# # number.py : Number-theoretic functions # # Part of the Python Cryptography Toolkit # # Written by Andrew M. Kuchling, Barry A. Warsaw, and others # # =================================================================== # The contents of this file are dedicated to the public domain. To # the extent that dedication to the public domain is not available, # everyone is granted a worldwide, perpetual, royalty-free, # non-exclusive license to exercise all rights associated with the # contents of this file for any purpose whatsoever. # No rights are reserved. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, # EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF # MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND # NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS # BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN # ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN # CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE # SOFTWARE. # =================================================================== # __revision__ = "$Id$" from Crypto.pct_warnings import GetRandomNumber_DeprecationWarning, PowmInsecureWarning from warnings import warn as _warn import math import sys from Crypto.Util.py3compat import * bignum = int try: from Crypto.PublicKey import _fastmath except ImportError: # For production, we are going to let import issues due to gmp/mpir shared # libraries not loading slide silently and use slowmath. If you'd rather # see an exception raised if _fastmath exists but cannot be imported, # uncomment the below # # from distutils.sysconfig import get_config_var # import inspect, os # _fm_path = os.path.normpath(os.path.dirname(os.path.abspath( # inspect.getfile(inspect.currentframe()))) # +"/../../PublicKey/_fastmath"+get_config_var("SO")) # if os.path.exists(_fm_path): # raise ImportError("While the _fastmath module exists, importing "+ # "it failed. This may point to the gmp or mpir shared library "+ # "not being in the path. _fastmath was found at "+_fm_path) _fastmath = None # You need libgmp v5 or later to get mpz_powm_sec. Warn if it's not available. if _fastmath is not None and not _fastmath.HAVE_DECL_MPZ_POWM_SEC: _warn("Not using mpz_powm_sec. You should rebuild using libgmp >= 5 to avoid timing attack vulnerability.", PowmInsecureWarning) # New functions from ._number_new import * # Commented out and replaced with faster versions below ## def long2str(n): ## s='' ## while n>0: ## s=chr(n & 255)+s ## n=n>>8 ## return s ## import types ## def str2long(s): ## if type(s)!=types.StringType: return s # Integers will be left alone ## return reduce(lambda x,y : x*256+ord(y), s, 0L) def size (N): """size(N:long) : int Returns the size of the number N in bits. """ bits = 0 while N >> bits: bits += 1 return bits def getRandomNumber(N, randfunc=None): """Deprecated. Use getRandomInteger or getRandomNBitInteger instead.""" warnings.warn("Crypto.Util.number.getRandomNumber has confusing semantics"+ "and has been deprecated. Use getRandomInteger or getRandomNBitInteger instead.", GetRandomNumber_DeprecationWarning) return getRandomNBitInteger(N, randfunc) def getRandomInteger(N, randfunc=None): """getRandomInteger(N:int, randfunc:callable):long Return a random number with at most N bits. 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. """ if randfunc is None: _import_Random() randfunc = Random.new().read S = randfunc(N>>3) odd_bits = N % 8 if odd_bits != 0: char = ord(randfunc(1)) >> (8-odd_bits) S = bchr(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 = getRandomInteger(bits, randfunc) while value > range_: value = getRandomInteger(bits, randfunc) return a + value def getRandomNBitInteger(N, randfunc=None): """getRandomInteger(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 = getRandomInteger (N-1, randfunc) value |= 2 ** (N-1) # Ensure high bit is set assert size(value) >= N return value def GCD(x,y): """GCD(x:long, y:long): long Return the GCD of x and y. """ x = abs(x) ; y = abs(y) while x > 0: x, y = y % x, x return y def inverse(u, v): """inverse(u:long, v:long):long Return the inverse of u mod v. """ u3, v3 = int(u), int(v) u1, v1 = 1, 0 while v3 > 0: q=divmod(u3, v3)[0] u1, v1 = v1, u1 - v1*q u3, v3 = v3, u3 - v3*q while u1<0: u1 = u1 + v return u1 # Given a number of bits to generate and a random generation function, # find a prime number of the appropriate size. def getPrime(N, randfunc=None): """getPrime(N:int, randfunc:callable):long Return a random N-bit prime number. If randfunc is omitted, then Random.new().read is used. """ if randfunc is None: _import_Random() randfunc = Random.new().read number=getRandomNBitInteger(N, randfunc) | 1 while (not isPrime(number, randfunc=randfunc)): number=number+2 return number 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. 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 >>= 1 tested = [] # we need to do at most n-2 rounds. for i in range (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 range (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-positive 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.getStrongPrime(int(N), int(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)/math.log(4))) # calculate range for X # lower_bound = sqrt(2) * 2^{511 + 128*x} # upper_bound = 2^{512 + 128*x} - 1 x = (N - 512) >> 7; # 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 = divmod(14142135623730950489 * (2 ** (511 + 128*x)), 10000000000000000000)[0] 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 = getRandomNBitInteger (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 range ((prime - offset) % prime, len (field), prime): field[j] = 1 # look for suitable p[i] starting at y result = 0 for j in range(len(field)): composite = field[j] # 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 candidate 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 useful 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: if GCD (e, divmod((X-1),2)[0]) != 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 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-positive 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(int(N), false_positive_prob, randfunc) 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)/math.log(4))) return _rabinMillerTest(N, rounds, randfunc) # Improved conversion functions contributed by Barry Warsaw, after # careful benchmarking import struct def long_to_bytes(n, blocksize=0): """long_to_bytes(n:long, blocksize:int) : string Convert a long integer to a byte string. If optional blocksize is given and greater than zero, pad the front of the byte string with binary zeros so that the length is a multiple of blocksize. """ # after much testing, this algorithm was deemed to be the fastest s = b('') n = int(n) pack = struct.pack while n > 0: s = pack('>I', n & 0xffffffff) + s n = n >> 32 # strip off leading zeros for i in range(len(s)): if s[i] != b('\000')[0]: break else: # only happens when n == 0 s = b('\000') i = 0 s = s[i:] # add back some pad bytes. this could be done more efficiently w.r.t. the # de-padding being done above, but sigh... if blocksize > 0 and len(s) % blocksize: s = (blocksize - len(s) % blocksize) * b('\000') + s return s def bytes_to_long(s): """bytes_to_long(string) : long Convert a byte string to a long integer. This is (essentially) the inverse of long_to_bytes(). """ acc = 0 unpack = struct.unpack length = len(s) if length % 4: extra = (4 - length % 4) s = b('\000') * extra + s length = length + extra for i in range(0, length, 4): acc = (acc << 32) + unpack('>I', s[i:i+4])[0] return acc # For backwards compatibility... import warnings def long2str(n, blocksize=0): warnings.warn("long2str() has been replaced by long_to_bytes()") return long_to_bytes(n, blocksize) def str2long(s): warnings.warn("str2long() has been replaced by bytes_to_long()") return bytes_to_long(s) def _import_Random(): # This is called in a function instead of at the module level in order to # avoid problems with recursive imports global Random, StrongRandom 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, 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, 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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, )