Spectral Analysis of Pollard Rho Collisions
| Venue: | Proc. of the 7th Algorithmic Number Theory Symposium (ANTS VII); Springer LNCS |
| Citations: | 9 - 0 self |
BibTeX
@INPROCEEDINGS{Miller_spectralanalysis,
author = {Stephen D. Miller and Ramarathnam Venkatesan},
title = {Spectral Analysis of Pollard Rho Collisions},
booktitle = {Proc. of the 7th Algorithmic Number Theory Symposium (ANTS VII); Springer LNCS},
year = {},
pages = {573--581}
}
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Abstract
Abstract. We show that the classical Pollard ρ algorithm for discrete logarithms produces a collision in expected time O ( √ n(log n) 3). This is the first nontrivial rigorous estimate for the collision probability for the unaltered Pollard ρ graph, and is close to the conjectured optimal bound of O ( √ n). The result is derived by showing that the mixing time for the random walk on this graph is O((log n) 3); without the squaring step in the Pollard ρ algorithm, the mixing time would be exponential in log n. The technique involves a spectral analysis of directed graphs, which captures the effect of the squaring step.
