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Pollard's rho algorithm
Pollard's rho algorithm is a special-purpose integer factorization algorithm. It was invented by John Pollard in 1975. It is particularly effective for a composite number having a small prime factor.
==Core ideas==
The ρ algorithm is based on Floyd's cycle-finding algorithm and on the observation that (as in the birthday problem) t random numbers ''x1'', ''x2, ... , ''xt in the range (n ) will contain a repetition with probability P > 0.5 if t > 1.177n1/2. The constant 1.177 comes from the more general result that if P is the probability that t random numbers in the range (n ) contain a repetition, then P > 1 - exp. Thus P > 0.5 provided 1/2 > exp, or t2 > 2nln 2, or t2 > 2n ln 2, or t > (2ln 2)1/2n1/2 = 1.177n1/2.
The ρ algorithm uses g(x), a polynomial modulo ''n'', as a generator of a pseudo-random sequence. (The most commonly used function is g(x) = x2 mod n.) Let's assume n = pq. The algorithm generates the sequence x1 = g(2), x2 = g(g(2)), x3 = g(g(g(2))), and so on. Two different sequences will in effect be running at the same time—the sequence and the sequence . Since p < n1/2, the latter sequence is likely to repeat earlier than the former sequence. The repetition of the mod p sequence will be detected by the fact that gcd(xk mod p - xm mod p, n) = p, where k < m. Once a repetition occurs, the sequence will cycle, because each term depends only on the previous one. The name ρ algorithm derives from the similarity in appearance between the Greek letter ρ and the directed graph formed by the values in the sequence and their successors. Once it is cycling, Floyd's cycle-finding algorithm will eventually detect a repetition. The algorithm succeeds whenever the sequence repeats before the sequence . The randomizing function g(x) must be a polynomial modulo n, so that it will work both modulo p and modulo n. That is, so that g(x mod p) ≡ g(x) (mod p).
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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