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

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 log2n, 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 

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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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