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A Birthday Paradox for Markov chains, with an optimal bound for collision in the Pollard Rho Algorithm for Discrete Logarithm
"... We show a Birthday Paradox for selfintersections 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 wit ..."
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We show a Birthday Paradox for selfintersections 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 Θ ( � G) steps. This is the first proof of the correct order bound which does not assume that every step of the algorithm produces an i.i.d. sample from G.
Collision Finding with Many Classical or Quantum Processors
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis, we investigate the cost of finding col ..."
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis, we investigate the cost of finding collisions in a blackbox function, a problem that is of fundamental importance in cryptanalysis. Inspired by the excellent performance of the heuristic rho method of collision finding, we define several new models of complexity that take into account the cost of moving information across a large space, and lay the groundwork for studying the performance of classical and quantum algorithms in these models. iii Acknowledgements I am deeply indebted to my supervisor, Dr. Michele Mosca, for introducing me to the subject of quantum information processing, and for the years of support and encouragement