Collision of random walks and a refined analysis of attacks on the discrete logarithm problem

Shuji Kijima, Ravi Montenegro

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Citations (Scopus)

Abstract

Some of the most efficient algorithms for finding the discrete logarithm involve pseudo-random implementations of Markov chains, with one or more “walks” proceeding until a collision occurs, i.e. some state is visited a second time. In this paper we develop a method for determining the expected time until the first collision. We use our technique to examine three methods for solving discrete-logarithm problems: Pollard’s Kangaroo, Pollard’s Rho, and a few versions of Gaudry-Schost. For the Kangaroo method we prove new and fairly precise matching upper and lower bounds. For the Rho method we prove the first rigorous non-trivial lower bound, and under a mild assumption show matching upper and lower bounds. Our Gaudry-Schost results are heuristic, but improve on the prior limited understanding of this method. We also give results for parallel versions of these algorithms.

Original languageEnglish
Title of host publicationPublic-Key Cryptography - PKC 2015 - 18th IACR International Conference on Practice and Theory in Public-Key Cryptography, Proceedings
EditorsJonathan Katz
PublisherSpringer Verlag
Pages127-149
Number of pages23
ISBN (Electronic)9783662464465
DOIs
Publication statusPublished - Jan 1 2015
Event18th IACR International Conference on Practice and Theory of Public-Key Cryptography, PKC 2015 - Gaithersburg, United States
Duration: Mar 30 2015Apr 1 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9020
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other18th IACR International Conference on Practice and Theory of Public-Key Cryptography, PKC 2015
CountryUnited States
CityGaithersburg
Period3/30/154/1/15

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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    Kijima, S., & Montenegro, R. (2015). Collision of random walks and a refined analysis of attacks on the discrete logarithm problem. In J. Katz (Ed.), Public-Key Cryptography - PKC 2015 - 18th IACR International Conference on Practice and Theory in Public-Key Cryptography, Proceedings (pp. 127-149). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9020). Springer Verlag. https://doi.org/10.1007/978-3-662-46447-2_6