sshprime.c

/*
 * Prime generation.
 */

#include <assert.h>
#include "ssh.h"

/*
 * This prime generation algorithm is pretty much cribbed from
 * OpenSSL. The algorithm is:
 * 
 *  - invent a B-bit random number and ensure the top and bottom
 *    bits are set (so it's definitely B-bit, and it's definitely
 *    odd)
 * 
 *  - see if it's coprime to all primes below 2^16; increment it by
 *    two until it is (this shouldn't take long in general)
 * 
 *  - perform the Miller-Rabin primality test enough times to
 *    ensure the probability of it being composite is 2^-80 or
 *    less
 * 
 *  - go back to square one if any M-R test fails.
 */

/*
 * The Miller-Rabin primality test is an extension to the Fermat
 * test. The Fermat test just checks that a^(p-1) == 1 mod p; this
 * is vulnerable to Carmichael numbers. Miller-Rabin considers how
 * that 1 is derived as well.
 * 
 * Lemma: if a^2 == 1 (mod p), and p is prime, then either a == 1
 * or a == -1 (mod p).
 * 
 *   Proof: p divides a^2-1, i.e. p divides (a+1)(a-1). Hence,
 *   since p is prime, either p divides (a+1) or p divides (a-1).
 *   But this is the same as saying that either a is congruent to
 *   -1 mod p or a is congruent to +1 mod p. []
 * 
 *   Comment: This fails when p is not prime. Consider p=mn, so
 *   that mn divides (a+1)(a-1). Now we could have m dividing (a+1)
 *   and n dividing (a-1), without the whole of mn dividing either.
 *   For example, consider a=10 and p=99. 99 = 9 * 11; 9 divides
 *   10-1 and 11 divides 10+1, so a^2 is congruent to 1 mod p
 *   without a having to be congruent to either 1 or -1.
 * 
 * So the Miller-Rabin test, as well as considering a^(p-1),
 * considers a^((p-1)/2), a^((p-1)/4), and so on as far as it can
 * go. In other words. we write p-1 as q * 2^k, with k as large as
 * possible (i.e. q must be odd), and we consider the powers
 * 
 *       a^(q*2^0)      a^(q*2^1)          ...  a^(q*2^(k-1))  a^(q*2^k)
 * i.e.  a^((n-1)/2^k)  a^((n-1)/2^(k-1))  ...  a^((n-1)/2)    a^(n-1)
 * 
 * If p is to be prime, the last of these must be 1. Therefore, by
 * the above lemma, the one before it must be either 1 or -1. And
 * _if_ it's 1, then the one before that must be either 1 or -1,
 * and so on ... In other words, we expect to see a trailing chain
 * of 1s preceded by a -1. (If we're unlucky, our trailing chain of
 * 1s will be as long as the list so we'll never get to see what
 * lies before it. This doesn't count as a test failure because it
 * hasn't _proved_ that p is not prime.)
 * 
 * For example, consider a=2 and p=1729. 1729 is a Carmichael
 * number: although it's not prime, it satisfies a^(p-1) == 1 mod p
 * for any a coprime to it. So the Fermat test wouldn't have a
 * problem with it at all, unless we happened to stumble on an a
 * which had a common factor.
 * 
 * So. 1729 - 1 equals 27 * 2^6. So we look at
 * 
 *     2^27 mod 1729 == 645
 *    2^108 mod 1729 == 1065
 *    2^216 mod 1729 == 1
 *    2^432 mod 1729 == 1
 *    2^864 mod 1729 == 1
 *   2^1728 mod 1729 == 1
 * 
 * We do have a trailing string of 1s, so the Fermat test would
 * have been happy. But this trailing string of 1s is preceded by
 * 1065; whereas if 1729 were prime, we'd expect to see it preceded
 * by -1 (i.e. 1728.). Guards! Seize this impostor.
 * 
 * (If we were unlucky, we might have tried a=16 instead of a=2;
 * now 16^27 mod 1729 == 1, so we would have seen a long string of
 * 1s and wouldn't have seen the thing _before_ the 1s. So, just
 * like the Fermat test, for a given p there may well exist values
 * of a which fail to show up its compositeness. So we try several,
 * just like the Fermat test. The difference is that Miller-Rabin
 * is not _in general_ fooled by Carmichael numbers.)
 * 
 * Put simply, then, the Miller-Rabin test requires us to:
 * 
 *  1. write p-1 as q * 2^k, with q odd
 *  2. compute z = (a^q) mod p.
 *  3. report success if z == 1 or z == -1.
 *  4. square z at most k-1 times, and report success if it becomes
 *     -1 at any point.
 *  5. report failure otherwise.
 * 
 * (We expect z to become -1 after at most k-1 squarings, because
 * if it became -1 after k squarings then a^(p-1) would fail to be
 * 1. And we don't need to investigate what happens after we see a
 * -1, because we _know_ that -1 squared is 1 modulo anything at
 * all, so after we've seen a -1 we can be sure of seeing nothing
 * but 1s.)
 */

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

#define NPRIMES (sizeof(primes) / sizeof(*primes))

/*
 * Generate a prime. We can deal with various extra properties of
 * the prime:
 * 
 *  - to speed up use in RSA, we can arrange to select a prime with
 *    the property (prime % modulus) != residue.
 * 
 *  - for use in DSA, we can arrange to select a prime which is one
 *    more than a multiple of a dirty great bignum. In this case
 *    `bits' gives the size of the factor by which we _multiply_
 *    that bignum, rather than the size of the whole number.
 *
 *  - for the basically cosmetic purposes of generating keys of the
 *    length actually specified rather than off by one bit, we permit
 *    the caller to provide an unsigned integer 'firstbits' which will
 *    match the top few bits of the returned prime. (That is, there
 *    will exist some n such that (returnvalue >> n) == firstbits.) If
 *    'firstbits' is not needed, specifying it to either 0 or 1 is
 *    an adequate no-op.
 */
Bignum primegen(int bits, int modulus, int residue, Bignum factor,
		int phase, progfn_t pfn, void *pfnparam, unsigned firstbits)
{
    int i, k, v, byte, bitsleft, check, checks, fbsize;
    unsigned long delta;
    unsigned long moduli[NPRIMES + 1];
    unsigned long residues[NPRIMES + 1];
    unsigned long multipliers[NPRIMES + 1];
    Bignum p, pm1, q, wqp, wqp2;
    int progress = 0;

    byte = 0;
    bitsleft = 0;

    fbsize = 0;
    while (firstbits >> fbsize)        /* work out how to align this */
        fbsize++;

  STARTOVER:

    pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress);

    /*
     * Generate a k-bit random number with top and bottom bits set.
     * Alternatively, if `factor' is nonzero, generate a k-bit
     * random number with the top bit set and the bottom bit clear,
     * multiply it by `factor', and add one.
     */
    p = bn_power_2(bits - 1);
    for (i = 0; i < bits; i++) {
	if (i == 0 || i == bits - 1) {
	    v = (i != 0 || !factor) ? 1 : 0;
        } else if (i >= bits - fbsize) {
            v = (firstbits >> (i - (bits - fbsize))) & 1;
        } else {
	    if (bitsleft <= 0)
		bitsleft = 8, byte = random_byte();
	    v = byte & 1;
	    byte >>= 1;
	    bitsleft--;
	}
	bignum_set_bit(p, i, v);
    }
    if (factor) {
	Bignum tmp = p;
	p = bigmul(tmp, factor);
	freebn(tmp);
	assert(bignum_bit(p, 0) == 0);
	bignum_set_bit(p, 0, 1);
    }

    /*
     * Ensure this random number is coprime to the first few
     * primes, by repeatedly adding either 2 or 2*factor to it
     * until it is.
     */
    for (i = 0; i < NPRIMES; i++) {
	moduli[i] = primes[i];
	residues[i] = bignum_mod_short(p, primes[i]);
	if (factor)
	    multipliers[i] = bignum_mod_short(factor, primes[i]);
	else
	    multipliers[i] = 1;
    }
    moduli[NPRIMES] = modulus;
    residues[NPRIMES] = (bignum_mod_short(p, (unsigned short) modulus)
			 + modulus - residue);
    if (factor)
	multipliers[NPRIMES] = bignum_mod_short(factor, modulus);
    else
	multipliers[NPRIMES] = 1;
    delta = 0;
    while (1) {
	for (i = 0; i < (sizeof(moduli) / sizeof(*moduli)); i++)
	    if (!((residues[i] + delta * multipliers[i]) % moduli[i]))
		break;
	if (i < (sizeof(moduli) / sizeof(*moduli))) {	/* we broke */
	    delta += 2;
	    if (delta > 65536) {
		freebn(p);
		goto STARTOVER;
	    }
	    continue;
	}
	break;
    }
    q = p;
    if (factor) {
	Bignum tmp;
	tmp = bignum_from_long(delta);
	p = bigmuladd(tmp, factor, q);
	freebn(tmp);
    } else {
	p = bignum_add_long(q, delta);
    }
    freebn(q);

    /*
     * Now apply the Miller-Rabin primality test a few times. First
     * work out how many checks are needed.
     */
    checks = 27;
    if (bits >= 150)
	checks = 18;
    if (bits >= 200)
	checks = 15;
    if (bits >= 250)
	checks = 12;
    if (bits >= 300)
	checks = 9;
    if (bits >= 350)
	checks = 8;
    if (bits >= 400)
	checks = 7;
    if (bits >= 450)
	checks = 6;
    if (bits >= 550)
	checks = 5;
    if (bits >= 650)
	checks = 4;
    if (bits >= 850)
	checks = 3;
    if (bits >= 1300)
	checks = 2;

    /*
     * Next, write p-1 as q*2^k.
     */
    for (k = 0; bignum_bit(p, k) == !k; k++)
	continue;	/* find first 1 bit in p-1 */
    q = bignum_rshift(p, k);
    /* And store p-1 itself, which we'll need. */
    pm1 = copybn(p);
    decbn(pm1);

    /*
     * Now, for each check ...
     */
    for (check = 0; check < checks; check++) {
	Bignum w;

	/*
	 * Invent a random number between 1 and p-1 inclusive.
	 */
	while (1) {
	    w = bn_power_2(bits - 1);
	    for (i = 0; i < bits; i++) {
		if (bitsleft <= 0)
		    bitsleft = 8, byte = random_byte();
		v = byte & 1;
		byte >>= 1;
		bitsleft--;
		bignum_set_bit(w, i, v);
	    }
	    bn_restore_invariant(w);
	    if (bignum_cmp(w, p) >= 0 || bignum_cmp(w, Zero) == 0) {
		freebn(w);
		continue;
	    }
	    break;
	}

	pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress);

	/*
	 * Compute w^q mod p.
	 */
	wqp = modpow(w, q, p);
	freebn(w);

	/*
	 * See if this is 1, or if it is -1, or if it becomes -1
	 * when squared at most k-1 times.
	 */
	if (bignum_cmp(wqp, One) == 0 || bignum_cmp(wqp, pm1) == 0) {
	    freebn(wqp);
	    continue;
	}
	for (i = 0; i < k - 1; i++) {
	    wqp2 = modmul(wqp, wqp, p);
	    freebn(wqp);
	    wqp = wqp2;
	    if (bignum_cmp(wqp, pm1) == 0)
		break;
	}
	if (i < k - 1) {
	    freebn(wqp);
	    continue;
	}

	/*
	 * It didn't. Therefore, w is a witness for the
	 * compositeness of p.
	 */
	freebn(wqp);
	freebn(p);
	freebn(pm1);
	freebn(q);
	goto STARTOVER;
    }

    /*
     * We have a prime!
     */
    freebn(q);
    freebn(pm1);
    return p;
}

/*
 * Invent a pair of values suitable for use as 'firstbits' in the
 * above function, such that their product is at least 2.
 *
 * This is used for generating both RSA and DSA keys which have
 * exactly the specified number of bits rather than one fewer - if you
 * generate an a-bit and a b-bit number completely at random and
 * multiply them together, you could end up with either an (ab-1)-bit
 * number or an (ab)-bit number. The former happens log(2)*2-1 of the
 * time (about 39%) and, though actually harmless, every time it
 * occurs it has a non-zero probability of sparking a user email along
 * the lines of 'Hey, I asked PuTTYgen for a 2048-bit key and I only
 * got 2047 bits! Bug!'
 */
void invent_firstbits(unsigned *one, unsigned *two)
{
    /*
     * Our criterion is that any number in the range [one,one+1)
     * multiplied by any number in the range [two,two+1) should have
     * the highest bit set. It should be clear that we can trivially
     * test this by multiplying the smallest values in each interval,
     * i.e. the ones we actually invented.
     */
    do {
        *one = 0x100 | random_byte();
        *two = 0x100 | random_byte();
    } while (*one * *two < 0x20000);
}