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# This file is part of pyphe.
#
# pyphe is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# pyphe is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with pyphe. If not, see <http://www.gnu.org/licenses/>.
import os
import random
from base64 import urlsafe_b64encode, urlsafe_b64decode
from binascii import hexlify, unhexlify
try:
import gmpy2
HAVE_GMP = True
except ImportError:
HAVE_GMP = False
try:
from Crypto.Util import number
HAVE_CRYPTO = True
except ImportError:
HAVE_CRYPTO = False
# GMP's powmod has greater overhead than Python's pow, but is faster.
# From a quick experiment on our machine, this seems to be the break even:
_USE_MOD_FROM_GMP_SIZE = (1 << (8*2))
def powmod(a, b, c):
"""
Uses GMP, if available, to do a^b mod c where a, b, c
are integers.
:return int: (a ** b) % c
"""
if a == 1:
return 1
if not HAVE_GMP or max(a, b, c) < _USE_MOD_FROM_GMP_SIZE:
return pow(a, b, c)
else:
return int(gmpy2.powmod(a, b, c))
def extended_euclidean_algorithm(a, b):
"""Extended Euclidean algorithm
Returns r, s, t such that r = s*a + t*b and r is gcd(a, b)
See <https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm>
"""
r0, r1 = a, b
s0, s1 = 1, 0
t0, t1 = 0, 1
while r1 != 0:
q = r0 // r1
r0, r1 = r1, r0 - q*r1
s0, s1 = s1, s0 - q*s1
t0, t1 = t1, t0 - q*t1
return r0, s0, t0
def invert(a, b):
"""
The multiplicitive inverse of a in the integers modulo b.
:return int: x, where a * x == 1 mod b
"""
if HAVE_GMP:
s = int(gmpy2.invert(a, b))
# according to documentation, gmpy2.invert might return 0 on
# non-invertible element, although it seems to actually raise an
# exception; for consistency, we always raise the exception
if s == 0:
raise ZeroDivisionError('invert() no inverse exists')
return s
else:
r, s, _ = extended_euclidean_algorithm(a, b)
if r != 1:
raise ZeroDivisionError('invert() no inverse exists')
return s % b
def getprimeover(N):
"""Return a random N-bit prime number using the System's best
Cryptographic random source.
Use GMP if available, otherwise fallback to PyCrypto
"""
if HAVE_GMP:
randfunc = random.SystemRandom()
r = gmpy2.mpz(randfunc.getrandbits(N))
r = gmpy2.bit_set(r, N - 1)
return int(gmpy2.next_prime(r))
elif HAVE_CRYPTO:
return number.getPrime(N, os.urandom)
else:
randfunc = random.SystemRandom()
n = randfunc.randrange(2**(N-1), 2**N) | 1
while not is_prime(n):
n += 2
return n
def isqrt(N):
""" returns the integer square root of N """
if HAVE_GMP:
return int(gmpy2.isqrt(N))
else:
return improved_i_sqrt(N)
def improved_i_sqrt(n):
""" taken from
http://stackoverflow.com/questions/15390807/integer-square-root-in-python
Thanks, mathmandan """
assert n >= 0
if n == 0:
return 0
i = n.bit_length() >> 1 # i = floor( (1 + floor(log_2(n))) / 2 )
m = 1 << i # m = 2^i
#
# Fact: (2^(i + 1))^2 > n, so m has at least as many bits
# as the floor of the square root of n.
#
# Proof: (2^(i+1))^2 = 2^(2i + 2) >= 2^(floor(log_2(n)) + 2)
# >= 2^(ceil(log_2(n) + 1) >= 2^(log_2(n) + 1) > 2^(log_2(n)) = n. QED.
#
while (m << i) > n: # (m<<i) = m*(2^i) = m*m
m >>= 1
i -= 1
d = n - (m << i) # d = n-m^2
for k in range(i-1, -1, -1):
j = 1 << k
new_diff = d - (((m<<1) | j) << k) # n-(m+2^k)^2 = n-m^2-2*m*2^k-2^(2k)
if new_diff >= 0:
d = new_diff
m |= j
return m
# base64 utils from jwcrypto
def base64url_encode(payload):
if not isinstance(payload, bytes):
payload = payload.encode('utf-8')
encode = urlsafe_b64encode(payload)
return encode.decode('utf-8').rstrip('=')
def base64url_decode(payload):
l = len(payload) % 4
if l == 2:
payload += '=='
elif l == 3:
payload += '='
elif l != 0:
raise ValueError('Invalid base64 string')
return urlsafe_b64decode(payload.encode('utf-8'))
def base64_to_int(source):
return int(hexlify(base64url_decode(source)), 16)
def int_to_base64(source):
assert source != 0
I = hex(source).rstrip("L").lstrip("0x")
return base64url_encode(unhexlify((len(I) % 2) * '0' + I))
# prime testing
first_primes = [
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, 10151, 10159,
10163, 10169, 10177, 10181, 10193, 10211, 10223, 10243, 10247, 10253,
10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321, 10331,
10333, 10337, 10343, 10357, 10369, 10391, 10399, 10427, 10429, 10433,
10453, 10457, 10459, 10463, 10477, 10487, 10499, 10501, 10513, 10529,
10531, 10559, 10567, 10589, 10597, 10601, 10607, 10613, 10627, 10631,
10639, 10651, 10657, 10663, 10667, 10687, 10691, 10709, 10711, 10723,
10729, 10733, 10739, 10753, 10771, 10781, 10789, 10799, 10831, 10837,
10847, 10853, 10859, 10861, 10867, 10883, 10889, 10891, 10903, 10909,
10937, 10939, 10949, 10957, 10973, 10979, 10987, 10993, 11003, 11027,
11047, 11057, 11059, 11069, 11071, 11083, 11087, 11093, 11113, 11117,
11119, 11131, 11149, 11159, 11161, 11171, 11173, 11177, 11197, 11213,
11239, 11243, 11251, 11257, 11261, 11273, 11279, 11287, 11299, 11311,
11317, 11321, 11329, 11351, 11353, 11369, 11383, 11393, 11399, 11411,
11423, 11437, 11443, 11447, 11467, 11471, 11483, 11489, 11491, 11497,
11503, 11519, 11527, 11549, 11551, 11579, 11587, 11593, 11597, 11617,
11621, 11633, 11657, 11677, 11681, 11689, 11699, 11701, 11717, 11719,
11731, 11743, 11777, 11779, 11783, 11789, 11801, 11807, 11813, 11821,
11827, 11831, 11833, 11839, 11863, 11867, 11887, 11897, 11903, 11909,
11923, 11927, 11933, 11939, 11941, 11953, 11959, 11969, 11971, 11981,
11987, 12007, 12011, 12037, 12041, 12043, 12049, 12071, 12073, 12097,
12101, 12107, 12109, 12113, 12119, 12143, 12149, 12157, 12161, 12163,
12197, 12203, 12211, 12227, 12239, 12241, 12251, 12253, 12263, 12269,
12277, 12281, 12289, 12301, 12323, 12329, 12343, 12347, 12373, 12377,
12379, 12391, 12401, 12409, 12413, 12421, 12433, 12437, 12451, 12457,
12473, 12479, 12487, 12491, 12497, 12503, 12511, 12517, 12527, 12539,
12541, 12547, 12553, 12569, 12577, 12583, 12589, 12601, 12611, 12613,
12619, 12637, 12641, 12647, 12653, 12659, 12671, 12689, 12697, 12703,
12713, 12721, 12739, 12743, 12757, 12763, 12781, 12791, 12799, 12809,
12821, 12823, 12829, 12841, 12853, 12889, 12893, 12899, 12907, 12911,
12917, 12919, 12923, 12941, 12953, 12959, 12967, 12973, 12979, 12983,
13001, 13003, 13007, 13009, 13033, 13037, 13043, 13049, 13063, 13093,
13099, 13103, 13109, 13121, 13127, 13147, 13151, 13159, 13163, 13171,
13177, 13183, 13187, 13217, 13219, 13229, 13241, 13249, 13259, 13267,
13291, 13297, 13309, 13313, 13327, 13331, 13337, 13339, 13367, 13381,
13397, 13399, 13411, 13417, 13421, 13441, 13451, 13457, 13463, 13469,
13477, 13487, 13499, 13513, 13523, 13537, 13553, 13567, 13577, 13591,
13597, 13613, 13619, 13627, 13633, 13649, 13669, 13679, 13681, 13687,
13691, 13693, 13697, 13709, 13711, 13721, 13723, 13729, 13751, 13757,
13759, 13763, 13781, 13789, 13799, 13807, 13829, 13831, 13841, 13859,
13873, 13877, 13879, 13883, 13901, 13903, 13907, 13913, 13921, 13931,
13933, 13963, 13967, 13997, 13999, 14009, 14011, 14029, 14033, 14051,
14057, 14071, 14081, 14083, 14087, 14107, 14143, 14149, 14153, 14159,
14173, 14177, 14197, 14207, 14221, 14243, 14249, 14251, 14281, 14293,
14303, 14321, 14323, 14327, 14341, 14347, 14369, 14387, 14389, 14401,
14407, 14411, 14419, 14423, 14431, 14437, 14447, 14449, 14461, 14479,
14489, 14503, 14519, 14533, 14537, 14543, 14549, 14551, 14557, 14561,
14563, 14591, 14593, 14621, 14627, 14629, 14633, 14639, 14653, 14657,
14669, 14683, 14699, 14713, 14717, 14723, 14731, 14737, 14741, 14747,
14753, 14759, 14767, 14771, 14779, 14783, 14797, 14813, 14821, 14827,
14831, 14843, 14851, 14867, 14869, 14879, 14887, 14891, 14897, 14923,
14929, 14939, 14947, 14951, 14957, 14969, 14983, 15013, 15017, 15031,
15053, 15061, 15073, 15077, 15083, 15091, 15101, 15107, 15121, 15131,
15137, 15139, 15149, 15161, 15173, 15187, 15193, 15199, 15217, 15227,
15233, 15241, 15259, 15263, 15269, 15271, 15277, 15287, 15289, 15299,
15307, 15313, 15319, 15329, 15331, 15349, 15359, 15361, 15373, 15377,
15383, 15391, 15401, 15413, 15427, 15439, 15443, 15451, 15461, 15467,
15473, 15493, 15497, 15511, 15527, 15541, 15551, 15559, 15569, 15581,
15583, 15601, 15607, 15619, 15629, 15641, 15643, 15647, 15649, 15661,
15667, 15671, 15679, 15683, 15727, 15731, 15733, 15737, 15739, 15749,
15761, 15767, 15773, 15787, 15791, 15797, 15803, 15809, 15817, 15823,
15859, 15877, 15881, 15887, 15889, 15901, 15907, 15913, 15919, 15923,
15937, 15959, 15971, 15973, 15991, 16001, 16007, 16033, 16057, 16061,
16063, 16067, 16069, 16073, 16087, 16091, 16097, 16103, 16111, 16127,
16139, 16141, 16183, 16187, 16189, 16193, 16217, 16223, 16229, 16231,
16249, 16253, 16267, 16273, 16301, 16319, 16333, 16339, 16349, 16361,
16363, 16369, 16381, 16411, 16417, 16421, 16427, 16433, 16447, 16451,
16453, 16477, 16481, 16487, 16493, 16519, 16529, 16547, 16553, 16561,
16567, 16573, 16603, 16607, 16619, 16631, 16633, 16649, 16651, 16657,
16661, 16673, 16691, 16693, 16699, 16703, 16729, 16741, 16747, 16759,
16763, 16787, 16811, 16823, 16829, 16831, 16843, 16871, 16879, 16883,
16889, 16901, 16903, 16921, 16927, 16931, 16937, 16943, 16963, 16979,
16981, 16987, 16993, 17011, 17021, 17027, 17029, 17033, 17041, 17047,
17053, 17077, 17093, 17099, 17107, 17117, 17123, 17137, 17159, 17167,
17183, 17189, 17191, 17203, 17207, 17209, 17231, 17239, 17257, 17291,
17293, 17299, 17317, 17321, 17327, 17333, 17341, 17351, 17359, 17377,
17383, 17387, 17389, 17393, 17401, 17417, 17419, 17431, 17443, 17449,
17467, 17471, 17477, 17483, 17489, 17491, 17497, 17509, 17519, 17539,
17551, 17569, 17573, 17579, 17581, 17597, 17599, 17609, 17623, 17627,
17657, 17659, 17669, 17681, 17683, 17707, 17713, 17729, 17737, 17747,
17749, 17761, 17783, 17789, 17791, 17807, 17827, 17837, 17839, 17851,
17863,
]
def miller_rabin(n, k):
"""Run the Miller-Rabin test on n with at most k iterations
Arguments:
n (int): number whose primality is to be tested
k (int): maximum number of iterations to run
Returns:
bool: If n is prime, then True is returned. Otherwise, False is
returned, except with probability less than 4**-k.
See <https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test>
"""
assert n > 3
# find r and d such that n-1 = 2^r × d
d = n-1
r = 0
while d % 2 == 0:
d //= 2
r += 1
assert n-1 == d * 2**r
assert d % 2 == 1
for _ in range(k): # each iteration divides risk of false prime by 4
a = random.randint(2, n-2) # choose a random witness
x = pow(a, d, n)
if x == 1 or x == n-1:
continue # go to next witness
for _ in range(1, r):
x = x*x % n
if x == n-1:
break # go to next witness
else:
return False
return True
def is_prime(n, mr_rounds=25):
"""Test whether n is probably prime
See <https://en.wikipedia.org/wiki/Primality_test#Probabilistic_tests>
Arguments:
n (int): the number to be tested
mr_rounds (int, optional): number of Miller-Rabin iterations to run;
defaults to 25 iterations, which is what the GMP library uses
Returns:
bool: when this function returns False, `n` is composite (not prime);
when it returns True, `n` is prime with overwhelming probability
"""
# as an optimization we quickly detect small primes using the list above
if n <= first_primes[-1]:
return n in first_primes
# for small dividors (relatively frequent), euclidean division is best
for p in first_primes:
if n % p == 0:
return False
# the actual generic test; give a false prime with probability 2⁻⁵⁰
return miller_rabin(n, mr_rounds)