# rmcardle/PuTTYNG

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 /* * Prime generation. */ #include #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, 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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); }
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