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# 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
#
# https://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.
"""
import rsa.common
import rsa.randnum
__all__ = ['getprime', 'are_relatively_prime']
def gcd(p: int, q: int) -> int:
"""Returns the greatest common divisor of p and q
>>> gcd(48, 180)
12
"""
while q != 0:
(p, q) = (q, p % q)
return p
def get_primality_testing_rounds(number: int) -> int:
"""Returns minimum number of rounds for Miller-Rabing primality testing,
based on number bitsize.
According to NIST FIPS 186-4, Appendix C, Table C.3, minimum number of
rounds of M-R testing, using an error probability of 2 ** (-100), for
different p, q bitsizes are:
* p, q bitsize: 512; rounds: 7
* p, q bitsize: 1024; rounds: 4
* p, q bitsize: 1536; rounds: 3
See: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
"""
# Calculate number bitsize.
bitsize = rsa.common.bit_size(number)
# Set number of rounds.
if bitsize >= 1536:
return 3
if bitsize >= 1024:
return 4
if bitsize >= 512:
return 7
# For smaller bitsizes, set arbitrary number of rounds.
return 10
def miller_rabin_primality_testing(n: int, k: int) -> bool:
"""Calculates whether n is composite (which is always correct) or prime
(which theoretically is incorrect with error probability 4**-k), by
applying Miller-Rabin primality testing.
For reference and implementation example, see:
https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
:param n: Integer to be tested for primality.
:type n: int
:param k: Number of rounds (witnesses) of Miller-Rabin testing.
:type k: int
:return: False if the number is composite, True if it's probably prime.
:rtype: bool
"""
# prevent potential infinite loop when d = 0
if n < 2:
return False
# Decompose (n - 1) to write it as (2 ** r) * d
# While d is even, divide it by 2 and increase the exponent.
d = n - 1
r = 0
while not (d & 1):
r += 1
d >>= 1
# Test k witnesses.
for _ in range(k):
# Generate random integer a, where 2 <= a <= (n - 2)
a = rsa.randnum.randint(n - 3) + 1
x = pow(a, d, n)
if x == 1 or x == n - 1:
continue
for _ in range(r - 1):
x = pow(x, 2, n)
if x == 1:
# n is composite.
return False
if x == n - 1:
# Exit inner loop and continue with next witness.
break
else:
# If loop doesn't break, n is composite.
return False
return True
def is_prime(number: int) -> bool:
"""Returns True if the number is prime, and False otherwise.
>>> is_prime(2)
True
>>> is_prime(42)
False
>>> is_prime(41)
True
"""
# Check for small numbers.
if number < 10:
return number in {2, 3, 5, 7}
# Check for even numbers.
if not (number & 1):
return False
# Calculate minimum number of rounds.
k = get_primality_testing_rounds(number)
# Run primality testing with (minimum + 1) rounds.
return miller_rabin_primality_testing(number, k + 1)
def getprime(nbits: int) -> int:
"""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
"""
assert nbits > 3 # the loop wil hang on too small numbers
while True:
integer = rsa.randnum.read_random_odd_int(nbits)
# Test for primeness
if is_prime(integer):
return integer
# Retry if not prime
def are_relatively_prime(a: int, b: int) -> bool:
"""Returns True if a and b are relatively prime, and False if they
are not.
>>> are_relatively_prime(2, 3)
True
>>> are_relatively_prime(2, 4)
False
"""
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 % 100 == 0 and count:
print('%i times' % count)
print('Doctests done')
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