Abstract
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 $\Theta(\sqrt{|G|})$ steps. Moreover, for the parallelized distinguished points algorithm on J processors we find that $\Theta(\sqrt{|G|}/J)$ 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.
Citation
Jeong Han Kim. Ravi Montenegro. Yuval Peres. Prasad Tetali. "A Birthday Paradox for Markov chains with an optimal bound for collision in the Pollard Rho algorithm for discrete logarithm." Ann. Appl. Probab. 20 (2) 495 - 521, April 2010. https://doi.org/10.1214/09-AAP625
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