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Paper 2010/617

Computing Discrete Logarithms in an Interval

Steven D. Galbraith and 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 $(1.714 + o(1)) \sqrt{N}$ 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 $(1.660 + o(1)) \sqrt{N}$ group operations. We give experimental results that show that the methods do work close to that predicted by the theoretical analysis.

Metadata
Available format(s)
PDF
Category
Public-key cryptography
Publication info
Published elsewhere. Submitted
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
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