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107 lines
2.7 KiB
Python
107 lines
2.7 KiB
Python
"""
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This script implements the Solovay-Strassen Primality test.
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This probabilistic primality test is based on Euler's criterion. It is similar
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to the Fermat test but uses quadratic residues. It can quickly identify
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composite numbers but may occasionally classify composite numbers as prime.
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More details and concepts about this can be found on:
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https://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test
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"""
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import random
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def jacobi_symbol(random_a: int, number: int) -> int:
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"""
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Calculate the Jacobi symbol. The Jacobi symbol is a generalization
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of the Legendre symbol, which can be used to simplify computations involving
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quadratic residues. The Jacobi symbol is used in primality tests, like the
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Solovay-Strassen test, because it helps determine if an integer is a
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quadratic residue modulo a given modulus, providing valuable information
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about the number's potential primality or compositeness.
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Parameters:
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random_a: A randomly chosen integer from 2 to n-2 (inclusive)
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number: The number that is tested for primality
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Returns:
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jacobi_symbol: The Jacobi symbol is a mathematical function
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used to determine whether an integer is a quadratic residue modulo
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another integer (usually prime) or not.
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>>> jacobi_symbol(2, 13)
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-1
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>>> jacobi_symbol(5, 19)
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1
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>>> jacobi_symbol(7, 14)
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0
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"""
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if random_a in (0, 1):
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return random_a
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random_a %= number
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t = 1
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while random_a != 0:
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while random_a % 2 == 0:
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random_a //= 2
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r = number % 8
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if r in (3, 5):
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t = -t
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random_a, number = number, random_a
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if random_a % 4 == number % 4 == 3:
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t = -t
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random_a %= number
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return t if number == 1 else 0
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def solovay_strassen(number: int, iterations: int) -> bool:
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"""
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Check whether the input number is prime or not using
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the Solovay-Strassen Primality test
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Parameters:
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number: The number that is tested for primality
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iterations: The number of times that the test is run
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which effects the accuracy
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Returns:
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result: True if number is probably prime and false
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if not
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>>> random.seed(10)
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>>> solovay_strassen(13, 5)
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True
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>>> solovay_strassen(9, 10)
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False
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>>> solovay_strassen(17, 15)
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True
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"""
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if number <= 1:
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return False
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if number <= 3:
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return True
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for _ in range(iterations):
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a = random.randint(2, number - 2)
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x = jacobi_symbol(a, number)
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y = pow(a, (number - 1) // 2, number)
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if x == 0 or y != x % number:
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return False
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return True
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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