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Efficiently calculate secure n-bit prime candidates for cryptographical uses in Python
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#!/usr/bin/env python3 | |
# -*- coding: utf-8 -*- | |
"""Calculate prime candidates for cryptography uses | |
License: This work is marked with CC0 1.0 Universal | |
<https://creativecommons.org/publicdomain/zero/1.0/>""" | |
import math | |
import secrets | |
import sys | |
from typing import Set # For older version compat | |
from warnings import filterwarnings as filter_warnings | |
primes: Set[int] = { | |
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, | |
} | |
def test_prime(n: int, k: int = 23) -> bool: | |
"""Test if a number is prime | |
Uses Fermat's Little Theorem & Miller-Rabin primality test | |
Also does simple checks such as if n<2 or n is a perfect square | |
then `n` is not prime""" | |
if n < 2 or math.isqrt(n)**2 == n: | |
return False | |
for p in primes: | |
if n < p * p: | |
return True | |
if n % p == 0: | |
return False | |
r, s = 0, n - 1 | |
while s % 2 == 0: | |
r += 1 | |
s //= 2 | |
for _ in range(k): | |
a: int = secrets.randbelow(n - 1) + 1 # [0; n) | |
x: int = pow(a, s, n) | |
if x == 1 or x == n - 1: | |
continue | |
for _ in range(r - 1): | |
x = pow(x, 2, n) | |
if x == n - 1: | |
break | |
else: | |
return False | |
return True | |
def generate_large_prime(n: int) -> int: | |
"""Generate a large n-bit prime number""" | |
found_prime: bool = False | |
p: int = 0 | |
while not found_prime: | |
p = secrets.randbits(n) # Generates a random n-bit number | |
p |= ( | |
1 << n - 1 | |
) | 1 # Ensures the n-bit length and assures it is an odd number | |
found_prime = test_prime(p) and test_prime((2 * p) + 1) and test_prime((p - 1) // 2) # Generate safe sophie germain primes | |
return p | |
def main() -> int: | |
"""Entry/main function""" | |
if len(sys.argv) != 2: | |
print( | |
"Please supply n bits to generate a prime candidate for", | |
file=sys.stderr, | |
) | |
return 1 | |
print(generate_large_prime(int(sys.argv[1]))) | |
return 0 | |
if __name__ == "__main__": | |
assert main.__annotations__.get("return") is int, "main() should return an integer" | |
filter_warnings("error", category=Warning) | |
raise SystemExit(main()) | |
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#!/usr/bin/env python3 | |
# -*- coding: utf-8 -*- | |
"""Calculate prime candidates for cryptography uses | |
License: This work is marked with CC0 1.0 Universal | |
<https://creativecommons.org/publicdomain/zero/1.0/>""" | |
import secrets | |
import sys | |
from warnings import filterwarnings as filter_warnings | |
import sympy | |
def generate_large_prime(n: int) -> int: | |
"""Generate a large n-bit prime number""" | |
found_prime: bool = False | |
p: int = 0 | |
while not found_prime: | |
p = secrets.randbits(n) # Generates a random n-bit number | |
p |= ( | |
1 << n - 1 | |
) | 1 # Ensures the n-bit length and assures it is an odd number | |
found_prime = ( | |
sympy.isprime(p) | |
and sympy.isprime((2 * p) + 1) | |
and sympy.isprime((p - 1) // 2) | |
) # Generate safe primes | |
return p | |
def main() -> int: | |
"""Entry/main function""" | |
if len(sys.argv) != 2: | |
print( | |
"Please supply n bits to generate a prime candidate for", | |
file=sys.stderr, | |
) | |
return 1 | |
print(generate_large_prime(int(sys.argv[1]))) | |
return 0 | |
if __name__ == "__main__": | |
assert main.__annotations__.get("return") is int, "main() should return an integer" | |
filter_warnings("error", category=Warning) | |
raise SystemExit(main()) |
Feel free to remove and test_prime((2 * p) + 1)
and whatnot if you just want primes, not safe primes
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Note that you can change the
primes
set to be smaller :)