We show a Birthday Paradox for self-intersections of Markov chains with uniform stationary distribution. As an application, we analyze Pollard’s Rho algorithm for finding the discrete logarithm in a cyclic group \\(G\\) and find that if the partition in the algorithm is given by a random oracle, then with high probability a collision occurs in \\(\\Theta (\\sqrt{|G|—\u221a})\\) steps. Moreover, for the parallelized distinguished points algorithm on \\(J\\) processors we find that \\(\\Theta (\\sqrt{|G|—\u221a}\/J)\\) steps suffices. These are the first proofs of the correct order bounds which do not assume that every step of the algorithm produces an i.i.d. sample from \\(G\\).<\/p>\n","protected":false},"excerpt":{"rendered":"
We show a Birthday Paradox for self-intersections of Markov chains with uniform stationary distribution. As an application, we analyze Pollard’s Rho algorithm for finding the discrete logarithm in a cyclic group and find that if the partition in the algorithm is given by a random oracle, then with high probability a collision occurs in steps. 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