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- Implemented a classical simulation of Shor’s Algorithm in `algorithms/cryptography/shor_algorithm.py` - The algorithm factors a composite number N without using quantum computing libraries - Uses modular exponentiation and order-finding to determine factors - Includes an example usage for testing
57 lines
1.6 KiB
Python
57 lines
1.6 KiB
Python
import math
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import random
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def gcd(a, b):
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"""Computes the greatest common divisor using Euclidean algorithm."""
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while b:
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a, b = b, a % b
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return a
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def modular_exponentiation(base, exp, mod):
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"""Computes (base^exp) % mod using fast modular exponentiation."""
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result = 1
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while exp > 0:
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if exp % 2 == 1:
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result = (result * base) % mod
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base = (base * base) % mod
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exp //= 2
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return result
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def find_order(a, N):
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"""Finds the smallest r such that a^r ≡ 1 (mod N)"""
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r = 1
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while modular_exponentiation(a, r, N) != 1:
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r += 1
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if r > N: # Prevent infinite loops
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return None
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return r
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def shor_algorithm(N):
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"""Simulates Shor’s Algorithm classically to factorize N."""
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if N % 2 == 0:
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return 2, N // 2 # Trivial case if N is even
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while True:
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a = random.randint(2, N - 1)
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factor = gcd(a, N)
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if factor > 1:
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return factor, N // factor # Lucky case: a and N are not coprime
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r = find_order(a, N)
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if r is None or r % 2 == 1:
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continue # Retry if order is not even
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factor1 = gcd(modular_exponentiation(a, r // 2, N) - 1, N)
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factor2 = gcd(modular_exponentiation(a, r // 2, N) + 1, N)
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if 1 < factor1 < N:
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return factor1, N // factor1
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if 1 < factor2 < N:
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return factor2, N // factor2
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# Example usage
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if __name__ == "__main__":
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N = 15 # You can test with 21, 35, 55, etc.
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factors = shor_algorithm(N)
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print(f"Factors of {N}: {factors}")
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