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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.The 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 [1, 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 [1, n] contain a repetition, then P > 1 - exp{ - t2/2n }. Thus P > 0.5 provided 1/2 > exp{ - t2/2n }, or t2 > 2nln 2, or t2 > 2n ln 2, or t > (2ln 2)1/2n1/2 = 1.177n1/2.The 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 timethe sequence {xk} and the sequence {xk mod p}. 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 algorithm derives from the similarity in appearance between the Greek letter 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 {xk mod p} repeats before the sequence {xk}. 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). https://en.wikipedia.org/wiki/Pollard%27s_rho_al...

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