
- 23327, 23333, 23339, 23357, 23369, 23371, 23399, 23417, 23431, 23447,\r
- 23459, 23473,\r
- 23497, 23509, 23531, 23537, 23539, 23549, 23557, 23561, 23563, 23567,\r
- 23581, 23593,\r
- 23599, 23603, 23609, 23623, 23627, 23629, 23633, 23663, 23669, 23671,\r
- 23677, 23687,\r
- 23689, 23719, 23741, 23743, 23747, 23753, 23761, 23767, 23773, 23789,\r
- 23801, 23813,\r
- 23819, 23827, 23831, 23833, 23857, 23869, 23873, 23879, 23887, 23893,\r
- 23899, 23909,\r
- 23911, 23917, 23929, 23957, 23971, 23977, 23981, 23993, 24001, 24007,\r
- 24019, 24023,\r
- 24029, 24043, 24049, 24061, 24071, 24077, 24083, 24091, 24097, 24103,\r
- 24107, 24109,\r
- 24113, 24121, 24133, 24137, 24151, 24169, 24179, 24181, 24197, 24203,\r
- 24223, 24229,\r
- 24239, 24247, 24251, 24281, 24317, 24329, 24337, 24359, 24371, 24373,\r
- 24379, 24391,\r
- 24407, 24413, 24419, 24421, 24439, 24443, 24469, 24473, 24481, 24499,\r
- 24509, 24517,\r
- 24527, 24533, 24547, 24551, 24571, 24593, 24611, 24623, 24631, 24659,\r
- 24671, 24677,\r
- 24683, 24691, 24697, 24709, 24733, 24749, 24763, 24767, 24781, 24793,\r
- 24799, 24809,\r
- 24821, 24841, 24847, 24851, 24859, 24877, 24889, 24907, 24917, 24919,\r
- 24923, 24943,\r
- 24953, 24967, 24971, 24977, 24979, 24989, 25013, 25031, 25033, 25037,\r
- 25057, 25073,\r
- 25087, 25097, 25111, 25117, 25121, 25127, 25147, 25153, 25163, 25169,\r
- 25171, 25183,\r
- 25189, 25219, 25229, 25237, 25243, 25247, 25253, 25261, 25301, 25303,\r
- 25307, 25309,\r
- 25321, 25339, 25343, 25349, 25357, 25367, 25373, 25391, 25409, 25411,\r
- 25423, 25439,\r
- 25447, 25453, 25457, 25463, 25469, 25471, 25523, 25537, 25541, 25561,\r
- 25577, 25579,\r
- 25583, 25589, 25601, 25603, 25609, 25621, 25633, 25639, 25643, 25657,\r
- 25667, 25673,\r
- 25679, 25693, 25703, 25717, 25733, 25741, 25747, 25759, 25763, 25771,\r
- 25793, 25799,\r
- 25801, 25819, 25841, 25847, 25849, 25867, 25873, 25889, 25903, 25913,\r
- 25919, 25931,\r
- 25933, 25939, 25943, 25951, 25969, 25981, 25997, 25999, 26003, 26017,\r
- 26021, 26029,\r
- 26041, 26053, 26083, 26099, 26107, 26111, 26113, 26119, 26141, 26153,\r
- 26161, 26171,\r
- 26177, 26183, 26189, 26203, 26209, 26227, 26237, 26249, 26251, 26261,\r
- 26263, 26267,\r
- 26293, 26297, 26309, 26317, 26321, 26339, 26347, 26357, 26371, 26387,\r
- 26393, 26399,\r
- 26407, 26417, 26423, 26431, 26437, 26449, 26459, 26479, 26489, 26497,\r
- 26501, 26513,\r
- 26539, 26557, 26561, 26573, 26591, 26597, 26627, 26633, 26641, 26647,\r
- 26669, 26681,\r
- 26683, 26687, 26693, 26699, 26701, 26711, 26713, 26717, 26723, 26729,\r
- 26731, 26737,\r
- 26759, 26777, 26783, 26801, 26813, 26821, 26833, 26839, 26849, 26861,\r
- 26863, 26879,\r
- 26881, 26891, 26893, 26903, 26921, 26927, 26947, 26951, 26953, 26959,\r
- 26981, 26987,\r
- 26993, 27011, 27017, 27031, 27043, 27059, 27061, 27067, 27073, 27077,\r
- 27091, 27103,\r
- 27107, 27109, 27127, 27143, 27179, 27191, 27197, 27211, 27239, 27241,\r
- 27253, 27259,\r
- 27271, 27277, 27281, 27283, 27299, 27329, 27337, 27361, 27367, 27397,\r
- 27407, 27409,\r
- 27427, 27431, 27437, 27449, 27457, 27479, 27481, 27487, 27509, 27527,\r
- 27529, 27539,\r
- 27541, 27551, 27581, 27583, 27611, 27617, 27631, 27647, 27653, 27673,\r
- 27689, 27691,\r
- 27697, 27701, 27733, 27737, 27739, 27743, 27749, 27751, 27763, 27767,\r
- 27773, 27779,\r
- 27791, 27793, 27799, 27803, 27809, 27817, 27823, 27827, 27847, 27851,\r
- 27883, 27893,\r
- 27901, 27917, 27919, 27941, 27943, 27947, 27953, 27961, 27967, 27983,\r
- 27997, 28001,\r
- 28019, 28027, 28031, 28051, 28057, 28069, 28081, 28087, 28097, 28099,\r
- 28109, 28111,\r
- 28123, 28151, 28163, 28181, 28183, 28201, 28211, 28219, 28229, 28277,\r
- 28279, 28283,\r
- 28289, 28297, 28307, 28309, 28319, 28349, 28351, 28387, 28393, 28403,\r
- 28409, 28411,\r
- 28429, 28433, 28439, 28447, 28463, 28477, 28493, 28499, 28513, 28517,\r
- 28537, 28541,\r
- 28547, 28549, 28559, 28571, 28573, 28579, 28591, 28597, 28603, 28607,\r
- 28619, 28621,\r
- 28627, 28631, 28643, 28649, 28657, 28661, 28663, 28669, 28687, 28697,\r
- 28703, 28711,\r
- 28723, 28729, 28751, 28753, 28759, 28771, 28789, 28793, 28807, 28813,\r
- 28817, 28837,\r
- 28843, 28859, 28867, 28871, 28879, 28901, 28909, 28921, 28927, 28933,\r
- 28949, 28961,\r
- 28979, 29009, 29017, 29021, 29023, 29027, 29033, 29059, 29063, 29077,\r
- 29101, 29123,\r
- 29129, 29131, 29137, 29147, 29153, 29167, 29173, 29179, 29191, 29201,\r
- 29207, 29209,\r
- 29221, 29231, 29243, 29251, 29269, 29287, 29297, 29303, 29311, 29327,\r
- 29333, 29339,\r
- 29347, 29363, 29383, 29387, 29389, 29399, 29401, 29411, 29423, 29429,\r
- 29437, 29443,\r
- 29453, 29473, 29483, 29501, 29527, 29531, 29537, 29567, 29569, 29573,\r
- 29581, 29587,\r
- 29599, 29611, 29629, 29633, 29641, 29663, 29669, 29671, 29683, 29717,\r
- 29723, 29741,\r
- 29753, 29759, 29761, 29789, 29803, 29819, 29833, 29837, 29851, 29863,\r
- 29867, 29873,\r
- 29879, 29881, 29917, 29921, 29927, 29947, 29959, 29983, 29989, 30011,\r
- 30013, 30029,\r
- 30047, 30059, 30071, 30089, 30091, 30097, 30103, 30109, 30113, 30119,\r
- 30133, 30137,\r
- 30139, 30161, 30169, 30181, 30187, 30197, 30203, 30211, 30223, 30241,\r
- 30253, 30259,\r
- 30269, 30271, 30293, 30307, 30313, 30319, 30323, 30341, 30347, 30367,\r
- 30389, 30391,\r
- 30403, 30427, 30431, 30449, 30467, 30469, 30491, 30493, 30497, 30509,\r
- 30517, 30529,\r
- 30539, 30553, 30557, 30559, 30577, 30593, 30631, 30637, 30643, 30649,\r
- 30661, 30671,\r
- 30677, 30689, 30697, 30703, 30707, 30713, 30727, 30757, 30763, 30773,\r
- 30781, 30803,\r
- 30809, 30817, 30829, 30839, 30841, 30851, 30853, 30859, 30869, 30871,\r
- 30881, 30893,\r
- 30911, 30931, 30937, 30941, 30949, 30971, 30977, 30983, 31013, 31019,\r
- 31033, 31039,\r
- 31051, 31063, 31069, 31079, 31081, 31091, 31121, 31123, 31139, 31147,\r
- 31151, 31153,\r
- 31159, 31177, 31181, 31183, 31189, 31193, 31219, 31223, 31231, 31237,\r
- 31247, 31249,\r
- 31253, 31259, 31267, 31271, 31277, 31307, 31319, 31321, 31327, 31333,\r
- 31337, 31357,\r
- 31379, 31387, 31391, 31393, 31397, 31469, 31477, 31481, 31489, 31511,\r
- 31513, 31517,\r
- 31531, 31541, 31543, 31547, 31567, 31573, 31583, 31601, 31607, 31627,\r
- 31643, 31649,\r
- 31657, 31663, 31667, 31687, 31699, 31721, 31723, 31727, 31729, 31741,\r
- 31751, 31769,\r
- 31771, 31793, 31799, 31817, 31847, 31849, 31859, 31873, 31883, 31891,\r
- 31907, 31957,\r
- 31963, 31973, 31981, 31991, 32003, 32009, 32027, 32029, 32051, 32057,\r
- 32059, 32063,\r
- 32069, 32077, 32083, 32089, 32099, 32117, 32119, 32141, 32143, 32159,\r
- 32173, 32183,\r
- 32189, 32191, 32203, 32213, 32233, 32237, 32251, 32257, 32261, 32297,\r
- 32299, 32303,\r
- 32309, 32321, 32323, 32327, 32341, 32353, 32359, 32363, 32369, 32371,\r
- 32377, 32381,\r
- 32401, 32411, 32413, 32423, 32429, 32441, 32443, 32467, 32479, 32491,\r
- 32497, 32503,\r
- 32507, 32531, 32533, 32537, 32561, 32563, 32569, 32573, 32579, 32587,\r
- 32603, 32609,\r
- 32611, 32621, 32633, 32647, 32653, 32687, 32693, 32707, 32713, 32717,\r
- 32719, 32749,\r
- 32771, 32779, 32783, 32789, 32797, 32801, 32803, 32831, 32833, 32839,\r
- 32843, 32869,\r
- 32887, 32909, 32911, 32917, 32933, 32939, 32941, 32957, 32969, 32971,\r
- 32983, 32987,\r
- 32993, 32999, 33013, 33023, 33029, 33037, 33049, 33053, 33071, 33073,\r
- 33083, 33091,\r
- 33107, 33113, 33119, 33149, 33151, 33161, 33179, 33181, 33191, 33199,\r
- 33203, 33211,\r
- 33223, 33247, 33287, 33289, 33301, 33311, 33317, 33329, 33331, 33343,\r
- 33347, 33349,\r
- 33353, 33359, 33377, 33391, 33403, 33409, 33413, 33427, 33457, 33461,\r
- 33469, 33479,\r
- 33487, 33493, 33503, 33521, 33529, 33533, 33547, 33563, 33569, 33577,\r
- 33581, 33587,\r
- 33589, 33599, 33601, 33613, 33617, 33619, 33623, 33629, 33637, 33641,\r
- 33647, 33679,\r
- 33703, 33713, 33721, 33739, 33749, 33751, 33757, 33767, 33769, 33773,\r
- 33791, 33797,\r
- 33809, 33811, 33827, 33829, 33851, 33857, 33863, 33871, 33889, 33893,\r
- 33911, 33923,\r
- 33931, 33937, 33941, 33961, 33967, 33997, 34019, 34031, 34033, 34039,\r
- 34057, 34061,\r
- 34123, 34127, 34129, 34141, 34147, 34157, 34159, 34171, 34183, 34211,\r
- 34213, 34217,\r
- 34231, 34253, 34259, 34261, 34267, 34273, 34283, 34297, 34301, 34303,\r
- 34313, 34319,\r
- 34327, 34337, 34351, 34361, 34367, 34369, 34381, 34403, 34421, 34429,\r
- 34439, 34457,\r
- 34469, 34471, 34483, 34487, 34499, 34501, 34511, 34513, 34519, 34537,\r
- 34543, 34549,\r
- 34583, 34589, 34591, 34603, 34607, 34613, 34631, 34649, 34651, 34667,\r
- 34673, 34679,\r
- 34687, 34693, 34703, 34721, 34729, 34739, 34747, 34757, 34759, 34763,\r
- 34781, 34807,\r
- 34819, 34841, 34843, 34847, 34849, 34871, 34877, 34883, 34897, 34913,\r
- 34919, 34939,\r
- 34949, 34961, 34963, 34981, 35023, 35027, 35051, 35053, 35059, 35069,\r
- 35081, 35083,\r
- 35089, 35099, 35107, 35111, 35117, 35129, 35141, 35149, 35153, 35159,\r
- 35171, 35201,\r
- 35221, 35227, 35251, 35257, 35267, 35279, 35281, 35291, 35311, 35317,\r
- 35323, 35327,\r
- 35339, 35353, 35363, 35381, 35393, 35401, 35407, 35419, 35423, 35437,\r
- 35447, 35449,\r
- 35461, 35491, 35507, 35509, 35521, 35527, 35531, 35533, 35537, 35543,\r
- 35569, 35573,\r
- 35591, 35593, 35597, 35603, 35617, 35671, 35677, 35729, 35731, 35747,\r
- 35753, 35759,\r
- 35771, 35797, 35801, 35803, 35809, 35831, 35837, 35839, 35851, 35863,\r
- 35869, 35879,\r
- 35897, 35899, 35911, 35923, 35933, 35951, 35963, 35969, 35977, 35983,\r
- 35993, 35999,\r
- 36007, 36011, 36013, 36017, 36037, 36061, 36067, 36073, 36083, 36097,\r
- 36107, 36109,\r
- 36131, 36137, 36151, 36161, 36187, 36191, 36209, 36217, 36229, 36241,\r
- 36251, 36263,\r
- 36269, 36277, 36293, 36299, 36307, 36313, 36319, 36341, 36343, 36353,\r
- 36373, 36383,\r
- 36389, 36433, 36451, 36457, 36467, 36469, 36473, 36479, 36493, 36497,\r
- 36523, 36527,\r
- 36529, 36541, 36551, 36559, 36563, 36571, 36583, 36587, 36599, 36607,\r
- 36629, 36637,\r
- 36643, 36653, 36671, 36677, 36683, 36691, 36697, 36709, 36713, 36721,\r
- 36739, 36749,\r
- 36761, 36767, 36779, 36781, 36787, 36791, 36793, 36809, 36821, 36833,\r
- 36847, 36857,\r
- 36871, 36877, 36887, 36899, 36901, 36913, 36919, 36923, 36929, 36931,\r
- 36943, 36947,\r
- 36973, 36979, 36997, 37003, 37013, 37019, 37021, 37039, 37049, 37057,\r
- 37061, 37087,\r
- 37097, 37117, 37123, 37139, 37159, 37171, 37181, 37189, 37199, 37201,\r
- 37217, 37223,\r
- 37243, 37253, 37273, 37277, 37307, 37309, 37313, 37321, 37337, 37339,\r
- 37357, 37361,\r
- 37363, 37369, 37379, 37397, 37409, 37423, 37441, 37447, 37463, 37483,\r
- 37489, 37493,\r
- 37501, 37507, 37511, 37517, 37529, 37537, 37547, 37549, 37561, 37567,\r
- 37571, 37573,\r
- 37579, 37589, 37591, 37607, 37619, 37633, 37643, 37649, 37657, 37663,\r
- 37691, 37693,\r
- 37699, 37717, 37747, 37781, 37783, 37799, 37811, 37813, 37831, 37847,\r
- 37853, 37861,\r
- 37871, 37879, 37889, 37897, 37907, 37951, 37957, 37963, 37967, 37987,\r
- 37991, 37993,\r
- 37997, 38011, 38039, 38047, 38053, 38069, 38083, 38113, 38119, 38149,\r
- 38153, 38167,\r
- 38177, 38183, 38189, 38197, 38201, 38219, 38231, 38237, 38239, 38261,\r
- 38273, 38281,\r
- 38287, 38299, 38303, 38317, 38321, 38327, 38329, 38333, 38351, 38371,\r
- 38377, 38393,\r
- 38431, 38447, 38449, 38453, 38459, 38461, 38501, 38543, 38557, 38561,\r
- 38567, 38569,\r
- 38593, 38603, 38609, 38611, 38629, 38639, 38651, 38653, 38669, 38671,\r
- 38677, 38693,\r
- 38699, 38707, 38711, 38713, 38723, 38729, 38737, 38747, 38749, 38767,\r
- 38783, 38791,\r
- 38803, 38821, 38833, 38839, 38851, 38861, 38867, 38873, 38891, 38903,\r
- 38917, 38921,\r
- 38923, 38933, 38953, 38959, 38971, 38977, 38993, 39019, 39023, 39041,\r
- 39043, 39047,\r
- 39079, 39089, 39097, 39103, 39107, 39113, 39119, 39133, 39139, 39157,\r
- 39161, 39163,\r
- 39181, 39191, 39199, 39209, 39217, 39227, 39229, 39233, 39239, 39241,\r
- 39251, 39293,\r
- 39301, 39313, 39317, 39323, 39341, 39343, 39359, 39367, 39371, 39373,\r
- 39383, 39397,\r
- 39409, 39419, 39439, 39443, 39451, 39461, 39499, 39503, 39509, 39511,\r
- 39521, 39541,\r
- 39551, 39563, 39569, 39581, 39607, 39619, 39623, 39631, 39659, 39667,\r
- 39671, 39679,\r
- 39703, 39709, 39719, 39727, 39733, 39749, 39761, 39769, 39779, 39791,\r
- 39799, 39821,\r
- 39827, 39829, 39839, 39841, 39847, 39857, 39863, 39869, 39877, 39883,\r
- 39887, 39901,\r
- 39929, 39937, 39953, 39971, 39979, 39983, 39989, 40009, 40013, 40031,\r
- 40037, 40039,\r
- 40063, 40087, 40093, 40099, 40111, 40123, 40127, 40129, 40151, 40153,\r
- 40163, 40169,\r
- 40177, 40189, 40193, 40213, 40231, 40237, 40241, 40253, 40277, 40283,\r
- 40289, 40343,\r
- 40351, 40357, 40361, 40387, 40423, 40427, 40429, 40433, 40459, 40471,\r
- 40483, 40487,\r
- 40493, 40499, 40507, 40519, 40529, 40531, 40543, 40559, 40577, 40583,\r
- 40591, 40597,\r
- 40609, 40627, 40637, 40639, 40693, 40697, 40699, 40709, 40739, 40751,\r
- 40759, 40763,\r
- 40771, 40787, 40801, 40813, 40819, 40823, 40829, 40841, 40847, 40849,\r
- 40853, 40867,\r
- 40879, 40883, 40897, 40903, 40927, 40933, 40939, 40949, 40961, 40973,\r
- 40993, 41011,\r
- 41017, 41023, 41039, 41047, 41051, 41057, 41077, 41081, 41113, 41117,\r
- 41131, 41141,\r
- 41143, 41149, 41161, 41177, 41179, 41183, 41189, 41201, 41203, 41213,\r
- 41221, 41227,\r
- 41231, 41233, 41243, 41257, 41263, 41269, 41281, 41299, 41333, 41341,\r
- 41351, 41357,\r
- 41381, 41387, 41389, 41399, 41411, 41413, 41443, 41453, 41467, 41479,\r
- 41491, 41507,\r
- 41513, 41519, 41521, 41539, 41543, 41549, 41579, 41593, 41597, 41603,\r
- 41609, 41611,\r
- 41617, 41621, 41627, 41641, 41647, 41651, 41659, 41669, 41681, 41687,\r
- 41719, 41729,\r
- 41737, 41759, 41761, 41771, 41777, 41801, 41809, 41813, 41843, 41849,\r
- 41851, 41863,\r
- 41879, 41887, 41893, 41897, 41903, 41911, 41927, 41941, 41947, 41953,\r
- 41957, 41959,\r
- 41969, 41981, 41983, 41999, 42013, 42017, 42019, 42023, 42043, 42061,\r
- 42071, 42073,\r
- 42083, 42089, 42101, 42131, 42139, 42157, 42169, 42179, 42181, 42187,\r
- 42193, 42197,\r
- 42209, 42221, 42223, 42227, 42239, 42257, 42281, 42283, 42293, 42299,\r
- 42307, 42323,\r
- 42331, 42337, 42349, 42359, 42373, 42379, 42391, 42397, 42403, 42407,\r
- 42409, 42433,\r
- 42437, 42443, 42451, 42457, 42461, 42463, 42467, 42473, 42487, 42491,\r
- 42499, 42509,\r
- 42533, 42557, 42569, 42571, 42577, 42589, 42611, 42641, 42643, 42649,\r
- 42667, 42677,\r
- 42683, 42689, 42697, 42701, 42703, 42709, 42719, 42727, 42737, 42743,\r
- 42751, 42767,\r
- 42773, 42787, 42793, 42797, 42821, 42829, 42839, 42841, 42853, 42859,\r
- 42863, 42899,\r
- 42901, 42923, 42929, 42937, 42943, 42953, 42961, 42967, 42979, 42989,\r
- 43003, 43013,\r
- 43019, 43037, 43049, 43051, 43063, 43067, 43093, 43103, 43117, 43133,\r
- 43151, 43159,\r
- 43177, 43189, 43201, 43207, 43223, 43237, 43261, 43271, 43283, 43291,\r
- 43313, 43319,\r
- 43321, 43331, 43391, 43397, 43399, 43403, 43411, 43427, 43441, 43451,\r
- 43457, 43481,\r
- 43487, 43499, 43517, 43541, 43543, 43573, 43577, 43579, 43591, 43597,\r
- 43607, 43609,\r
- 43613, 43627, 43633, 43649, 43651, 43661, 43669, 43691, 43711, 43717,\r
- 43721, 43753,\r
- 43759, 43777, 43781, 43783, 43787, 43789, 43793, 43801, 43853, 43867,\r
- 43889, 43891,\r
- 43913, 43933, 43943, 43951, 43961, 43963, 43969, 43973, 43987, 43991,\r
- 43997, 44017,\r
- 44021, 44027, 44029, 44041, 44053, 44059, 44071, 44087, 44089, 44101,\r
- 44111, 44119,\r
- 44123, 44129, 44131, 44159, 44171, 44179, 44189, 44201, 44203, 44207,\r
- 44221, 44249,\r
- 44257, 44263, 44267, 44269, 44273, 44279, 44281, 44293, 44351, 44357,\r
- 44371, 44381,\r
- 44383, 44389, 44417, 44449, 44453, 44483, 44491, 44497, 44501, 44507,\r
- 44519, 44531,\r
- 44533, 44537, 44543, 44549, 44563, 44579, 44587, 44617, 44621, 44623,\r
- 44633, 44641,\r
- 44647, 44651, 44657, 44683, 44687, 44699, 44701, 44711, 44729, 44741,\r
- 44753, 44771,\r
- 44773, 44777, 44789, 44797, 44809, 44819, 44839, 44843, 44851, 44867,\r
- 44879, 44887,\r
- 44893, 44909, 44917, 44927, 44939, 44953, 44959, 44963, 44971, 44983,\r
- 44987, 45007,\r
- 45013, 45053, 45061, 45077, 45083, 45119, 45121, 45127, 45131, 45137,\r
- 45139, 45161,\r
- 45179, 45181, 45191, 45197, 45233, 45247, 45259, 45263, 45281, 45289,\r
- 45293, 45307,\r
- 45317, 45319, 45329, 45337, 45341, 45343, 45361, 45377, 45389, 45403,\r
- 45413, 45427,\r
- 45433, 45439, 45481, 45491, 45497, 45503, 45523, 45533, 45541, 45553,\r
- 45557, 45569,\r
- 45587, 45589, 45599, 45613, 45631, 45641, 45659, 45667, 45673, 45677,\r
- 45691, 45697,\r
- 45707, 45737, 45751, 45757, 45763, 45767, 45779, 45817, 45821, 45823,\r
- 45827, 45833,\r
- 45841, 45853, 45863, 45869, 45887, 45893, 45943, 45949, 45953, 45959,\r
- 45971, 45979,\r
- 45989, 46021, 46027, 46049, 46051, 46061, 46073, 46091, 46093, 46099,\r
- 46103, 46133,\r
- 46141, 46147, 46153, 46171, 46181, 46183, 46187, 46199, 46219, 46229,\r
- 46237, 46261,\r
- 46271, 46273, 46279, 46301, 46307, 46309, 46327, 46337, 46349, 46351,\r
- 46381, 46399,\r
- 46411, 46439, 46441, 46447, 46451, 46457, 46471, 46477, 46489, 46499,\r
- 46507, 46511,\r
- 46523, 46549, 46559, 46567, 46573, 46589, 46591, 46601, 46619, 46633,\r
- 46639, 46643,\r
- 46649, 46663, 46679, 46681, 46687, 46691, 46703, 46723, 46727, 46747,\r
- 46751, 46757,\r
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- 46867, 46877,\r
- 46889, 46901, 46919, 46933, 46957, 46993, 46997, 47017, 47041, 47051,\r
- 47057, 47059,\r
- 47087, 47093, 47111, 47119, 47123, 47129, 47137, 47143, 47147, 47149,\r
- 47161, 47189,\r
- 47207, 47221, 47237, 47251, 47269, 47279, 47287, 47293, 47297, 47303,\r
- 47309, 47317,\r
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- 47431, 47441,\r
- 47459, 47491, 47497, 47501, 47507, 47513, 47521, 47527, 47533, 47543,\r
- 47563, 47569,\r
- 47581, 47591, 47599, 47609, 47623, 47629, 47639, 47653, 47657, 47659,\r
- 47681, 47699,\r
- 47701, 47711, 47713, 47717, 47737, 47741, 47743, 47777, 47779, 47791,\r
- 47797, 47807,\r
- 47809, 47819, 47837, 47843, 47857, 47869, 47881, 47903, 47911, 47917,\r
- 47933, 47939,\r
- 47947, 47951, 47963, 47969, 47977, 47981, 48017, 48023, 48029, 48049,\r
- 48073, 48079,\r
- 48091, 48109, 48119, 48121, 48131, 48157, 48163, 48179, 48187, 48193,\r
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- 48481, 48487,\r
- 48491, 48497, 48523, 48527, 48533, 48539, 48541, 48563, 48571, 48589,\r
- 48593, 48611,\r
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- 48751, 48757,\r
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- 48847, 48857,\r
- 48859, 48869, 48871, 48883, 48889, 48907, 48947, 48953, 48973, 48989,\r
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- 60527, 60539,\r
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- 60659, 60661,\r
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- 60773, 60779,\r
- 60793, 60811, 60821, 60859, 60869, 60887, 60889, 60899, 60901, 60913,\r
- 60917, 60919,\r
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- 61051, 61057,\r
- 61091, 61099, 61121, 61129, 61141, 61151, 61153, 61169, 61211, 61223,\r
- 61231, 61253,\r
- 61261, 61283, 61291, 61297, 61331, 61333, 61339, 61343, 61357, 61363,\r
- 61379, 61381,\r
- 61403, 61409, 61417, 61441, 61463, 61469, 61471, 61483, 61487, 61493,\r
- 61507, 61511,\r
- 61519, 61543, 61547, 61553, 61559, 61561, 61583, 61603, 61609, 61613,\r
- 61627, 61631,\r
- 61637, 61643, 61651, 61657, 61667, 61673, 61681, 61687, 61703, 61717,\r
- 61723, 61729,\r
- 61751, 61757, 61781, 61813, 61819, 61837, 61843, 61861, 61871, 61879,\r
- 61909, 61927,\r
- 61933, 61949, 61961, 61967, 61979, 61981, 61987, 61991, 62003, 62011,\r
- 62017, 62039,\r
- 62047, 62053, 62057, 62071, 62081, 62099, 62119, 62129, 62131, 62137,\r
- 62141, 62143,\r
- 62171, 62189, 62191, 62201, 62207, 62213, 62219, 62233, 62273, 62297,\r
- 62299, 62303,\r
- 62311, 62323, 62327, 62347, 62351, 62383, 62401, 62417, 62423, 62459,\r
- 62467, 62473,\r
- 62477, 62483, 62497, 62501, 62507, 62533, 62539, 62549, 62563, 62581,\r
- 62591, 62597,\r
- 62603, 62617, 62627, 62633, 62639, 62653, 62659, 62683, 62687, 62701,\r
- 62723, 62731,\r
- 62743, 62753, 62761, 62773, 62791, 62801, 62819, 62827, 62851, 62861,\r
- 62869, 62873,\r
- 62897, 62903, 62921, 62927, 62929, 62939, 62969, 62971, 62981, 62983,\r
- 62987, 62989,\r
- 63029, 63031, 63059, 63067, 63073, 63079, 63097, 63103, 63113, 63127,\r
- 63131, 63149,\r
- 63179, 63197, 63199, 63211, 63241, 63247, 63277, 63281, 63299, 63311,\r
- 63313, 63317,\r
- 63331, 63337, 63347, 63353, 63361, 63367, 63377, 63389, 63391, 63397,\r
- 63409, 63419,\r
- 63421, 63439, 63443, 63463, 63467, 63473, 63487, 63493, 63499, 63521,\r
- 63527, 63533,\r
- 63541, 63559, 63577, 63587, 63589, 63599, 63601, 63607, 63611, 63617,\r
- 63629, 63647,\r
- 63649, 63659, 63667, 63671, 63689, 63691, 63697, 63703, 63709, 63719,\r
- 63727, 63737,\r
- 63743, 63761, 63773, 63781, 63793, 63799, 63803, 63809, 63823, 63839,\r
- 63841, 63853,\r
- 63857, 63863, 63901, 63907, 63913, 63929, 63949, 63977, 63997, 64007,\r
- 64013, 64019,\r
- 64033, 64037, 64063, 64067, 64081, 64091, 64109, 64123, 64151, 64153,\r
- 64157, 64171,\r
- 64187, 64189, 64217, 64223, 64231, 64237, 64271, 64279, 64283, 64301,\r
- 64303, 64319,\r
- 64327, 64333, 64373, 64381, 64399, 64403, 64433, 64439, 64451, 64453,\r
- 64483, 64489,\r
- 64499, 64513, 64553, 64567, 64577, 64579, 64591, 64601, 64609, 64613,\r
- 64621, 64627,\r
- 64633, 64661, 64663, 64667, 64679, 64693, 64709, 64717, 64747, 64763,\r
- 64781, 64783,\r
- 64793, 64811, 64817, 64849, 64853, 64871, 64877, 64879, 64891, 64901,\r
- 64919, 64921,\r
- 64927, 64937, 64951, 64969, 64997, 65003, 65011, 65027, 65029, 65033,\r
- 65053, 65063,\r
- 65071, 65089, 65099, 65101, 65111, 65119, 65123, 65129, 65141, 65147,\r
- 65167, 65171,\r
- 65173, 65179, 65183, 65203, 65213, 65239, 65257, 65267, 65269, 65287,\r
- 65293, 65309,\r
- 65323, 65327, 65353, 65357, 65371, 65381, 65393, 65407, 65413, 65419,\r
- 65423, 65437,\r
- 65447, 65449, 65479, 65497, 65519, 65521,\r
-};\r
-\r
-#define NPRIMES (sizeof(primes) / sizeof(*primes))\r
-\r
-/*\r
- * Generate a prime. We can deal with various extra properties of\r
- * the prime:\r
- * \r
- * - to speed up use in RSA, we can arrange to select a prime with\r
- * the property (prime % modulus) != residue.\r
- * \r
- * - for use in DSA, we can arrange to select a prime which is one\r
- * more than a multiple of a dirty great bignum. In this case\r
- * `bits' gives the size of the factor by which we _multiply_\r
- * that bignum, rather than the size of the whole number.\r
- */\r
-Bignum primegen(int bits, int modulus, int residue, Bignum factor,\r
- int phase, progfn_t pfn, void *pfnparam)\r
-{\r
- int i, k, v, byte, bitsleft, check, checks;\r
- unsigned long delta;\r
- unsigned long moduli[NPRIMES + 1];\r
- unsigned long residues[NPRIMES + 1];\r
- unsigned long multipliers[NPRIMES + 1];\r
- Bignum p, pm1, q, wqp, wqp2;\r
- int progress = 0;\r
-\r
- byte = 0;\r
- bitsleft = 0;\r
-\r
- STARTOVER:\r
-\r
- pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress);\r
-\r
- /*\r
- * Generate a k-bit random number with top and bottom bits set.\r
- * Alternatively, if `factor' is nonzero, generate a k-bit\r
- * random number with the top bit set and the bottom bit clear,\r
- * multiply it by `factor', and add one.\r
- */\r
- p = bn_power_2(bits - 1);\r
- for (i = 0; i < bits; i++) {\r
- if (i == 0 || i == bits - 1)\r
- v = (i != 0 || !factor) ? 1 : 0;\r
- else {\r
- if (bitsleft <= 0)\r
- bitsleft = 8, byte = random_byte();\r
- v = byte & 1;\r
- byte >>= 1;\r
- bitsleft--;\r
- }\r
- bignum_set_bit(p, i, v);\r
- }\r
- if (factor) {\r
- Bignum tmp = p;\r
- p = bigmul(tmp, factor);\r
- freebn(tmp);\r
- assert(bignum_bit(p, 0) == 0);\r
- bignum_set_bit(p, 0, 1);\r
- }\r
-\r
- /*\r
- * Ensure this random number is coprime to the first few\r
- * primes, by repeatedly adding either 2 or 2*factor to it\r
- * until it is.\r
- */\r
- for (i = 0; i < NPRIMES; i++) {\r
- moduli[i] = primes[i];\r
- residues[i] = bignum_mod_short(p, primes[i]);\r
- if (factor)\r
- multipliers[i] = bignum_mod_short(factor, primes[i]);\r
- else\r
- multipliers[i] = 1;\r
- }\r
- moduli[NPRIMES] = modulus;\r
- residues[NPRIMES] = (bignum_mod_short(p, (unsigned short) modulus)\r
- + modulus - residue);\r
- if (factor)\r
- multipliers[NPRIMES] = bignum_mod_short(factor, modulus);\r
- else\r
- multipliers[NPRIMES] = 1;\r
- delta = 0;\r
- while (1) {\r
- for (i = 0; i < (sizeof(moduli) / sizeof(*moduli)); i++)\r
- if (!((residues[i] + delta * multipliers[i]) % moduli[i]))\r
- break;\r
- if (i < (sizeof(moduli) / sizeof(*moduli))) { /* we broke */\r
- delta += 2;\r
- if (delta > 65536) {\r
- freebn(p);\r
- goto STARTOVER;\r
- }\r
- continue;\r
- }\r
- break;\r
- }\r
- q = p;\r
- if (factor) {\r
- Bignum tmp;\r
- tmp = bignum_from_long(delta);\r
- p = bigmuladd(tmp, factor, q);\r
- freebn(tmp);\r
- } else {\r
- p = bignum_add_long(q, delta);\r
- }\r
- freebn(q);\r
-\r
- /*\r
- * Now apply the Miller-Rabin primality test a few times. First\r
- * work out how many checks are needed.\r
- */\r
- checks = 27;\r
- if (bits >= 150)\r
- checks = 18;\r
- if (bits >= 200)\r
- checks = 15;\r
- if (bits >= 250)\r
- checks = 12;\r
- if (bits >= 300)\r
- checks = 9;\r
- if (bits >= 350)\r
- checks = 8;\r
- if (bits >= 400)\r
- checks = 7;\r
- if (bits >= 450)\r
- checks = 6;\r
- if (bits >= 550)\r
- checks = 5;\r
- if (bits >= 650)\r
- checks = 4;\r
- if (bits >= 850)\r
- checks = 3;\r
- if (bits >= 1300)\r
- checks = 2;\r
-\r
- /*\r
- * Next, write p-1 as q*2^k.\r
- */\r
- for (k = 0; bignum_bit(p, k) == !k; k++)\r
- continue; /* find first 1 bit in p-1 */\r
- q = bignum_rshift(p, k);\r
- /* And store p-1 itself, which we'll need. */\r
- pm1 = copybn(p);\r
- decbn(pm1);\r
-\r
- /*\r
- * Now, for each check ...\r
- */\r
- for (check = 0; check < checks; check++) {\r
- Bignum w;\r
-\r
- /*\r
- * Invent a random number between 1 and p-1 inclusive.\r
- */\r
- while (1) {\r
- w = bn_power_2(bits - 1);\r
- for (i = 0; i < bits; i++) {\r
- if (bitsleft <= 0)\r
- bitsleft = 8, byte = random_byte();\r
- v = byte & 1;\r
- byte >>= 1;\r
- bitsleft--;\r
- bignum_set_bit(w, i, v);\r
- }\r
- bn_restore_invariant(w);\r
- if (bignum_cmp(w, p) >= 0 || bignum_cmp(w, Zero) == 0) {\r
- freebn(w);\r
- continue;\r
- }\r
- break;\r
- }\r
-\r
- pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress);\r
-\r
- /*\r
- * Compute w^q mod p.\r
- */\r
- wqp = modpow(w, q, p);\r
- freebn(w);\r
-\r
- /*\r
- * See if this is 1, or if it is -1, or if it becomes -1\r
- * when squared at most k-1 times.\r
- */\r
- if (bignum_cmp(wqp, One) == 0 || bignum_cmp(wqp, pm1) == 0) {\r
- freebn(wqp);\r
- continue;\r
- }\r
- for (i = 0; i < k - 1; i++) {\r
- wqp2 = modmul(wqp, wqp, p);\r
- freebn(wqp);\r
- wqp = wqp2;\r
- if (bignum_cmp(wqp, pm1) == 0)\r
- break;\r
- }\r
- if (i < k - 1) {\r
- freebn(wqp);\r
- continue;\r
- }\r
-\r
- /*\r
- * It didn't. Therefore, w is a witness for the\r
- * compositeness of p.\r
- */\r
- freebn(wqp);\r
- freebn(p);\r
- freebn(pm1);\r
- freebn(q);\r
- goto STARTOVER;\r
- }\r
-\r
- /*\r
- * We have a prime!\r
- */\r
- freebn(q);\r
- freebn(pm1);\r
- return p;\r
-}\r
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