diff --git a/lib/Crypto/PublicKey/_RSA.py b/lib/Crypto/PublicKey/_RSA.py index 9267615..89ab5bb 100644 --- a/lib/Crypto/PublicKey/_RSA.py +++ b/lib/Crypto/PublicKey/_RSA.py @@ -37,6 +37,7 @@ def generate_py(bits, randfunc, progress_func=None): the progress of the key generation. """ obj=RSAobj() + obj.e = 65537L # Generate the prime factors of n if progress_func: @@ -46,8 +47,8 @@ def generate_py(bits, randfunc, progress_func=None): # Note that q might be one bit longer than p if somebody specifies an odd # number of bits for the key. (Why would anyone do that? You don't get # more security.) - p = pubkey.getPrime(bits/2, randfunc) - q = pubkey.getPrime(bits - (bits/2), randfunc) + p = pubkey.getStrongPrime(bits/2, obj.e, 1e-12, randfunc) + q = pubkey.getStrongPrime(bits - (bits/2), obj.e, 1e-12, randfunc) # p shall be smaller than q (for calc of u) if p > q: @@ -60,7 +61,6 @@ def generate_py(bits, randfunc, progress_func=None): obj.u = pubkey.inverse(obj.p, obj.q) obj.n = obj.p*obj.q - obj.e = 65537L if progress_func: progress_func('d\n') obj.d=pubkey.inverse(obj.e, (obj.p-1)*(obj.q-1)) diff --git a/lib/Crypto/SelfTest/Util/test_number.py b/lib/Crypto/SelfTest/Util/test_number.py index b7aced4..80913a3 100644 --- a/lib/Crypto/SelfTest/Util/test_number.py +++ b/lib/Crypto/SelfTest/Util/test_number.py @@ -226,6 +226,44 @@ class MiscTests(unittest.TestCase): self.assertEqual((a, b, int(math.floor(float(a) / b))), (a, b, number.floor_div(a, b))) + def test_getStrongPrime(self): + """Util.number.getStrongPrime""" + self.assertRaises(ValueError, number.getStrongPrime, 256) + self.assertRaises(ValueError, number.getStrongPrime, 513) + bits = 512 + x = number.getStrongPrime(bits) + self.assertNotEqual(x % 2, 0) + self.assertEqual(x > (1 << bits-1)-1, 1) + self.assertEqual(x < (1 << bits), 1) + e = 2**16+1 + x = number.getStrongPrime(bits, e) + self.assertEqual(number.GCD(x-1, e), 1) + self.assertNotEqual(x % 2, 0) + self.assertEqual(x > (1 << bits-1)-1, 1) + self.assertEqual(x < (1 << bits), 1) + e = 2**16+2 + x = number.getStrongPrime(bits, e) + self.assertEqual(number.GCD((x-1)/2, e), 1) + self.assertNotEqual(x % 2, 0) + self.assertEqual(x > (1 << bits-1)-1, 1) + self.assertEqual(x < (1 << bits), 1) + + def test_isPrime(self): + """Util.number.isPrime""" + self.assertEqual(number.isPrime(2), True) + self.assertEqual(number.isPrime(3), True) + self.assertEqual(number.isPrime(4), False) + self.assertEqual(number.isPrime(2**1279-1), True) + # test some known gmp pseudo-primes taken from + # http://www.trnicely.net/misc/mpzspsp.html + for composite in (43 * 127 * 211, 61 * 151 * 211, 15259 * 30517, + 346141 * 692281, 1007119 * 2014237, 3589477 * 7178953, + 4859419 * 9718837, 2730439 * 5460877, + 245127919 * 490255837, 963939391 * 1927878781, + 4186358431 * 8372716861, 1576820467 * 3153640933): + self.assertEqual(number.isPrime(long(composite)), False) + + def get_tests(config={}): from Crypto.SelfTest.st_common import list_test_cases return list_test_cases(MiscTests) diff --git a/lib/Crypto/Util/number.py b/lib/Crypto/Util/number.py index 4a71eae..9a7fbdc 100644 --- a/lib/Crypto/Util/number.py +++ b/lib/Crypto/Util/number.py @@ -26,6 +26,8 @@ __revision__ = "$Id$" +import math + bignum = long try: from Crypto.PublicKey import _fastmath @@ -52,21 +54,18 @@ def size (N): """size(N:long) : int Returns the size of the number N in bits. """ - bits, power = 0,1L - while N >= power: + bits = 0 + while N >> bits: bits += 1 - power = power << 1 return bits + def getRandomNumber(N, randfunc=None): """getRandomNumber(N:int, randfunc:callable):long - Return a random N-bit number. + Return a random number with at most N bits. If randfunc is omitted, then Random.new().read is used. - NOTE: Confusingly, this function does NOT return N random bits; It returns - a random N-bit number, i.e. a random number between 2**(N-1) and (2**N)-1. - This function is for internal use only and may be renamed or removed in the future. """ @@ -80,6 +79,35 @@ def getRandomNumber(N, randfunc=None): char = ord(randfunc(1)) >> (8-odd_bits) S = chr(char) + S value = bytes_to_long(S) + return value + +def getRandomRange(a, b, randfunc=None): + """getRandomRange(a:int, b:int, randfunc:callable):long + Return a random number n so that a <= n < b. + + If randfunc is omitted, then Random.new().read is used. + + This function is for internal use only and may be renamed or removed in + the future. + """ + range_ = b - a - 1 + bits = size(range_) + value = getRandomNumber(bits, randfunc) + while value > range_: + value = getRandomNumber(bits, randfunc) + return a + value + +def getRandomNBitNumber(N, randfunc=None): + """getRandomNumber(N:int, randfunc:callable):long + Return a random number with exactly N-bits, i.e. a random number + between 2**(N-1) and (2**N)-1. + + If randfunc is omitted, then Random.new().read is used. + + This function is for internal use only and may be renamed or removed in + the future. + """ + value = getRandomNumber (N-1, randfunc) value |= 2L ** (N-1) # Ensure high bit is set assert size(value) >= N return value @@ -94,7 +122,7 @@ def GCD(x,y): return y def inverse(u, v): - """inverse(u:long, u:long):long + """inverse(u:long, v:long):long Return the inverse of u mod v. """ u3, v3 = long(u), long(v) @@ -120,67 +148,207 @@ def getPrime(N, randfunc=None): _import_Random() randfunc = Random.new().read - number=getRandomNumber(N, randfunc) | 1 + number=getRandomNBitNumber(N, randfunc) | 1 while (not isPrime(number, randfunc=randfunc)): number=number+2 return number -def isPrime(N, randfunc=None): - """isPrime(N:long, randfunc:callable):bool - Return true if N is prime. +def _rabinMillerTest(n, rounds, randfunc=None): + """_rabinMillerTest(n:long, rounds:int, randfunc:callable):int + Tests if n is prime. + Returns 0 when n is definitly composite. + Returns 1 when n is probably prime. + Returns 2 when n is definitly prime. + If randfunc is omitted, then Random.new().read is used. - """ - _import_Random() - if randfunc is None: - randfunc = Random.new().read - randint = StrongRandom(randfunc=randfunc).randint + This function is for internal use only and may be renamed or removed in + the future. + """ + # check special cases (n==2, n even, n < 2) + if n < 3 or (n & 1) == 0: + return n == 2 + # n might be very large so it might be beneficial to precalculate n-1 + n_1 = n - 1 + # determine m and b so that 2**b * m = n - 1 and b maximal + b = 0 + m = n_1 + while (m & 1) == 0: + b += 1 + m //= 2 - if N == 1: - return 0 - if N in sieve: - return 1 - for i in sieve: - if (N % i)==0: + tested = [] + # we need to do at most n-2 rounds. + for i in xrange (min (rounds, n-2)): + # randomly choose a < n and make sure it hasn't been tested yet + a = getRandomRange (2, n, randfunc) + while a in tested: + a = getRandomRange (2, n, randfunc) + tested.append (a) + # do the rabin-miller test + z = pow (a, m, n) # (a**m) % n + if z == 1 or z == n_1: + continue + composite = 1 + for r in xrange (b): + z = (z * z) % n + if z == 1: + return 0 + elif z == n_1: + composite = 0 + break + if composite: return 0 + return 1 + +def getStrongPrime(N, e=0, false_positive_prob=1e-6, randfunc=None): + """getStrongPrime(N:int, e:int, false_positive_prob:float, randfunc:callable):long + Return a random strong N-bit prime number. + In this context p is a strong prime if p-1 and p+1 have at + least one large prime factor. + N should be a multiple of 128 and > 512. + + If e is provided the returned prime p-1 will be coprime to e + and thus suitable for RSA where e is the public exponent. + + The optional false_positive_prob is the statistical probability + that true is returned even though it is not (pseudo-prime). + It defaults to 1e-6 (less than 1:1000000). + Note that the real probability of a false-positiv is far less. This is + just the mathematically provable limit. + + randfunc should take a single int parameter and return that + many random bytes as a string. + If randfunc is omitted, then Random.new().read is used. + """ + # This function was implemented following the + # instructions found in the paper: + # "FAST GENERATION OF RANDOM, STRONG RSA PRIMES" + # by Robert D. Silverman + # RSA Laboratories + # May 17, 1997 + # which by the time of writing could be freely downloaded here: + # http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.17.2713&rep=rep1&type=pdf # Use the accelerator if available if _fastmath is not None: - return _fastmath.isPrime(N) - - # Compute the highest bit that's set in N - N1 = N - 1L - n = 1L - while (n> 1L - - # Rabin-Miller test - for c in sieve[:7]: - a=long(c) ; d=1L ; t=n - while (t): # Iterate over the bits in N1 - x=(d*d) % N - if x==1L and d!=1L and d!=N1: - return 0 # Square root of 1 found - if N1 & t: - d=(x*a) % N + return _fastmath.getStrongPrime(long(N), long(e), false_positive_prob, randfunc) + + if (N < 512) or ((N % 128) != 0): + raise ValueError ("bits must be multiple of 128 and > 512") + + rabin_miller_rounds = int(math.ceil(-math.log(false_positive_prob, 4))) + + # calculate range for X + # lower_bound = sqrt(2) * 2^{511 + 128*x} + # upper_bound = 2^{512 + 128*x} - 1 + x = (N - 512) // 128; + # We need to approximate the sqrt(2) in the lower_bound by an integer + # expression because floating point math overflows with these numbers + lower_bound = (14142135623730950489 * (2 ** (511 + 128*x)) / + 10000000000000000000) + upper_bound = (1 << (512 + 128*x)) - 1 + # Randomly choose X in calculated range + X = getRandomRange (lower_bound, upper_bound, randfunc) + + # generate p1 and p2 + p = [0, 0] + for i in (0, 1): + # randomly choose 101-bit y + y = getRandomNBitNumber (101, randfunc) + + # initialize the field for sieving + field = [0] * 5 * len (sieve_base) + # sieve the field + for prime in sieve_base: + offset = y % prime + for j in xrange ((prime - offset) % prime, len (field), prime): + field[j] = 1 + + # look for suitable p[i] starting at y + result = 0 + for j, composite in enumerate (field): + # look for next canidate + if composite: + continue + tmp = y + j + result = _rabinMillerTest (tmp, rabin_miller_rounds) + if result > 0: + p[i] = tmp + break + if result == 0: + raise RuntimeError ("Couln't find prime in field. " + "Developer: Increase field_size") + + # Calculate R + # R = (p2^{-1} mod p1) * p2 - (p1^{-1} mod p2) * p1 + tmp1 = inverse (p[1], p[0]) * p[1] # (p2^-1 mod p1)*p2 + tmp2 = inverse (p[0], p[1]) * p[0] # (p1^-1 mod p2)*p1 + R = tmp1 - tmp2 # (p2^-1 mod p1)*p2 - (p1^-1 mod p2)*p1 + + # search for final prime number starting by Y0 + # Y0 = X + (R - X mod p1p2) + increment = p[0] * p[1] + X = X + (R - (X % increment)) + while 1: + is_possible_prime = 1 + # first check canidate against sieve_base + for prime in sieve_base: + if (X % prime) == 0: + is_possible_prime = 0 + break + # if e is given make sure that e and X-1 are coprime + # this is not necessarily a strong prime criterion but usefull when + # creating them for RSA where the p-1 and q-1 should be coprime to + # the public exponent e + if e and is_possible_prime: + if e & 1: + if GCD (e, X-1) != 1: + is_possible_prime = 0 else: - d=x - t = t >> 1L - if d!=1L: - return 0 - return 1 + if GCD (e, (X-1)/2) != 1: + is_possible_prime = 0 + # do some Rabin-Miller-Tests + if is_possible_prime: + result = _rabinMillerTest (X, rabin_miller_rounds) + if result > 0: + break + X += increment + # abort when X has more bits than requested + # TODO: maybe we shouldn't abort but rather start over. + if X >= 1 << N: + raise RuntimeError ("Couln't find prime in field. " + "Developer: Increase field_size") + return X -# Small primes used for checking primality; these are all the primes -# less than 256. This should be enough to eliminate most of the odd -# numbers before needing to do a Rabin-Miller test at all. +def isPrime(N, false_positive_prob=1e-6, randfunc=None): + """isPrime(N:long, false_positive_prob:float, randfunc:callable):bool + Return true if N is prime. + + The optional false_positive_prob is the statistical probability + that true is returned even though it is not (pseudo-prime). + It defaults to 1e-6 (less than 1:1000000). + Note that the real probability of a false-positiv is far less. This is + just the mathematically provable limit. + + If randfunc is omitted, then Random.new().read is used. + """ + if _fastmath is not None: + return _fastmath.isPrime(long(N), false_positive_prob, randfunc) -sieve=[2, 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] + if N < 3 or N & 1 == 0: + return N == 2 + for p in sieve_base: + if N == p: + return 1 + if N % p == 0: + return 0 + + rounds = int(math.ceil(-math.log(false_positive_prob, 4))) + return _rabinMillerTest(N, rounds, randfunc) + # Improved conversion functions contributed by Barry Warsaw, after # careful benchmarking @@ -248,3 +416,1011 @@ def _import_Random(): from Crypto import Random from Crypto.Random.random import StrongRandom + + +# The first 10000 primes used for checking primality. +# This should be enough to eliminate most of the odd +# numbers before needing to do a Rabin-Miller test at all. +sieve_base = ( + 2, 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, 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100469, +100483, 100493, 100501, 100511, 100517, 100519, 100523, 100537, 100547, 100549, +100559, 100591, 100609, 100613, 100621, 100649, 100669, 100673, 100693, 100699, +100703, 100733, 100741, 100747, 100769, 100787, 100799, 100801, 100811, 100823, +100829, 100847, 100853, 100907, 100913, 100927, 100931, 100937, 100943, 100957, +100981, 100987, 100999, 101009, 101021, 101027, 101051, 101063, 101081, 101089, +101107, 101111, 101113, 101117, 101119, 101141, 101149, 101159, 101161, 101173, +101183, 101197, 101203, 101207, 101209, 101221, 101267, 101273, 101279, 101281, +101287, 101293, 101323, 101333, 101341, 101347, 101359, 101363, 101377, 101383, +101399, 101411, 101419, 101429, 101449, 101467, 101477, 101483, 101489, 101501, +101503, 101513, 101527, 101531, 101533, 101537, 101561, 101573, 101581, 101599, +101603, 101611, 101627, 101641, 101653, 101663, 101681, 101693, 101701, 101719, +101723, 101737, 101741, 101747, 101749, 101771, 101789, 101797, 101807, 101833, +101837, 101839, 101863, 101869, 101873, 101879, 101891, 101917, 101921, 101929, +101939, 101957, 101963, 101977, 101987, 101999, 102001, 102013, 102019, 102023, +102031, 102043, 102059, 102061, 102071, 102077, 102079, 102101, 102103, 102107, +102121, 102139, 102149, 102161, 102181, 102191, 102197, 102199, 102203, 102217, +102229, 102233, 102241, 102251, 102253, 102259, 102293, 102299, 102301, 102317, +102329, 102337, 102359, 102367, 102397, 102407, 102409, 102433, 102437, 102451, +102461, 102481, 102497, 102499, 102503, 102523, 102533, 102539, 102547, 102551, +102559, 102563, 102587, 102593, 102607, 102611, 102643, 102647, 102653, 102667, +102673, 102677, 102679, 102701, 102761, 102763, 102769, 102793, 102797, 102811, +102829, 102841, 102859, 102871, 102877, 102881, 102911, 102913, 102929, 102931, +102953, 102967, 102983, 103001, 103007, 103043, 103049, 103067, 103069, 103079, +103087, 103091, 103093, 103099, 103123, 103141, 103171, 103177, 103183, 103217, +103231, 103237, 103289, 103291, 103307, 103319, 103333, 103349, 103357, 103387, +103391, 103393, 103399, 103409, 103421, 103423, 103451, 103457, 103471, 103483, +103511, 103529, 103549, 103553, 103561, 103567, 103573, 103577, 103583, 103591, +103613, 103619, 103643, 103651, 103657, 103669, 103681, 103687, 103699, 103703, +103723, 103769, 103787, 103801, 103811, 103813, 103837, 103841, 103843, 103867, +103889, 103903, 103913, 103919, 103951, 103963, 103967, 103969, 103979, 103981, +103991, 103993, 103997, 104003, 104009, 104021, 104033, 104047, 104053, 104059, +104087, 104089, 104107, 104113, 104119, 104123, 104147, 104149, 104161, 104173, +104179, 104183, 104207, 104231, 104233, 104239, 104243, 104281, 104287, 104297, +104309, 104311, 104323, 104327, 104347, 104369, 104381, 104383, 104393, 104399, +104417, 104459, 104471, 104473, 104479, 104491, 104513, 104527, 104537, 104543, +104549, 104551, 104561, 104579, 104593, 104597, 104623, 104639, 104651, 104659, +104677, 104681, 104683, 104693, 104701, 104707, 104711, 104717, 104723, 104729, +) + diff --git a/pct-speedtest.py b/pct-speedtest.py index 911ffc2..87472af 100644 --- a/pct-speedtest.py +++ b/pct-speedtest.py @@ -27,6 +27,7 @@ import time import os import sys +from Crypto.PublicKey import RSA from Crypto.Cipher import AES, ARC2, ARC4, Blowfish, CAST, DES3, DES, XOR from Crypto.Hash import MD2, MD4, MD5, SHA256, SHA try: @@ -79,6 +80,17 @@ class Benchmark: sys.stdout.write("%.2f %s\n" % (value, units)) sys.stdout.flush() + def test_pubkey_setup(self, pubkey_name, module, key_bytes): + self.announce_start("%s pubkey setup" % (pubkey_name,)) + keys = self.random_keys(key_bytes)[:5] + + t0 = time.time() + for k in keys: + module.generate(key_bytes*8) + t = time.time() + pubkey_setups_per_second = len(keys) / (t - t0) + self.announce_result(pubkey_setups_per_second, "Keys/sec") + def test_key_setup(self, cipher_name, module, key_bytes, mode): self.announce_start("%s key setup" % (cipher_name,)) @@ -152,6 +164,11 @@ class Benchmark: self.announce_result(hash_speed / 10**6, "MBps") def run(self): + pubkey_specs = [ + ("RSA(1024)", RSA, 1024/8), + ("RSA(2048)", RSA, 2048/8), + ("RSA(4096)", RSA, 4096/8), + ] block_specs = [ ("DES", DES, 8), ("DES3", DES3, 24), @@ -179,6 +196,9 @@ class Benchmark: if RIPEMD is not None: hash_specs += [("RIPEMD", RIPEMD)] + for pubkey_name, module, key_bytes in pubkey_specs: + self.test_pubkey_setup(pubkey_name, module, key_bytes) + for cipher_name, module, key_bytes in block_specs: self.test_key_setup(cipher_name, module, key_bytes, module.MODE_CBC) self.test_encryption("%s-CBC" % (cipher_name,), module, key_bytes, module.MODE_CBC) diff --git a/src/_fastmath.c b/src/_fastmath.c index 0c036e3..5fbe18a 100755 --- a/src/_fastmath.c +++ b/src/_fastmath.c @@ -33,6 +33,11 @@ #include /* for conversions */ #include +#define SIEVE_BASE_SIZE (sizeof (sieve_base) / sizeof (sieve_base[0])) + +static unsigned int sieve_base[10000]; +static int rabinMillerTest (mpz_t n, int rounds, PyObject *randfunc); + static void longObjToMPZ (mpz_t m, PyLongObject * p) { @@ -772,13 +777,18 @@ rsaKey_has_private (rsaKey * key, PyObject * args) static PyObject * -isPrime (PyObject * self, PyObject * args) +isPrime (PyObject * self, PyObject * args, PyObject * kwargs) { - PyObject *l; + unsigned int i, rounds; + double false_positive_prob=1e-6; + PyObject *l, *randfunc=NULL; mpz_t n; int result; + static char *kwlist[] = {"N", "false_positive_prob", "randfunc", NULL}; - if (!PyArg_ParseTuple (args, "O!", &PyLong_Type, &l)) + if (!PyArg_ParseTupleAndKeywords (args, kwargs, "O!|dO:isPrime", kwlist, + &PyLong_Type, &l, &false_positive_prob, + &randfunc)) { return NULL; } @@ -786,33 +796,589 @@ isPrime (PyObject * self, PyObject * args) longObjToMPZ (n, (PyLongObject *) l); Py_BEGIN_ALLOW_THREADS; - result = mpz_probab_prime_p(n, 5); - Py_END_ALLOW_THREADS; + /* first check if n is known to be prime and do some trail division */ + for (i = 0; i < SIEVE_BASE_SIZE; ++i) + { + if (mpz_cmp_ui (n, sieve_base[i]) == 0) + { + result = 2; + goto cleanup; + } + if (mpz_divisible_ui_p (n, sieve_base[i])) + { + result = 0; + goto cleanup; + } + } + /* do some rounds of Rabin-Miller-Tests */ + rounds = (unsigned int)ceil (-log (false_positive_prob) / log (4)); + Py_BLOCK_THREADS; + result = rabinMillerTest(n, rounds, randfunc); + Py_UNBLOCK_THREADS; +cleanup: mpz_clear (n); + Py_END_ALLOW_THREADS; - if (result == 0) { + if (result == 0) + { Py_INCREF(Py_False); return Py_False; - } else { + } else { Py_INCREF(Py_True); return Py_True; } } + +inline size_t size (mpz_t n) +{ + return mpz_sizeinbase (n, 2); +} + +void bytes_to_mpz (mpz_t result, const unsigned char *bytes, size_t size) +{ + unsigned long int i; + mpz_t tmp; + mpz_init (tmp); + Py_BEGIN_ALLOW_THREADS; + mpz_set_ui (result, 0); + for (i = 0; i < size; ++i) + { + /* get current byte */ + mpz_set_ui (tmp, (unsigned long int)bytes[i]); + /* left shift and add */ + mpz_mul_2exp (tmp, tmp, 8 * i); + mpz_add (result, result, tmp); + } + mpz_clear (tmp); + Py_END_ALLOW_THREADS; +} + + +/* Returns a new reference to a rng from the Crypto.Random module. */ +static PyObject * +getRNG () +{ + /* PyModule_GetDict, PyDict_GetItemString return a borrowed ref */ + PyObject *module, *module_dict, *new_func, *rng; + + module = PyImport_ImportModule ("Crypto.Random"); + if (!module) + return 0; + module_dict = PyModule_GetDict (module); + Py_DECREF (module); + new_func = PyDict_GetItemString (module_dict, "new"); + if (!PyCallable_Check (new_func)) + { + PyErr_SetString (PyExc_RuntimeError, + "Cryptor.Random.new is not callable."); + return 0; + } + rng = PyObject_CallObject (new_func, NULL); + return rng; +} + + +/* Sets n to a rangom number with at most `bits` bits . + * If randfunc is provided it should be a callable which takes a single int + * parameter and return as many random bytes as a python string. + * Returns 1 on success + * Returns 0 on error (most likly a python error) + * The thread should be holding the GIL. The function does nothing to check + * this. (PyGILState_Ensure() was only introduced in python2.3 and we want to + * support 2.1) + */ +static int +getRandomNumber (mpz_t n, unsigned long int bits, PyObject *randfunc_) +{ + PyObject *arglist, *randfunc, *rng=NULL, *rand_bytes=NULL; + int return_val = 1; + unsigned long int bytes = bits / 8; + unsigned long int odd_bits = bits % 8; + /* generate 1 to 8 bits too many. + we will remove them later by right-shifting */ + bytes++; + /* we need to handle the cases where randfunc is NULL or None */ + if ((randfunc_ == NULL) || (randfunc_ == Py_None)) + { + rng = getRNG(); + if (!rng) + { + return_val = 0; + goto cleanup; + } + randfunc = PyObject_GetAttrString (rng, "read"); + } + else + { + randfunc = randfunc_; + } + + if (!PyCallable_Check (randfunc)) + { + PyErr_SetString (PyExc_TypeError, "randfunc must be callable"); + return_val = 0; + goto cleanup; + } + + arglist = Py_BuildValue ("(l)", (long int)bytes); + rand_bytes = PyObject_CallObject (randfunc, arglist); + Py_DECREF (arglist); + if (!PyString_Check (rand_bytes)) + { + PyErr_SetString (PyExc_TypeError, + "randfunc must return a string of random bytes"); + return_val = 0; + goto cleanup; + } + + bytes_to_mpz (n, PyString_AsString(rand_bytes), bytes); + /* remove superflous bits by right-shifting */ + mpz_fdiv_q_2exp (n, n, 8 - odd_bits); + +cleanup: + Py_XDECREF (rand_bytes); + if (rng) + { + Py_XDECREF (randfunc); + Py_DECREF (rng); + } + return return_val; +} + + +/* Sets n to a rangom number with exactly `bits` random bits. + * randfunc should be either NULL, PyNone or a callable which takes a single + * integer parameter and return as many random bytes as a python string. + * Returns 1 on success + * Returns 0 on error (most likly a python error) + * The thread should be holding the GIL. The function does nothing to check + * this. (PyGILState_Ensure() was only introduced in python2.3 and we want to + * support 2.1) + */ +static int +getRandomNBitNumber (mpz_t n, unsigned long int bits, PyObject *randfunc) +{ + if (!getRandomNumber (n, bits, randfunc)) + return 0; + /* set the MSB to ensure n really has the correct number of bits. */ + mpz_setbit (n, bits); + return 1; +} + + +/* Sets n to a rangom number so that lower_bound <= n < upper_bound . + * If randfunc is provided it should be a callable which takes a single int + * parameter and return as many random bytes as a python string. + * Returns 1 on success + * Returns 0 on error (most likly a python error) + * The thread should be holding the GIL. The function does nothing to check + * this. (PyGILState_Ensure() was only introduced in python2.3 and we want to + * support 2.1) + */ +static int +getRandomRange (mpz_t n, mpz_t lower_bound, mpz_t upper_bound, + PyObject *randfunc) +{ + size_t bits; + mpz_t range; + mpz_init (range); + mpz_sub (range, upper_bound, lower_bound); + mpz_sub_ui (range, range, 1); + bits = size (range); + + do + { + if (!getRandomNumber (n, bits, randfunc)) + { + mpz_clear (range); + return 0; + } + } while (mpz_cmp (n, range) > 0); + + mpz_clear (range); + mpz_add (n, n, lower_bound); + return 1; +} + + + +static void +sieve_field (char *field, unsigned long int field_size, mpz_t start) +{ + mpz_t mpz_offset; + unsigned int offset; + unsigned int i, j; + + mpz_init (mpz_offset); + + for (i = 0; i < SIEVE_BASE_SIZE; ++i) + { + mpz_mod_ui (mpz_offset, start, sieve_base[i]); + offset = mpz_get_ui (mpz_offset); + for (j = (sieve_base[i] - offset) % sieve_base[i]; j < field_size; j += sieve_base[i]) + { + field[j] = 1; + } + } + + mpz_clear (mpz_offset); +} + + +#define MAX_RABIN_MILLER_ROUNDS 255 + +/* Tests if n is prime. + * Returns 0 when n is definitly composite. + * Returns 1 when n is probably prime. + * every round reduces the chance of a false positive be at least 1/4. + * + * If randfunc is omitted, then the python version Random.new().read is used. + * + * The thread should be holding the GIL. The function does nothing to check + * this. (PyGILState_Ensure() was only introduced in python2.3 and we want to + * support 2.1) + */ +static int +rabinMillerTest (mpz_t n, int rounds, PyObject *randfunc) +{ + int base_was_tested; + unsigned long int i, j, b, composite, return_val=1; + mpz_t a, m, z, n_1, tmp; + mpz_t tested[MAX_RABIN_MILLER_ROUNDS]; + + if (rounds > MAX_RABIN_MILLER_ROUNDS) + { + PyErr_WarnEx (PyExc_RuntimeWarning, + "rounds to Rabin-Miller-Test exceeds maximum. " + "rounds will be set to the maximum.\n" + "Go complain to the devs about it if you like.", 1); + rounds = MAX_RABIN_MILLER_ROUNDS; + } + + Py_BEGIN_ALLOW_THREADS; + if ((mpz_tstbit (n, 0) == 0) || (mpz_cmp_ui (n, 3) < 0)) + return (mpz_cmp_ui (n, 2) == 0); + + mpz_init (tmp); + mpz_init (n_1); + mpz_init (a); + mpz_init (m); + mpz_init (z); + mpz_sub_ui (n_1, n, 1); + b = mpz_scan1 (n_1, 0); + mpz_fdiv_q_2exp (m, n_1, b); + + if (mpz_fits_ulong_p (n) && (mpz_get_ui (n) - 2 < rounds)) + rounds = mpz_get_ui (n) - 2; + for (i = 0; i < rounds; ++i) + { + mpz_set_ui (tmp, 2); + do + { + base_was_tested = 0; + Py_BLOCK_THREADS; + if (!getRandomRange (a, tmp, n, randfunc)) + { + return_val = -1; + Py_UNBLOCK_THREADS; + goto cleanup; + } + Py_UNBLOCK_THREADS; + for (j = 0; j < i; j++) + { + if (mpz_cmp (a, tested[j]) == 0) + { + base_was_tested = 1; + break; + } + } + } while (base_was_tested); + mpz_init_set (tested[i], a); + mpz_powm (z, a, m, n); + if ((mpz_cmp_ui (z, 1) == 0) || (mpz_cmp (z, n_1) == 0)) + continue; + composite = 1; + for (j = 0; j < b; ++j) + { + /* z = (z * z) % n */ + mpz_mul (z, z, z); + mpz_mod (z, z, n); + if (mpz_cmp_ui (z, 1) == 0) + { + return_val = 0; + goto cleanup; + } + else if (mpz_cmp (z, n_1) == 0) + { + composite = 0; + break; + } + } + + if (composite) + { + return_val = 0; + goto cleanup; + } + } + +cleanup: + mpz_clear (tmp); + mpz_clear (n_1); + mpz_clear (a); + mpz_clear (m); + mpz_clear (z); + Py_END_ALLOW_THREADS; + return return_val; +} + + +/* getStrongPrime() generates a number p which is with a high probability a + * prime for which the following is true: + * p+1 has at least one large prime factor + * p-1 has at least one large prime factor + * This functions was implemented following the instructions found in the paper + * "FAST GENERATION OF RANDOM, STRONG RSA PRIMES" + * by Robert D. Silverman + * RSA Laboratories + * May 17, 1997 + * which by the time of writing could be obtained here: + * http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.17.2713&rep=rep1&type=pdf + */ +static PyObject * +getStrongPrime (PyObject *self, PyObject *args, PyObject *kwargs) +{ + unsigned long int i, j, result, bits, x, e=0; + mpz_t p[2], y[2], R, X; + mpz_t tmp[2], lower_bound, upper_bound, range, increment; + mpf_t tmp_bound; + char *field; + double false_positive_prob; + int rabin_miller_rounds, is_possible_prime, error = 0; + PyObject *prime, *randfunc=NULL; + static char *kwlist[] = {"N", "e", "false_positive_prob", "randfunc", NULL}; + unsigned long int base_size = SIEVE_BASE_SIZE; + unsigned long int field_size = 5 * base_size; + + if (!PyArg_ParseTupleAndKeywords (args, kwargs, "l|ldO:getStrongPrime", + kwlist, &bits, &e, &false_positive_prob, + &randfunc)) + { + return NULL; + } + + if ((bits < 512) || ((bits % 128) != 0)) + { + PyErr_SetString (PyExc_ValueError, + "bits must be multiple of 128 and > 512"); + return NULL; + } + + Py_BEGIN_ALLOW_THREADS; + rabin_miller_rounds = (int)ceil (-log (false_positive_prob) / log (4)); + /* The variable names y, p, X and R correspond to + * the names in the paper mentioned above + */ + mpz_init2 (y[0], 101); + mpz_init2 (y[1], 101); + mpz_init2 (p[0], 121); + mpz_init2 (p[1], 121); + mpz_init2 (X, bits); + mpz_init2 (R, bits); + mpz_init2 (increment, 242); + mpz_init (tmp[0]); + mpz_init (tmp[1]); + mpz_init2 (lower_bound, bits); + mpz_init2 (upper_bound, bits); + mpf_init (tmp_bound); + mpz_init (range); + + /* calculate range for X + * lower_bound = sqrt(2) * 2^{511 + 128*x} + * upper_bound = 2^{512 + 128*x} - 1 + */ + x = (bits - 512) / 128; + mpf_sqrt_ui (tmp_bound, 2); + mpf_mul_2exp (tmp_bound, tmp_bound, 511 + 128 * x); + mpf_ceil (tmp_bound, tmp_bound); + mpz_set_f (lower_bound, tmp_bound); + mpz_set_ui (upper_bound, 1); + mpz_mul_2exp (upper_bound, upper_bound, 512 + 128 * x); + mpz_sub_ui (upper_bound, upper_bound, 1); + mpz_sub (range, upper_bound, lower_bound); + + /* Randomly choose X, y[0] and y[1] */ + Py_BLOCK_THREADS; + int res = 1; + res &= getRandomRange (X, lower_bound, upper_bound, randfunc); + res &= getRandomNBitNumber (y[0], 101, randfunc); + res &= getRandomNBitNumber (y[1], 101, randfunc); + Py_UNBLOCK_THREADS; + if (!res) + { + error = 1; + goto cleanup; + } + + /* generate p1 and p2 */ + for (i = 0; i < 2; ++i) + { + /* generate p[i] */ + /* initialize the field for sieving */ + field = (char*)calloc (field_size, 1); + sieve_field (field, field_size, y[i]); + + result = 0; + for (j = 0; j < field_size; ++j) + { + /* look for next canidate */ + if (field[j]) + continue; + mpz_add_ui (tmp[0], y[i], j); + Py_BLOCK_THREADS; + result = rabinMillerTest(tmp[0], rabin_miller_rounds, randfunc); + Py_UNBLOCK_THREADS; + if (result > 0) + break; + else if (result < 0) + { + error = 1; + goto cleanup; + } + } + free (field); + if (!result) + { + error = 1; + Py_BLOCK_THREADS; + PyErr_SetString (PyExc_RuntimeError, + "Couln't find prime in field. " + "Developer: Increase field_size"); + /* unblock threads because we acquire the GIL after cleanup */ + Py_UNBLOCK_THREADS; + goto cleanup; + } + mpz_set (p[i], tmp[0]); + } + + /* Calculate R + * R = (p2^{-1} mod p1) * p2 - (p1^{-1} mod p2) * p1 + */ + mpz_invert (tmp[0], p[1], p[0]); /* p2^-1 mod p1 */ + mpz_invert (tmp[1], p[0], p[1]); /* p1^-1 mod p2 */ + mpz_mul (tmp[0], tmp[0], p[1]); /* (p2^-1 mod p1)*p2 */ + mpz_mul (tmp[1], tmp[1], p[0]); /* (p1^-1 mod p2)*p1 */ + mpz_sub (R, tmp[0], tmp[1]); /* (p2^-1 mod p1)*p2 - (p1^-1 mod p2)*p1 */ + + /* search for final prime number starting by Y0 + * Y0 = X + (R - X mod p1p2) + */ + mpz_mul (increment, p[0], p[1]); /* p1 * p2 */ + mpz_mod (tmp[0], X, increment); /* X mod (p1*p2) */ + mpz_sub (tmp[1], R, tmp[0]); /* R - X mod (p1*p2) */ + mpz_add (X, X, tmp[1]); /* X + (R - X mod (p1*p2)) */ + while (1) + { + is_possible_prime = 1; + /* first check canidate against sieve_base */ + for (j = 0; j < base_size; ++j) + { + if (mpz_divisible_ui_p (X, sieve_base[j])) + { + is_possible_prime = 0; + break; + } + } + /* if e is given check for some more constrains */ + if (e && is_possible_prime) + { + /* if e is odd make sure that e and X-1 are coprime */ + if (e & 1) + { + mpz_sub_ui (tmp[0], X, 1); + if (mpz_gcd_ui (NULL, tmp[0], e) != 1) + is_possible_prime = 0; + } + /* if e is even make sure that e and (X-1)/2 are coprime. */ + else + { + mpz_sub_ui (tmp[0], X, 1); + mpz_divexact_ui (tmp[0], tmp[0], 2); + if (mpz_gcd_ui (NULL, tmp[0], e) != 1) + is_possible_prime = 0; + } + } + /* let gmp do some Rabin-Miller-Tests */ + if (is_possible_prime) + { + Py_BLOCK_THREADS; + result = rabinMillerTest(X, rabin_miller_rounds, randfunc); + Py_UNBLOCK_THREADS; + if (result > 0) + break; + else if (result < 0) + { + error = 1; + goto cleanup; + } + } + mpz_add (X, X, increment); + /* abort when X is larger than upper_bound */ + /* TODO: maybe we shouldn't abort but rather start over. + * X probably was an unfortunate choice. */ + if (mpz_cmp (X, upper_bound) >= 0) + { + error = 1; + Py_BLOCK_THREADS; + PyErr_SetString (PyExc_RuntimeError, + "Couln't find prime in field. " + "Developer: Increase field_size"); + /* unblock threads because we acquire the GIL after cleanup */ + Py_UNBLOCK_THREADS; + goto cleanup; + } + } + +cleanup: + mpz_clear (range); + mpz_clear (increment); + mpz_clear (upper_bound); + mpz_clear (lower_bound); + mpz_clear (R); + mpz_clear (tmp[1]); + mpz_clear (tmp[0]); + mpz_clear (p[1]); + mpz_clear (p[0]); + mpz_clear (y[1]); + mpz_clear (y[0]); + /* mpzToLongObj uses Python API so we must acquire the GIL */ + Py_END_ALLOW_THREADS; + if (error) + prime = NULL; + else + prime = mpzToLongObj (X); + mpz_clear (X); + return prime; +} + + + static PyMethodDef _fastmath__methods__[] = { {"dsa_construct", dsaKey_new, METH_VARARGS}, {"rsa_construct", rsaKey_new, METH_VARARGS}, - {"isPrime", isPrime, METH_VARARGS}, + {"isPrime", (PyCFunction)isPrime, METH_VARARGS | METH_KEYWORDS}, + {"getStrongPrime", (PyCFunction)getStrongPrime, METH_VARARGS | METH_KEYWORDS}, {NULL, NULL} }; void init_fastmath (void) { - PyObject *_fastmath_module; - PyObject *_fastmath_dict; + PyObject *_fastmath_module; + PyObject *_fastmath_dict; rsaKeyType.ob_type = &PyType_Type; dsaKeyType.ob_type = &PyType_Type; @@ -821,3 +1387,1010 @@ init_fastmath (void) fastmathError = PyErr_NewException ("_fastmath.error", NULL, NULL); PyDict_SetItemString (_fastmath_dict, "error", fastmathError); } + + + +/* The first 10000 primes to be used as a base for sieving */ +static unsigned int sieve_base[10000] = { + 2, 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, 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