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
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
|
# -*- coding: utf-8 -*-
#
# Copyright 2011 Sybren A. Stüvel <sybren@stuvel.eu>
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
'''Numerical functions related to primes.
Implementation based on the book Algorithm Design by Michael T. Goodrich and
Roberto Tamassia, 2002.
'''
__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
:returns: -1, 0 or 1
'''
assert a > 0
assert b > 0
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, 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
# The implemented algorithm using the Jacobi witness function has error
# probability q <= 0.5, according to Goodrich et. al
#
# q = 0.5
# t = int(math.ceil(k / log(1 / q, 2)))
# So t = k / log(2, 2) = k / 1 = k
# this means we can use range(k) rather than range(t)
for _ in range(k):
x = rsa.randnum.randint(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)
False
>>> is_prime(41)
True
'''
return randomized_primality_testing(number, 6)
def getprime(nbits):
'''Returns a prime number that can be stored in 'nbits' bits.
>>> p = getprime(128)
>>> is_prime(p-1)
False
>>> is_prime(p)
True
>>> is_prime(p+1)
False
>>> from rsa import common
>>> common.bit_size(p) == 128
True
'''
while True:
integer = rsa.randnum.read_random_int(nbits)
# Make sure it's odd
integer |= 1
# Test for primeness
if is_prime(integer):
return integer
# Retry if not prime
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)
if __name__ == '__main__':
print('Running doctests 1000x or until failure')
import doctest
for count in range(1000):
(failures, tests) = doctest.testmod()
if failures:
break
if count and count % 100 == 0:
print('%i times' % count)
print('Doctests done')
|
