1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
|
'''Numerical functions related to primes.'''
__all__ = [ 'getprime', 'are_relatively_prime']
import rsa.randnum
def gcd(p, q):
"""Returns the greatest common divisor of p and q
>>> gcd(48, 180)
12
"""
while q != 0:
if p < q: (p,q) = (q,p)
(p,q) = (q, p % q)
return p
def jacobi(a, b):
"""Calculates the value of the Jacobi symbol (a/b) where both a and b are
positive integers, and b is odd
"""
if a == 0: return 0
result = 1
while a > 1:
if a & 1:
if ((a-1)*(b-1) >> 2) & 1:
result = -result
a, b = b % a, a
else:
if (((b * b) - 1) >> 3) & 1:
result = -result
a >>= 1
if a == 0: return 0
return result
def jacobi_witness(x, n):
"""Returns False if n is an Euler pseudo-prime with base x, and
True otherwise.
"""
j = jacobi(x, n) % n
f = pow(x, (n-1)/2, n)
if j == f: return False
return True
def randomized_primality_testing(n, k):
"""Calculates whether n is composite (which is always correct) or
prime (which is incorrect with error probability 2**-k)
Returns False if the number is composite, and True if it's
probably prime.
"""
# 50% of Jacobi-witnesses can report compositness of non-prime numbers
for i in range(k):
x = rsa.randnum.randint(1, n-1)
if jacobi_witness(x, n): return False
return True
def is_prime(number):
"""Returns True if the number is prime, and False otherwise.
>>> is_prime(42)
0
>>> is_prime(41)
1
"""
if randomized_primality_testing(number, 6):
# Prime, according to Jacobi
return True
# Not prime
return False
def getprime(nbits):
"""Returns a prime number of max. 'math.ceil(nbits/8)*8' bits. In
other words: nbits is rounded up to whole bytes.
>>> p = getprime(8)
>>> is_prime(p-1)
0
>>> is_prime(p)
1
>>> is_prime(p+1)
0
"""
while True:
integer = rsa.randnum.read_random_int(nbits)
# Make sure it's odd
integer |= 1
# Test for primeness
if is_prime(integer): break
# Retry if not prime
return integer
def are_relatively_prime(a, b):
"""Returns True if a and b are relatively prime, and False if they
are not.
>>> are_relatively_prime(2, 3)
1
>>> are_relatively_prime(2, 4)
0
"""
d = gcd(a, b)
return (d == 1)
|