Birthday paradox
After O(√p) steps of random walk mod p, collision expected. gcd of collision difference with n gives p.
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Floyd's cycle detection
Tortoise + hare: x_i and x_{2i}. When they collide mod p → gcd yields p.
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Brent's improvement
Fewer gcd computations. About 25% faster in practice.
Complexity
Expected O(N^(1/4) · polylog). Finds smallest prime factor first.