### Abstract

We show that the classical Pollard p 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 p 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 p 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.

Original language | English (US) |
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Title of host publication | Algorithmic Number Theory - 7th International Symposium, ANTS-VII, Proceedings |

Publisher | Springer Verlag |

Pages | 573-581 |

Number of pages | 9 |

ISBN (Print) | 3540360751, 9783540360759 |

DOIs | |

State | Published - 2006 |

Event | 7th International Symposium on Algorithmic Number Theory, ANTS-VII - Berlin, Germany Duration: Jul 23 2006 → Jul 28 2006 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4076 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 7th International Symposium on Algorithmic Number Theory, ANTS-VII |
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Country | Germany |

City | Berlin |

Period | 7/23/06 → 7/28/06 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Keywords

- Collision time
- Discrete logarithm
- Expander graph
- Mixing time
- Pollard Rho algorithm
- Random walk
- Spectral analysis

## Fingerprint Dive into the research topics of 'Spectral analysis of Pollard Rho collisions'. Together they form a unique fingerprint.

## Cite this

Miller, S. D., & Venkatesan, R. (2006). Spectral analysis of Pollard Rho collisions. In

*Algorithmic Number Theory - 7th International Symposium, ANTS-VII, Proceedings*(pp. 573-581). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4076 LNCS). Springer Verlag. https://doi.org/10.1007/11792086_40