A Birthday Paradox for Markov Chains, with an Optimal Bound for Collision in the Pollard Rho Algorithm for Discrete Logarithm

@inproceedings{Kim2008ABP,
  title={A Birthday Paradox for Markov Chains, with an Optimal Bound for Collision in the Pollard Rho Algorithm for Discrete Logarithm},
  author={Jeong Han Kim and Ravi Montenegro and Yuval Peres and Prasad Tetali},
  booktitle={ANTS},
  year={2008}
}
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 ( √|G|) steps. Moreover, for the parallelized distinguished points algorithm on J processors we find that ( √|G|/J ) steps suffices. These are the first proofs… CONTINUE READING
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