pe.old/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.
*/
Bignum primegen(int bits, int modulus, int residue, Bignum factor,
int phase, progfn_t pfn, void *pfnparam)
{
int i, k, v, byte, bitsleft, check, checks;
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;
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 (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;
}