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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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