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  1. /*
  2. * Prime generation.
  3. */
  4.  
  5. #include <assert.h>
  6. #include "ssh.h"
  7.  
  8. /*
  9. * This prime generation algorithm is pretty much cribbed from
  10. * OpenSSL. The algorithm is:
  11. *
  12. * - invent a B-bit random number and ensure the top and bottom
  13. * bits are set (so it's definitely B-bit, and it's definitely
  14. * odd)
  15. *
  16. * - see if it's coprime to all primes below 2^16; increment it by
  17. * two until it is (this shouldn't take long in general)
  18. *
  19. * - perform the Miller-Rabin primality test enough times to
  20. * ensure the probability of it being composite is 2^-80 or
  21. * less
  22. *
  23. * - go back to square one if any M-R test fails.
  24. */
  25.  
  26. /*
  27. * The Miller-Rabin primality test is an extension to the Fermat
  28. * test. The Fermat test just checks that a^(p-1) == 1 mod p; this
  29. * is vulnerable to Carmichael numbers. Miller-Rabin considers how
  30. * that 1 is derived as well.
  31. *
  32. * Lemma: if a^2 == 1 (mod p), and p is prime, then either a == 1
  33. * or a == -1 (mod p).
  34. *
  35. * Proof: p divides a^2-1, i.e. p divides (a+1)(a-1). Hence,
  36. * since p is prime, either p divides (a+1) or p divides (a-1).
  37. * But this is the same as saying that either a is congruent to
  38. * -1 mod p or a is congruent to +1 mod p. []
  39. *
  40. * Comment: This fails when p is not prime. Consider p=mn, so
  41. * that mn divides (a+1)(a-1). Now we could have m dividing (a+1)
  42. * and n dividing (a-1), without the whole of mn dividing either.
  43. * For example, consider a=10 and p=99. 99 = 9 * 11; 9 divides
  44. * 10-1 and 11 divides 10+1, so a^2 is congruent to 1 mod p
  45. * without a having to be congruent to either 1 or -1.
  46. *
  47. * So the Miller-Rabin test, as well as considering a^(p-1),
  48. * considers a^((p-1)/2), a^((p-1)/4), and so on as far as it can
  49. * go. In other words. we write p-1 as q * 2^k, with k as large as
  50. * possible (i.e. q must be odd), and we consider the powers
  51. *
  52. * a^(q*2^0) a^(q*2^1) ... a^(q*2^(k-1)) a^(q*2^k)
  53. * i.e. a^((n-1)/2^k) a^((n-1)/2^(k-1)) ... a^((n-1)/2) a^(n-1)
  54. *
  55. * If p is to be prime, the last of these must be 1. Therefore, by
  56. * the above lemma, the one before it must be either 1 or -1. And
  57. * _if_ it's 1, then the one before that must be either 1 or -1,
  58. * and so on ... In other words, we expect to see a trailing chain
  59. * of 1s preceded by a -1. (If we're unlucky, our trailing chain of
  60. * 1s will be as long as the list so we'll never get to see what
  61. * lies before it. This doesn't count as a test failure because it
  62. * hasn't _proved_ that p is not prime.)
  63. *
  64. * For example, consider a=2 and p=1729. 1729 is a Carmichael
  65. * number: although it's not prime, it satisfies a^(p-1) == 1 mod p
  66. * for any a coprime to it. So the Fermat test wouldn't have a
  67. * problem with it at all, unless we happened to stumble on an a
  68. * which had a common factor.
  69. *
  70. * So. 1729 - 1 equals 27 * 2^6. So we look at
  71. *
  72. * 2^27 mod 1729 == 645
  73. * 2^108 mod 1729 == 1065
  74. * 2^216 mod 1729 == 1
  75. * 2^432 mod 1729 == 1
  76. * 2^864 mod 1729 == 1
  77. * 2^1728 mod 1729 == 1
  78. *
  79. * We do have a trailing string of 1s, so the Fermat test would
  80. * have been happy. But this trailing string of 1s is preceded by
  81. * 1065; whereas if 1729 were prime, we'd expect to see it preceded
  82. * by -1 (i.e. 1728.). Guards! Seize this impostor.
  83. *
  84. * (If we were unlucky, we might have tried a=16 instead of a=2;
  85. * now 16^27 mod 1729 == 1, so we would have seen a long string of
  86. * 1s and wouldn't have seen the thing _before_ the 1s. So, just
  87. * like the Fermat test, for a given p there may well exist values
  88. * of a which fail to show up its compositeness. So we try several,
  89. * just like the Fermat test. The difference is that Miller-Rabin
  90. * is not _in general_ fooled by Carmichael numbers.)
  91. *
  92. * Put simply, then, the Miller-Rabin test requires us to:
  93. *
  94. * 1. write p-1 as q * 2^k, with q odd
  95. * 2. compute z = (a^q) mod p.
  96. * 3. report success if z == 1 or z == -1.
  97. * 4. square z at most k-1 times, and report success if it becomes
  98. * -1 at any point.
  99. * 5. report failure otherwise.
  100. *
  101. * (We expect z to become -1 after at most k-1 squarings, because
  102. * if it became -1 after k squarings then a^(p-1) would fail to be
  103. * 1. And we don't need to investigate what happens after we see a
  104. * -1, because we _know_ that -1 squared is 1 modulo anything at
  105. * all, so after we've seen a -1 we can be sure of seeing nothing
  106. * but 1s.)
  107. */
  108.  
  109. /*
  110. * The first few odd primes.
  111. *
  112. * import sys
  113. * def sieve(n):
  114. * z = []
  115. * list = []
  116. * for i in range(n): z.append(1)
  117. * for i in range(2,n):
  118. * if z[i]:
  119. * list.append(i)
  120. * for j in range(i,n,i): z[j] = 0
  121. * return list
  122. * list = sieve(65535)
  123. * for i in list[1:]: sys.stdout.write("%d," % i)
  124. */
  125. static const unsigned short primes[] = {
  126. 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
  127. 71, 73, 79, 83, 89, 97, 101,
  128. 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173,
  129. 179, 181, 191, 193,
  130. 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271,
  131. 277, 281, 283, 293,
  132. 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383,
  133. 389, 397, 401, 409,
  134. 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491,
  135. 499, 503, 509, 521,
  136. 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613,
  137. 617, 619, 631, 641,
  138. 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733,
  139. 739, 743, 751, 757,
  140. 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857,
  141. 859, 863, 877, 881,
  142. 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983,
  143. 991, 997, 1009,
  144. 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087,
  145. 1091, 1093,
  146. 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187,
  147. 1193, 1201,
  148. 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289,
  149. 1291, 1297,
  150. 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409,
  151. 1423, 1427,
  152. 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489,
  153. 1493, 1499,
  154. 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597,
  155. 1601, 1607,
  156. 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697,
  157. 1699, 1709,
  158. 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801,
  159. 1811, 1823,
  160. 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913,
  161. 1931, 1933,
  162. 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027,
  163. 2029, 2039,
  164. 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131,
  165. 2137, 2141,
  166. 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251,
  167. 2267, 2269,
  168. 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351,
  169. 2357, 2371,
  170. 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447,
  171. 2459, 2467,
  172. 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591,
  173. 2593, 2609,
  174. 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689,
  175. 2693, 2699,
  176. 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789,
  177. 2791, 2797,
  178. 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897,
  179. 2903, 2909,
  180. 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019,
  181. 3023, 3037,
  182. 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163,
  183. 3167, 3169,
  184. 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259,
  185. 3271, 3299,
  186. 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371,
  187. 3373, 3389,
  188. 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499,
  189. 3511, 3517,
  190. 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593,
  191. 3607, 3613,
  192. 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701,
  193. 3709, 3719,
  194. 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823,
  195. 3833, 3847,
  196. 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929,
  197. 3931, 3943,
  198. 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051,
  199. 4057, 4073,
  200. 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159,
  201. 4177, 4201,
  202. 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273,
  203. 4283, 4289,
  204. 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421,
  205. 4423, 4441,
  206. 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523,
  207. 4547, 4549,
  208. 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651,
  209. 4657, 4663,
  210. 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787,
  211. 4789, 4793,
  212. 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919,
  213. 4931, 4933,
  214. 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009,
  215. 5011, 5021,
  216. 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119,
  217. 5147, 5153,
  218. 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273,
  219. 5279, 5281,
  220. 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407,
  221. 5413, 5417,
  222. 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503,
  223. 5507, 5519,
  224. 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641,
  225. 5647, 5651,
  226. 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741,
  227. 5743, 5749,
  228. 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851,
  229. 5857, 5861,
  230. 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987,
  231. 6007, 6011,
  232. 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113,
  233. 6121, 6131,
  234. 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229,
  235. 6247, 6257,
  236. 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337,
  237. 6343, 6353,
  238. 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469,
  239. 6473, 6481,
  240. 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599,
  241. 6607, 6619,
  242. 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719,
  243. 6733, 6737,
  244. 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841,
  245. 6857, 6863,
  246. 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967,
  247. 6971, 6977,
  248. 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079,
  249. 7103, 7109,
  250. 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219,
  251. 7229, 7237,
  252. 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351,
  253. 7369, 7393,
  254. 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507,
  255. 7517, 7523,
  256. 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591,
  257. 7603, 7607,
  258. 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717,
  259. 7723, 7727,
  260. 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867,
  261. 7873, 7877,
  262. 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993,
  263. 8009, 8011,
  264. 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117,
  265. 8123, 8147,
  266. 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243,
  267. 8263, 8269,
  268. 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377,
  269. 8387, 8389,
  270. 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527,
  271. 8537, 8539,
  272. 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647,
  273. 8663, 8669,
  274. 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747,
  275. 8753, 8761,
  276. 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863,
  277. 8867, 8887,
  278. 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007,
  279. 9011, 9013,
  280. 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137,
  281. 9151, 9157,
  282. 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257,
  283. 9277, 9281,
  284. 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391,
  285. 9397, 9403,
  286. 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479,
  287. 9491, 9497,
  288. 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629,
  289. 9631, 9643,
  290. 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749,
  291. 9767, 9769,
  292. 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859,
  293. 9871, 9883,
  294. 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973, 10007,
  295. 10009, 10037,
  296. 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099, 10103, 10111,
  297. 10133, 10139,
  298. 10141, 10151, 10159, 10163, 10169, 10177, 10181, 10193, 10211, 10223,
  299. 10243, 10247,
  300. 10253, 10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321,
  301. 10331, 10333,
  302. 10337, 10343, 10357, 10369, 10391, 10399, 10427, 10429, 10433, 10453,
  303. 10457, 10459,
  304. 10463, 10477, 10487, 10499, 10501, 10513, 10529, 10531, 10559, 10567,
  305. 10589, 10597,
  306. 10601, 10607, 10613, 10627, 10631, 10639, 10651, 10657, 10663, 10667,
  307. 10687, 10691,
  308. 10709, 10711, 10723, 10729, 10733, 10739, 10753, 10771, 10781, 10789,
  309. 10799, 10831,
  310. 10837, 10847, 10853, 10859, 10861, 10867, 10883, 10889, 10891, 10903,
  311. 10909, 10937,
  312. 10939, 10949, 10957, 10973, 10979, 10987, 10993, 11003, 11027, 11047,
  313. 11057, 11059,
  314. 11069, 11071, 11083, 11087, 11093, 11113, 11117, 11119, 11131, 11149,
  315. 11159, 11161,
  316. 11171, 11173, 11177, 11197, 11213, 11239, 11243, 11251, 11257, 11261,
  317. 11273, 11279,
  318. 11287, 11299, 11311, 11317, 11321, 11329, 11351, 11353, 11369, 11383,
  319. 11393, 11399,
  320. 11411, 11423, 11437, 11443, 11447, 11467, 11471, 11483, 11489, 11491,
  321. 11497, 11503,
  322. 11519, 11527, 11549, 11551, 11579, 11587, 11593, 11597, 11617, 11621,
  323. 11633, 11657,
  324. 11677, 11681, 11689, 11699, 11701, 11717, 11719, 11731, 11743, 11777,
  325. 11779, 11783,
  326. 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833, 11839, 11863,
  327. 11867, 11887,
  328. 11897, 11903, 11909, 11923, 11927, 11933, 11939, 11941, 11953, 11959,
  329. 11969, 11971,
  330. 11981, 11987, 12007, 12011, 12037, 12041, 12043, 12049, 12071, 12073,
  331. 12097, 12101,
  332. 12107, 12109, 12113, 12119, 12143, 12149, 12157, 12161, 12163, 12197,
  333. 12203, 12211,
  334. 12227, 12239, 12241, 12251, 12253, 12263, 12269, 12277, 12281, 12289,
  335. 12301, 12323,
  336. 12329, 12343, 12347, 12373, 12377, 12379, 12391, 12401, 12409, 12413,
  337. 12421, 12433,
  338. 12437, 12451, 12457, 12473, 12479, 12487, 12491, 12497, 12503, 12511,
  339. 12517, 12527,
  340. 12539, 12541, 12547, 12553, 12569, 12577, 12583, 12589, 12601, 12611,
  341. 12613, 12619,
  342. 12637, 12641, 12647, 12653, 12659, 12671, 12689, 12697, 12703, 12713,
  343. 12721, 12739,
  344. 12743, 12757, 12763, 12781, 12791, 12799, 12809, 12821, 12823, 12829,
  345. 12841, 12853,
  346. 12889, 12893, 12899, 12907, 12911, 12917, 12919, 12923, 12941, 12953,
  347. 12959, 12967,
  348. 12973, 12979, 12983, 13001, 13003, 13007, 13009, 13033, 13037, 13043,
  349. 13049, 13063,
  350. 13093, 13099, 13103, 13109, 13121, 13127, 13147, 13151, 13159, 13163,
  351. 13171, 13177,
  352. 13183, 13187, 13217, 13219, 13229, 13241, 13249, 13259, 13267, 13291,
  353. 13297, 13309,
  354. 13313, 13327, 13331, 13337, 13339, 13367, 13381, 13397, 13399, 13411,
  355. 13417, 13421,
  356. 13441, 13451, 13457, 13463, 13469, 13477, 13487, 13499, 13513, 13523,
  357. 13537, 13553,
  358. 13567, 13577, 13591, 13597, 13613, 13619, 13627, 13633, 13649, 13669,
  359. 13679, 13681,
  360. 13687, 13691, 13693, 13697, 13709, 13711, 13721, 13723, 13729, 13751,
  361. 13757, 13759,
  362. 13763, 13781, 13789, 13799, 13807, 13829, 13831, 13841, 13859, 13873,
  363. 13877, 13879,
  364. 13883, 13901, 13903, 13907, 13913, 13921, 13931, 13933, 13963, 13967,
  365. 13997, 13999,
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  716. 35221, 35227, 35251, 35257, 35267, 35279, 35281, 35291, 35311, 35317,
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  718. 35339, 35353, 35363, 35381, 35393, 35401, 35407, 35419, 35423, 35437,
  719. 35447, 35449,
  720. 35461, 35491, 35507, 35509, 35521, 35527, 35531, 35533, 35537, 35543,
  721. 35569, 35573,
  722. 35591, 35593, 35597, 35603, 35617, 35671, 35677, 35729, 35731, 35747,
  723. 35753, 35759,
  724. 35771, 35797, 35801, 35803, 35809, 35831, 35837, 35839, 35851, 35863,
  725. 35869, 35879,
  726. 35897, 35899, 35911, 35923, 35933, 35951, 35963, 35969, 35977, 35983,
  727. 35993, 35999,
  728. 36007, 36011, 36013, 36017, 36037, 36061, 36067, 36073, 36083, 36097,
  729. 36107, 36109,
  730. 36131, 36137, 36151, 36161, 36187, 36191, 36209, 36217, 36229, 36241,
  731. 36251, 36263,
  732. 36269, 36277, 36293, 36299, 36307, 36313, 36319, 36341, 36343, 36353,
  733. 36373, 36383,
  734. 36389, 36433, 36451, 36457, 36467, 36469, 36473, 36479, 36493, 36497,
  735. 36523, 36527,
  736. 36529, 36541, 36551, 36559, 36563, 36571, 36583, 36587, 36599, 36607,
  737. 36629, 36637,
  738. 36643, 36653, 36671, 36677, 36683, 36691, 36697, 36709, 36713, 36721,
  739. 36739, 36749,
  740. 36761, 36767, 36779, 36781, 36787, 36791, 36793, 36809, 36821, 36833,
  741. 36847, 36857,
  742. 36871, 36877, 36887, 36899, 36901, 36913, 36919, 36923, 36929, 36931,
  743. 36943, 36947,
  744. 36973, 36979, 36997, 37003, 37013, 37019, 37021, 37039, 37049, 37057,
  745. 37061, 37087,
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  747. 37217, 37223,
  748. 37243, 37253, 37273, 37277, 37307, 37309, 37313, 37321, 37337, 37339,
  749. 37357, 37361,
  750. 37363, 37369, 37379, 37397, 37409, 37423, 37441, 37447, 37463, 37483,
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  756. 37699, 37717, 37747, 37781, 37783, 37799, 37811, 37813, 37831, 37847,
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  758. 37871, 37879, 37889, 37897, 37907, 37951, 37957, 37963, 37967, 37987,
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  760. 37997, 38011, 38039, 38047, 38053, 38069, 38083, 38113, 38119, 38149,
  761. 38153, 38167,
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  763. 38273, 38281,
  764. 38287, 38299, 38303, 38317, 38321, 38327, 38329, 38333, 38351, 38371,
  765. 38377, 38393,
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  767. 38567, 38569,
  768. 38593, 38603, 38609, 38611, 38629, 38639, 38651, 38653, 38669, 38671,
  769. 38677, 38693,
  770. 38699, 38707, 38711, 38713, 38723, 38729, 38737, 38747, 38749, 38767,
  771. 38783, 38791,
  772. 38803, 38821, 38833, 38839, 38851, 38861, 38867, 38873, 38891, 38903,
  773. 38917, 38921,
  774. 38923, 38933, 38953, 38959, 38971, 38977, 38993, 39019, 39023, 39041,
  775. 39043, 39047,
  776. 39079, 39089, 39097, 39103, 39107, 39113, 39119, 39133, 39139, 39157,
  777. 39161, 39163,
  778. 39181, 39191, 39199, 39209, 39217, 39227, 39229, 39233, 39239, 39241,
  779. 39251, 39293,
  780. 39301, 39313, 39317, 39323, 39341, 39343, 39359, 39367, 39371, 39373,
  781. 39383, 39397,
  782. 39409, 39419, 39439, 39443, 39451, 39461, 39499, 39503, 39509, 39511,
  783. 39521, 39541,
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  785. 39671, 39679,
  786. 39703, 39709, 39719, 39727, 39733, 39749, 39761, 39769, 39779, 39791,
  787. 39799, 39821,
  788. 39827, 39829, 39839, 39841, 39847, 39857, 39863, 39869, 39877, 39883,
  789. 39887, 39901,
  790. 39929, 39937, 39953, 39971, 39979, 39983, 39989, 40009, 40013, 40031,
  791. 40037, 40039,
  792. 40063, 40087, 40093, 40099, 40111, 40123, 40127, 40129, 40151, 40153,
  793. 40163, 40169,
  794. 40177, 40189, 40193, 40213, 40231, 40237, 40241, 40253, 40277, 40283,
  795. 40289, 40343,
  796. 40351, 40357, 40361, 40387, 40423, 40427, 40429, 40433, 40459, 40471,
  797. 40483, 40487,
  798. 40493, 40499, 40507, 40519, 40529, 40531, 40543, 40559, 40577, 40583,
  799. 40591, 40597,
  800. 40609, 40627, 40637, 40639, 40693, 40697, 40699, 40709, 40739, 40751,
  801. 40759, 40763,
  802. 40771, 40787, 40801, 40813, 40819, 40823, 40829, 40841, 40847, 40849,
  803. 40853, 40867,
  804. 40879, 40883, 40897, 40903, 40927, 40933, 40939, 40949, 40961, 40973,
  805. 40993, 41011,
  806. 41017, 41023, 41039, 41047, 41051, 41057, 41077, 41081, 41113, 41117,
  807. 41131, 41141,
  808. 41143, 41149, 41161, 41177, 41179, 41183, 41189, 41201, 41203, 41213,
  809. 41221, 41227,
  810. 41231, 41233, 41243, 41257, 41263, 41269, 41281, 41299, 41333, 41341,
  811. 41351, 41357,
  812. 41381, 41387, 41389, 41399, 41411, 41413, 41443, 41453, 41467, 41479,
  813. 41491, 41507,
  814. 41513, 41519, 41521, 41539, 41543, 41549, 41579, 41593, 41597, 41603,
  815. 41609, 41611,
  816. 41617, 41621, 41627, 41641, 41647, 41651, 41659, 41669, 41681, 41687,
  817. 41719, 41729,
  818. 41737, 41759, 41761, 41771, 41777, 41801, 41809, 41813, 41843, 41849,
  819. 41851, 41863,
  820. 41879, 41887, 41893, 41897, 41903, 41911, 41927, 41941, 41947, 41953,
  821. 41957, 41959,
  822. 41969, 41981, 41983, 41999, 42013, 42017, 42019, 42023, 42043, 42061,
  823. 42071, 42073,
  824. 42083, 42089, 42101, 42131, 42139, 42157, 42169, 42179, 42181, 42187,
  825. 42193, 42197,
  826. 42209, 42221, 42223, 42227, 42239, 42257, 42281, 42283, 42293, 42299,
  827. 42307, 42323,
  828. 42331, 42337, 42349, 42359, 42373, 42379, 42391, 42397, 42403, 42407,
  829. 42409, 42433,
  830. 42437, 42443, 42451, 42457, 42461, 42463, 42467, 42473, 42487, 42491,
  831. 42499, 42509,
  832. 42533, 42557, 42569, 42571, 42577, 42589, 42611, 42641, 42643, 42649,
  833. 42667, 42677,
  834. 42683, 42689, 42697, 42701, 42703, 42709, 42719, 42727, 42737, 42743,
  835. 42751, 42767,
  836. 42773, 42787, 42793, 42797, 42821, 42829, 42839, 42841, 42853, 42859,
  837. 42863, 42899,
  838. 42901, 42923, 42929, 42937, 42943, 42953, 42961, 42967, 42979, 42989,
  839. 43003, 43013,
  840. 43019, 43037, 43049, 43051, 43063, 43067, 43093, 43103, 43117, 43133,
  841. 43151, 43159,
  842. 43177, 43189, 43201, 43207, 43223, 43237, 43261, 43271, 43283, 43291,
  843. 43313, 43319,
  844. 43321, 43331, 43391, 43397, 43399, 43403, 43411, 43427, 43441, 43451,
  845. 43457, 43481,
  846. 43487, 43499, 43517, 43541, 43543, 43573, 43577, 43579, 43591, 43597,
  847. 43607, 43609,
  848. 43613, 43627, 43633, 43649, 43651, 43661, 43669, 43691, 43711, 43717,
  849. 43721, 43753,
  850. 43759, 43777, 43781, 43783, 43787, 43789, 43793, 43801, 43853, 43867,
  851. 43889, 43891,
  852. 43913, 43933, 43943, 43951, 43961, 43963, 43969, 43973, 43987, 43991,
  853. 43997, 44017,
  854. 44021, 44027, 44029, 44041, 44053, 44059, 44071, 44087, 44089, 44101,
  855. 44111, 44119,
  856. 44123, 44129, 44131, 44159, 44171, 44179, 44189, 44201, 44203, 44207,
  857. 44221, 44249,
  858. 44257, 44263, 44267, 44269, 44273, 44279, 44281, 44293, 44351, 44357,
  859. 44371, 44381,
  860. 44383, 44389, 44417, 44449, 44453, 44483, 44491, 44497, 44501, 44507,
  861. 44519, 44531,
  862. 44533, 44537, 44543, 44549, 44563, 44579, 44587, 44617, 44621, 44623,
  863. 44633, 44641,
  864. 44647, 44651, 44657, 44683, 44687, 44699, 44701, 44711, 44729, 44741,
  865. 44753, 44771,
  866. 44773, 44777, 44789, 44797, 44809, 44819, 44839, 44843, 44851, 44867,
  867. 44879, 44887,
  868. 44893, 44909, 44917, 44927, 44939, 44953, 44959, 44963, 44971, 44983,
  869. 44987, 45007,
  870. 45013, 45053, 45061, 45077, 45083, 45119, 45121, 45127, 45131, 45137,
  871. 45139, 45161,
  872. 45179, 45181, 45191, 45197, 45233, 45247, 45259, 45263, 45281, 45289,
  873. 45293, 45307,
  874. 45317, 45319, 45329, 45337, 45341, 45343, 45361, 45377, 45389, 45403,
  875. 45413, 45427,
  876. 45433, 45439, 45481, 45491, 45497, 45503, 45523, 45533, 45541, 45553,
  877. 45557, 45569,
  878. 45587, 45589, 45599, 45613, 45631, 45641, 45659, 45667, 45673, 45677,
  879. 45691, 45697,
  880. 45707, 45737, 45751, 45757, 45763, 45767, 45779, 45817, 45821, 45823,
  881. 45827, 45833,
  882. 45841, 45853, 45863, 45869, 45887, 45893, 45943, 45949, 45953, 45959,
  883. 45971, 45979,
  884. 45989, 46021, 46027, 46049, 46051, 46061, 46073, 46091, 46093, 46099,
  885. 46103, 46133,
  886. 46141, 46147, 46153, 46171, 46181, 46183, 46187, 46199, 46219, 46229,
  887. 46237, 46261,
  888. 46271, 46273, 46279, 46301, 46307, 46309, 46327, 46337, 46349, 46351,
  889. 46381, 46399,
  890. 46411, 46439, 46441, 46447, 46451, 46457, 46471, 46477, 46489, 46499,
  891. 46507, 46511,
  892. 46523, 46549, 46559, 46567, 46573, 46589, 46591, 46601, 46619, 46633,
  893. 46639, 46643,
  894. 46649, 46663, 46679, 46681, 46687, 46691, 46703, 46723, 46727, 46747,
  895. 46751, 46757,
  896. 46769, 46771, 46807, 46811, 46817, 46819, 46829, 46831, 46853, 46861,
  897. 46867, 46877,
  898. 46889, 46901, 46919, 46933, 46957, 46993, 46997, 47017, 47041, 47051,
  899. 47057, 47059,
  900. 47087, 47093, 47111, 47119, 47123, 47129, 47137, 47143, 47147, 47149,
  901. 47161, 47189,
  902. 47207, 47221, 47237, 47251, 47269, 47279, 47287, 47293, 47297, 47303,
  903. 47309, 47317,
  904. 47339, 47351, 47353, 47363, 47381, 47387, 47389, 47407, 47417, 47419,
  905. 47431, 47441,
  906. 47459, 47491, 47497, 47501, 47507, 47513, 47521, 47527, 47533, 47543,
  907. 47563, 47569,
  908. 47581, 47591, 47599, 47609, 47623, 47629, 47639, 47653, 47657, 47659,
  909. 47681, 47699,
  910. 47701, 47711, 47713, 47717, 47737, 47741, 47743, 47777, 47779, 47791,
  911. 47797, 47807,
  912. 47809, 47819, 47837, 47843, 47857, 47869, 47881, 47903, 47911, 47917,
  913. 47933, 47939,
  914. 47947, 47951, 47963, 47969, 47977, 47981, 48017, 48023, 48029, 48049,
  915. 48073, 48079,
  916. 48091, 48109, 48119, 48121, 48131, 48157, 48163, 48179, 48187, 48193,
  917. 48197, 48221,
  918. 48239, 48247, 48259, 48271, 48281, 48299, 48311, 48313, 48337, 48341,
  919. 48353, 48371,
  920. 48383, 48397, 48407, 48409, 48413, 48437, 48449, 48463, 48473, 48479,
  921. 48481, 48487,
  922. 48491, 48497, 48523, 48527, 48533, 48539, 48541, 48563, 48571, 48589,
  923. 48593, 48611,
  924. 48619, 48623, 48647, 48649, 48661, 48673, 48677, 48679, 48731, 48733,
  925. 48751, 48757,
  926. 48761, 48767, 48779, 48781, 48787, 48799, 48809, 48817, 48821, 48823,
  927. 48847, 48857,
  928. 48859, 48869, 48871, 48883, 48889, 48907, 48947, 48953, 48973, 48989,
  929. 48991, 49003,
  930. 49009, 49019, 49031, 49033, 49037, 49043, 49057, 49069, 49081, 49103,
  931. 49109, 49117,
  932. 49121, 49123, 49139, 49157, 49169, 49171, 49177, 49193, 49199, 49201,
  933. 49207, 49211,
  934. 49223, 49253, 49261, 49277, 49279, 49297, 49307, 49331, 49333, 49339,
  935. 49363, 49367,
  936. 49369, 49391, 49393, 49409, 49411, 49417, 49429, 49433, 49451, 49459,
  937. 49463, 49477,
  938. 49481, 49499, 49523, 49529, 49531, 49537, 49547, 49549, 49559, 49597,
  939. 49603, 49613,
  940. 49627, 49633, 49639, 49663, 49667, 49669, 49681, 49697, 49711, 49727,
  941. 49739, 49741,
  942. 49747, 49757, 49783, 49787, 49789, 49801, 49807, 49811, 49823, 49831,
  943. 49843, 49853,
  944. 49871, 49877, 49891, 49919, 49921, 49927, 49937, 49939, 49943, 49957,
  945. 49991, 49993,
  946. 49999, 50021, 50023, 50033, 50047, 50051, 50053, 50069, 50077, 50087,
  947. 50093, 50101,
  948. 50111, 50119, 50123, 50129, 50131, 50147, 50153, 50159, 50177, 50207,
  949. 50221, 50227,
  950. 50231, 50261, 50263, 50273, 50287, 50291, 50311, 50321, 50329, 50333,
  951. 50341, 50359,
  952. 50363, 50377, 50383, 50387, 50411, 50417, 50423, 50441, 50459, 50461,
  953. 50497, 50503,
  954. 50513, 50527, 50539, 50543, 50549, 50551, 50581, 50587, 50591, 50593,
  955. 50599, 50627,
  956. 50647, 50651, 50671, 50683, 50707, 50723, 50741, 50753, 50767, 50773,
  957. 50777, 50789,
  958. 50821, 50833, 50839, 50849, 50857, 50867, 50873, 50891, 50893, 50909,
  959. 50923, 50929,
  960. 50951, 50957, 50969, 50971, 50989, 50993, 51001, 51031, 51043, 51047,
  961. 51059, 51061,
  962. 51071, 51109, 51131, 51133, 51137, 51151, 51157, 51169, 51193, 51197,
  963. 51199, 51203,
  964. 51217, 51229, 51239, 51241, 51257, 51263, 51283, 51287, 51307, 51329,
  965. 51341, 51343,
  966. 51347, 51349, 51361, 51383, 51407, 51413, 51419, 51421, 51427, 51431,
  967. 51437, 51439,
  968. 51449, 51461, 51473, 51479, 51481, 51487, 51503, 51511, 51517, 51521,
  969. 51539, 51551,
  970. 51563, 51577, 51581, 51593, 51599, 51607, 51613, 51631, 51637, 51647,
  971. 51659, 51673,
  972. 51679, 51683, 51691, 51713, 51719, 51721, 51749, 51767, 51769, 51787,
  973. 51797, 51803,
  974. 51817, 51827, 51829, 51839, 51853, 51859, 51869, 51871, 51893, 51899,
  975. 51907, 51913,
  976. 51929, 51941, 51949, 51971, 51973, 51977, 51991, 52009, 52021, 52027,
  977. 52051, 52057,
  978. 52067, 52069, 52081, 52103, 52121, 52127, 52147, 52153, 52163, 52177,
  979. 52181, 52183,
  980. 52189, 52201, 52223, 52237, 52249, 52253, 52259, 52267, 52289, 52291,
  981. 52301, 52313,
  982. 52321, 52361, 52363, 52369, 52379, 52387, 52391, 52433, 52453, 52457,
  983. 52489, 52501,
  984. 52511, 52517, 52529, 52541, 52543, 52553, 52561, 52567, 52571, 52579,
  985. 52583, 52609,
  986. 52627, 52631, 52639, 52667, 52673, 52691, 52697, 52709, 52711, 52721,
  987. 52727, 52733,
  988. 52747, 52757, 52769, 52783, 52807, 52813, 52817, 52837, 52859, 52861,
  989. 52879, 52883,
  990. 52889, 52901, 52903, 52919, 52937, 52951, 52957, 52963, 52967, 52973,
  991. 52981, 52999,
  992. 53003, 53017, 53047, 53051, 53069, 53077, 53087, 53089, 53093, 53101,
  993. 53113, 53117,
  994. 53129, 53147, 53149, 53161, 53171, 53173, 53189, 53197, 53201, 53231,
  995. 53233, 53239,
  996. 53267, 53269, 53279, 53281, 53299, 53309, 53323, 53327, 53353, 53359,
  997. 53377, 53381,
  998. 53401, 53407, 53411, 53419, 53437, 53441, 53453, 53479, 53503, 53507,
  999. 53527, 53549,
  1000. 53551, 53569, 53591, 53593, 53597, 53609, 53611, 53617, 53623, 53629,
  1001. 53633, 53639,
  1002. 53653, 53657, 53681, 53693, 53699, 53717, 53719, 53731, 53759, 53773,
  1003. 53777, 53783,
  1004. 53791, 53813, 53819, 53831, 53849, 53857, 53861, 53881, 53887, 53891,
  1005. 53897, 53899,
  1006. 53917, 53923, 53927, 53939, 53951, 53959, 53987, 53993, 54001, 54011,
  1007. 54013, 54037,
  1008. 54049, 54059, 54083, 54091, 54101, 54121, 54133, 54139, 54151, 54163,
  1009. 54167, 54181,
  1010. 54193, 54217, 54251, 54269, 54277, 54287, 54293, 54311, 54319, 54323,
  1011. 54331, 54347,
  1012. 54361, 54367, 54371, 54377, 54401, 54403, 54409, 54413, 54419, 54421,
  1013. 54437, 54443,
  1014. 54449, 54469, 54493, 54497, 54499, 54503, 54517, 54521, 54539, 54541,
  1015. 54547, 54559,
  1016. 54563, 54577, 54581, 54583, 54601, 54617, 54623, 54629, 54631, 54647,
  1017. 54667, 54673,
  1018. 54679, 54709, 54713, 54721, 54727, 54751, 54767, 54773, 54779, 54787,
  1019. 54799, 54829,
  1020. 54833, 54851, 54869, 54877, 54881, 54907, 54917, 54919, 54941, 54949,
  1021. 54959, 54973,
  1022. 54979, 54983, 55001, 55009, 55021, 55049, 55051, 55057, 55061, 55073,
  1023. 55079, 55103,
  1024. 55109, 55117, 55127, 55147, 55163, 55171, 55201, 55207, 55213, 55217,
  1025. 55219, 55229,
  1026. 55243, 55249, 55259, 55291, 55313, 55331, 55333, 55337, 55339, 55343,
  1027. 55351, 55373,
  1028. 55381, 55399, 55411, 55439, 55441, 55457, 55469, 55487, 55501, 55511,
  1029. 55529, 55541,
  1030. 55547, 55579, 55589, 55603, 55609, 55619, 55621, 55631, 55633, 55639,
  1031. 55661, 55663,
  1032. 55667, 55673, 55681, 55691, 55697, 55711, 55717, 55721, 55733, 55763,
  1033. 55787, 55793,
  1034. 55799, 55807, 55813, 55817, 55819, 55823, 55829, 55837, 55843, 55849,
  1035. 55871, 55889,
  1036. 55897, 55901, 55903, 55921, 55927, 55931, 55933, 55949, 55967, 55987,
  1037. 55997, 56003,
  1038. 56009, 56039, 56041, 56053, 56081, 56087, 56093, 56099, 56101, 56113,
  1039. 56123, 56131,
  1040. 56149, 56167, 56171, 56179, 56197, 56207, 56209, 56237, 56239, 56249,
  1041. 56263, 56267,
  1042. 56269, 56299, 56311, 56333, 56359, 56369, 56377, 56383, 56393, 56401,
  1043. 56417, 56431,
  1044. 56437, 56443, 56453, 56467, 56473, 56477, 56479, 56489, 56501, 56503,
  1045. 56509, 56519,
  1046. 56527, 56531, 56533, 56543, 56569, 56591, 56597, 56599, 56611, 56629,
  1047. 56633, 56659,
  1048. 56663, 56671, 56681, 56687, 56701, 56711, 56713, 56731, 56737, 56747,
  1049. 56767, 56773,
  1050. 56779, 56783, 56807, 56809, 56813, 56821, 56827, 56843, 56857, 56873,
  1051. 56891, 56893,
  1052. 56897, 56909, 56911, 56921, 56923, 56929, 56941, 56951, 56957, 56963,
  1053. 56983, 56989,
  1054. 56993, 56999, 57037, 57041, 57047, 57059, 57073, 57077, 57089, 57097,
  1055. 57107, 57119,
  1056. 57131, 57139, 57143, 57149, 57163, 57173, 57179, 57191, 57193, 57203,
  1057. 57221, 57223,
  1058. 57241, 57251, 57259, 57269, 57271, 57283, 57287, 57301, 57329, 57331,
  1059. 57347, 57349,
  1060. 57367, 57373, 57383, 57389, 57397, 57413, 57427, 57457, 57467, 57487,
  1061. 57493, 57503,
  1062. 57527, 57529, 57557, 57559, 57571, 57587, 57593, 57601, 57637, 57641,
  1063. 57649, 57653,
  1064. 57667, 57679, 57689, 57697, 57709, 57713, 57719, 57727, 57731, 57737,
  1065. 57751, 57773,
  1066. 57781, 57787, 57791, 57793, 57803, 57809, 57829, 57839, 57847, 57853,
  1067. 57859, 57881,
  1068. 57899, 57901, 57917, 57923, 57943, 57947, 57973, 57977, 57991, 58013,
  1069. 58027, 58031,
  1070. 58043, 58049, 58057, 58061, 58067, 58073, 58099, 58109, 58111, 58129,
  1071. 58147, 58151,
  1072. 58153, 58169, 58171, 58189, 58193, 58199, 58207, 58211, 58217, 58229,
  1073. 58231, 58237,
  1074. 58243, 58271, 58309, 58313, 58321, 58337, 58363, 58367, 58369, 58379,
  1075. 58391, 58393,
  1076. 58403, 58411, 58417, 58427, 58439, 58441, 58451, 58453, 58477, 58481,
  1077. 58511, 58537,
  1078. 58543, 58549, 58567, 58573, 58579, 58601, 58603, 58613, 58631, 58657,
  1079. 58661, 58679,
  1080. 58687, 58693, 58699, 58711, 58727, 58733, 58741, 58757, 58763, 58771,
  1081. 58787, 58789,
  1082. 58831, 58889, 58897, 58901, 58907, 58909, 58913, 58921, 58937, 58943,
  1083. 58963, 58967,
  1084. 58979, 58991, 58997, 59009, 59011, 59021, 59023, 59029, 59051, 59053,
  1085. 59063, 59069,
  1086. 59077, 59083, 59093, 59107, 59113, 59119, 59123, 59141, 59149, 59159,
  1087. 59167, 59183,
  1088. 59197, 59207, 59209, 59219, 59221, 59233, 59239, 59243, 59263, 59273,
  1089. 59281, 59333,
  1090. 59341, 59351, 59357, 59359, 59369, 59377, 59387, 59393, 59399, 59407,
  1091. 59417, 59419,
  1092. 59441, 59443, 59447, 59453, 59467, 59471, 59473, 59497, 59509, 59513,
  1093. 59539, 59557,
  1094. 59561, 59567, 59581, 59611, 59617, 59621, 59627, 59629, 59651, 59659,
  1095. 59663, 59669,
  1096. 59671, 59693, 59699, 59707, 59723, 59729, 59743, 59747, 59753, 59771,
  1097. 59779, 59791,
  1098. 59797, 59809, 59833, 59863, 59879, 59887, 59921, 59929, 59951, 59957,
  1099. 59971, 59981,
  1100. 59999, 60013, 60017, 60029, 60037, 60041, 60077, 60083, 60089, 60091,
  1101. 60101, 60103,
  1102. 60107, 60127, 60133, 60139, 60149, 60161, 60167, 60169, 60209, 60217,
  1103. 60223, 60251,
  1104. 60257, 60259, 60271, 60289, 60293, 60317, 60331, 60337, 60343, 60353,
  1105. 60373, 60383,
  1106. 60397, 60413, 60427, 60443, 60449, 60457, 60493, 60497, 60509, 60521,
  1107. 60527, 60539,
  1108. 60589, 60601, 60607, 60611, 60617, 60623, 60631, 60637, 60647, 60649,
  1109. 60659, 60661,
  1110. 60679, 60689, 60703, 60719, 60727, 60733, 60737, 60757, 60761, 60763,
  1111. 60773, 60779,
  1112. 60793, 60811, 60821, 60859, 60869, 60887, 60889, 60899, 60901, 60913,
  1113. 60917, 60919,
  1114. 60923, 60937, 60943, 60953, 60961, 61001, 61007, 61027, 61031, 61043,
  1115. 61051, 61057,
  1116. 61091, 61099, 61121, 61129, 61141, 61151, 61153, 61169, 61211, 61223,
  1117. 61231, 61253,
  1118. 61261, 61283, 61291, 61297, 61331, 61333, 61339, 61343, 61357, 61363,
  1119. 61379, 61381,
  1120. 61403, 61409, 61417, 61441, 61463, 61469, 61471, 61483, 61487, 61493,
  1121. 61507, 61511,
  1122. 61519, 61543, 61547, 61553, 61559, 61561, 61583, 61603, 61609, 61613,
  1123. 61627, 61631,
  1124. 61637, 61643, 61651, 61657, 61667, 61673, 61681, 61687, 61703, 61717,
  1125. 61723, 61729,
  1126. 61751, 61757, 61781, 61813, 61819, 61837, 61843, 61861, 61871, 61879,
  1127. 61909, 61927,
  1128. 61933, 61949, 61961, 61967, 61979, 61981, 61987, 61991, 62003, 62011,
  1129. 62017, 62039,
  1130. 62047, 62053, 62057, 62071, 62081, 62099, 62119, 62129, 62131, 62137,
  1131. 62141, 62143,
  1132. 62171, 62189, 62191, 62201, 62207, 62213, 62219, 62233, 62273, 62297,
  1133. 62299, 62303,
  1134. 62311, 62323, 62327, 62347, 62351, 62383, 62401, 62417, 62423, 62459,
  1135. 62467, 62473,
  1136. 62477, 62483, 62497, 62501, 62507, 62533, 62539, 62549, 62563, 62581,
  1137. 62591, 62597,
  1138. 62603, 62617, 62627, 62633, 62639, 62653, 62659, 62683, 62687, 62701,
  1139. 62723, 62731,
  1140. 62743, 62753, 62761, 62773, 62791, 62801, 62819, 62827, 62851, 62861,
  1141. 62869, 62873,
  1142. 62897, 62903, 62921, 62927, 62929, 62939, 62969, 62971, 62981, 62983,
  1143. 62987, 62989,
  1144. 63029, 63031, 63059, 63067, 63073, 63079, 63097, 63103, 63113, 63127,
  1145. 63131, 63149,
  1146. 63179, 63197, 63199, 63211, 63241, 63247, 63277, 63281, 63299, 63311,
  1147. 63313, 63317,
  1148. 63331, 63337, 63347, 63353, 63361, 63367, 63377, 63389, 63391, 63397,
  1149. 63409, 63419,
  1150. 63421, 63439, 63443, 63463, 63467, 63473, 63487, 63493, 63499, 63521,
  1151. 63527, 63533,
  1152. 63541, 63559, 63577, 63587, 63589, 63599, 63601, 63607, 63611, 63617,
  1153. 63629, 63647,
  1154. 63649, 63659, 63667, 63671, 63689, 63691, 63697, 63703, 63709, 63719,
  1155. 63727, 63737,
  1156. 63743, 63761, 63773, 63781, 63793, 63799, 63803, 63809, 63823, 63839,
  1157. 63841, 63853,
  1158. 63857, 63863, 63901, 63907, 63913, 63929, 63949, 63977, 63997, 64007,
  1159. 64013, 64019,
  1160. 64033, 64037, 64063, 64067, 64081, 64091, 64109, 64123, 64151, 64153,
  1161. 64157, 64171,
  1162. 64187, 64189, 64217, 64223, 64231, 64237, 64271, 64279, 64283, 64301,
  1163. 64303, 64319,
  1164. 64327, 64333, 64373, 64381, 64399, 64403, 64433, 64439, 64451, 64453,
  1165. 64483, 64489,
  1166. 64499, 64513, 64553, 64567, 64577, 64579, 64591, 64601, 64609, 64613,
  1167. 64621, 64627,
  1168. 64633, 64661, 64663, 64667, 64679, 64693, 64709, 64717, 64747, 64763,
  1169. 64781, 64783,
  1170. 64793, 64811, 64817, 64849, 64853, 64871, 64877, 64879, 64891, 64901,
  1171. 64919, 64921,
  1172. 64927, 64937, 64951, 64969, 64997, 65003, 65011, 65027, 65029, 65033,
  1173. 65053, 65063,
  1174. 65071, 65089, 65099, 65101, 65111, 65119, 65123, 65129, 65141, 65147,
  1175. 65167, 65171,
  1176. 65173, 65179, 65183, 65203, 65213, 65239, 65257, 65267, 65269, 65287,
  1177. 65293, 65309,
  1178. 65323, 65327, 65353, 65357, 65371, 65381, 65393, 65407, 65413, 65419,
  1179. 65423, 65437,
  1180. 65447, 65449, 65479, 65497, 65519, 65521,
  1181. };
  1182.  
  1183. #define NPRIMES (sizeof(primes) / sizeof(*primes))
  1184.  
  1185. /*
  1186. * Generate a prime. We can deal with various extra properties of
  1187. * the prime:
  1188. *
  1189. * - to speed up use in RSA, we can arrange to select a prime with
  1190. * the property (prime % modulus) != residue.
  1191. *
  1192. * - for use in DSA, we can arrange to select a prime which is one
  1193. * more than a multiple of a dirty great bignum. In this case
  1194. * `bits' gives the size of the factor by which we _multiply_
  1195. * that bignum, rather than the size of the whole number.
  1196. */
  1197. Bignum primegen(int bits, int modulus, int residue, Bignum factor,
  1198. int phase, progfn_t pfn, void *pfnparam)
  1199. {
  1200. int i, k, v, byte, bitsleft, check, checks;
  1201. unsigned long delta;
  1202. unsigned long moduli[NPRIMES + 1];
  1203. unsigned long residues[NPRIMES + 1];
  1204. unsigned long multipliers[NPRIMES + 1];
  1205. Bignum p, pm1, q, wqp, wqp2;
  1206. int progress = 0;
  1207.  
  1208. byte = 0;
  1209. bitsleft = 0;
  1210.  
  1211. STARTOVER:
  1212.  
  1213. pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress);
  1214.  
  1215. /*
  1216. * Generate a k-bit random number with top and bottom bits set.
  1217. * Alternatively, if `factor' is nonzero, generate a k-bit
  1218. * random number with the top bit set and the bottom bit clear,
  1219. * multiply it by `factor', and add one.
  1220. */
  1221. p = bn_power_2(bits - 1);
  1222. for (i = 0; i < bits; i++) {
  1223. if (i == 0 || i == bits - 1)
  1224. v = (i != 0 || !factor) ? 1 : 0;
  1225. else {
  1226. if (bitsleft <= 0)
  1227. bitsleft = 8, byte = random_byte();
  1228. v = byte & 1;
  1229. byte >>= 1;
  1230. bitsleft--;
  1231. }
  1232. bignum_set_bit(p, i, v);
  1233. }
  1234. if (factor) {
  1235. Bignum tmp = p;
  1236. p = bigmul(tmp, factor);
  1237. freebn(tmp);
  1238. assert(bignum_bit(p, 0) == 0);
  1239. bignum_set_bit(p, 0, 1);
  1240. }
  1241.  
  1242. /*
  1243. * Ensure this random number is coprime to the first few
  1244. * primes, by repeatedly adding either 2 or 2*factor to it
  1245. * until it is.
  1246. */
  1247. for (i = 0; i < NPRIMES; i++) {
  1248. moduli[i] = primes[i];
  1249. residues[i] = bignum_mod_short(p, primes[i]);
  1250. if (factor)
  1251. multipliers[i] = bignum_mod_short(factor, primes[i]);
  1252. else
  1253. multipliers[i] = 1;
  1254. }
  1255. moduli[NPRIMES] = modulus;
  1256. residues[NPRIMES] = (bignum_mod_short(p, (unsigned short) modulus)
  1257. + modulus - residue);
  1258. if (factor)
  1259. multipliers[NPRIMES] = bignum_mod_short(factor, modulus);
  1260. else
  1261. multipliers[NPRIMES] = 1;
  1262. delta = 0;
  1263. while (1) {
  1264. for (i = 0; i < (sizeof(moduli) / sizeof(*moduli)); i++)
  1265. if (!((residues[i] + delta * multipliers[i]) % moduli[i]))
  1266. break;
  1267. if (i < (sizeof(moduli) / sizeof(*moduli))) { /* we broke */
  1268. delta += 2;
  1269. if (delta > 65536) {
  1270. freebn(p);
  1271. goto STARTOVER;
  1272. }
  1273. continue;
  1274. }
  1275. break;
  1276. }
  1277. q = p;
  1278. if (factor) {
  1279. Bignum tmp;
  1280. tmp = bignum_from_long(delta);
  1281. p = bigmuladd(tmp, factor, q);
  1282. freebn(tmp);
  1283. } else {
  1284. p = bignum_add_long(q, delta);
  1285. }
  1286. freebn(q);
  1287.  
  1288. /*
  1289. * Now apply the Miller-Rabin primality test a few times. First
  1290. * work out how many checks are needed.
  1291. */
  1292. checks = 27;
  1293. if (bits >= 150)
  1294. checks = 18;
  1295. if (bits >= 200)
  1296. checks = 15;
  1297. if (bits >= 250)
  1298. checks = 12;
  1299. if (bits >= 300)
  1300. checks = 9;
  1301. if (bits >= 350)
  1302. checks = 8;
  1303. if (bits >= 400)
  1304. checks = 7;
  1305. if (bits >= 450)
  1306. checks = 6;
  1307. if (bits >= 550)
  1308. checks = 5;
  1309. if (bits >= 650)
  1310. checks = 4;
  1311. if (bits >= 850)
  1312. checks = 3;
  1313. if (bits >= 1300)
  1314. checks = 2;
  1315.  
  1316. /*
  1317. * Next, write p-1 as q*2^k.
  1318. */
  1319. for (k = 0; bignum_bit(p, k) == !k; k++)
  1320. continue; /* find first 1 bit in p-1 */
  1321. q = bignum_rshift(p, k);
  1322. /* And store p-1 itself, which we'll need. */
  1323. pm1 = copybn(p);
  1324. decbn(pm1);
  1325.  
  1326. /*
  1327. * Now, for each check ...
  1328. */
  1329. for (check = 0; check < checks; check++) {
  1330. Bignum w;
  1331.  
  1332. /*
  1333. * Invent a random number between 1 and p-1 inclusive.
  1334. */
  1335. while (1) {
  1336. w = bn_power_2(bits - 1);
  1337. for (i = 0; i < bits; i++) {
  1338. if (bitsleft <= 0)
  1339. bitsleft = 8, byte = random_byte();
  1340. v = byte & 1;
  1341. byte >>= 1;
  1342. bitsleft--;
  1343. bignum_set_bit(w, i, v);
  1344. }
  1345. bn_restore_invariant(w);
  1346. if (bignum_cmp(w, p) >= 0 || bignum_cmp(w, Zero) == 0) {
  1347. freebn(w);
  1348. continue;
  1349. }
  1350. break;
  1351. }
  1352.  
  1353. pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress);
  1354.  
  1355. /*
  1356. * Compute w^q mod p.
  1357. */
  1358. wqp = modpow(w, q, p);
  1359. freebn(w);
  1360.  
  1361. /*
  1362. * See if this is 1, or if it is -1, or if it becomes -1
  1363. * when squared at most k-1 times.
  1364. */
  1365. if (bignum_cmp(wqp, One) == 0 || bignum_cmp(wqp, pm1) == 0) {
  1366. freebn(wqp);
  1367. continue;
  1368. }
  1369. for (i = 0; i < k - 1; i++) {
  1370. wqp2 = modmul(wqp, wqp, p);
  1371. freebn(wqp);
  1372. wqp = wqp2;
  1373. if (bignum_cmp(wqp, pm1) == 0)
  1374. break;
  1375. }
  1376. if (i < k - 1) {
  1377. freebn(wqp);
  1378. continue;
  1379. }
  1380.  
  1381. /*
  1382. * It didn't. Therefore, w is a witness for the
  1383. * compositeness of p.
  1384. */
  1385. freebn(wqp);
  1386. freebn(p);
  1387. freebn(pm1);
  1388. freebn(q);
  1389. goto STARTOVER;
  1390. }
  1391.  
  1392. /*
  1393. * We have a prime!
  1394. */
  1395. freebn(q);
  1396. freebn(pm1);
  1397. return p;
  1398. }
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