Paper 2010/617

Computing Discrete Logarithms in an Interval

Steven D. Galbraith, John M. Pollard, and Raminder S. Ruprai

Abstract

The discrete logarithm problem in an interval of size $N$ in a group $G$ is: Given $g, h \in G$ and an integer $ N$ to find an integer $0 \le n \le N$, if it exists, such that $h = g^n$. Previously the best low-storage algorithm to solve this problem was the van Oorschot and Wiener version of the Pollard kangaroo method. The heuristic average case running time of this method is $(2 + o(1)) \sqrt{N}$ group operations. We present two new low-storage algorithms for the discrete logarithm problem in an interval of size $N$. The first algorithm is based on the Pollard kangaroo method, but uses 4 kangaroos instead of the usual two. We explain why this algorithm has heuristic average case expected running time of $(1.715 + o(1)) \sqrt{N}$ group operations. The second algorithm is based on the Gaudry-Schost algorithm and the ideas of our first algorithm. We explain why this algorithm has heuristic average case expected running time of $(1.661 + o(1)) \sqrt{N}$ group operations. We give experimental results that show that the methods do work close to that predicted by the theoretical analysis. This is a revised version since the published paper that contains a corrected proof of Theorem 6 (the statement of Theorem 6 is unchanged). We thank Ravi Montenegro for pointing out the errors.

Metadata
Available format(s)
PDF
Category
Public-key cryptography
Publication info
Published elsewhere. MINOR revision.Math. Comp., 82, No. 282 (2013) 1181-1195.
DOI
10.1090/S0025-5718-2012-02641-X
Keywords
discrete logarithm problem (DLP)
Contact author(s)
S Galbraith @ math auckland ac nz
History
2018-11-23: revised
2010-12-08: received
See all versions
Short URL
https://ia.cr/2010/617
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2010/617,
      author = {Steven D.  Galbraith and John M.  Pollard and Raminder S.  Ruprai},
      title = {Computing Discrete Logarithms in an Interval},
      howpublished = {Cryptology ePrint Archive, Paper 2010/617},
      year = {2010},
      doi = {10.1090/S0025-5718-2012-02641-X},
      note = {\url{https://eprint.iacr.org/2010/617}},
      url = {https://eprint.iacr.org/2010/617}
}
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