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Designs, Codes and Cryptography

, Volume 63, Issue 1, pp 1–13 | Cite as

A low-memory algorithm for finding short product representations in finite groups

  • Gaetan Bisson
  • Andrew V. Sutherland
Open Access
Article

Abstract

We describe a space-efficient algorithm for solving a generalization of the subset sum problem in a finite group G, using a Pollard-ρ approach. Given an element z and a sequence of elements S, our algorithm attempts to find a subsequence of S whose product in G is equal to z. For a random sequence S of length d log2 n, where n = #G and d ≥ 2 is a constant, we find that its expected running time is \({O(\sqrt{n}\,{\rm log}\,n)}\) group operations (we give a rigorous proof for d > 4), and it only needs to store O(1) group elements. We consider applications to class groups of imaginary quadratic fields, and to finding isogenies between elliptic curves over a finite field.

Keywords

Short product Generic group algorithm Pollard-rho Isogeny search 

Mathematics Subject Classification (2000)

20D60 11R29 

Notes

Acknowledgments

The authors are indebted to Andrew Shallue for his kind help and advice in putting our result in the context of subset sum problems, and to Steven Galbraith for his useful feedback on an early draft of this paper.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.LORIAVandœuvre-lès-NancyFrance
  2. 2.Eindhoven University of TechnologyEindhovenThe Netherlands
  3. 3.Massachusetts Institute of TechnologyCambridgeUSA

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