Pollard Rho Analysis:

By the birthday-paradox, the sequence then must repeat after steps in expectation. As will be shown below, a similar sequence of for a prime factor of n will also repeat in steps or steps in expectation because the greatest value of the smallest prime factor of n is less than .

Let p be a nontrivial factor of n, then the sequence induces a corresponding sequence modulo p where

Thus, although the sequence is not being computed explicitly, it is well defined and obeys the same recurrence as the sequence . By similar reasoning as for the original sequence, the sequence repeats in . Consider the figure below for the illustration.