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inria-00560256, version 1

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

Gaetan Bisson 1, 2, Andrew V. Sutherland 3

Designs, Codes and Cryptography (2011)

Abstract: We describe a space-efficient algorithm for solving a generalization of the subset sum problem in a finite group G, using a Pollard-rho 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 log_2 n, where n=#G and d >= 2 is a constant, we find that its expected running time is O(sqrt(n) 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.

  • 1:  Eindhoven Institute for the Protection of Systems and Information (EIPSI)
  • Technische Universiteit Eindhoven
  • 2:  CARAMEL (INRIA Nancy - Grand Est / LORIA)
  • INRIA – CNRS : UMR7503 – Université de Lorraine
  • 3:  Department of Mathematics [MIT]
  • Massachussetts Institute of Technology (MIT)
  • Domain : Computer Science/Cryptography and Security
    Mathematics/Number Theory
    Mathematics/Group Theory
  • Comment : 12 pages
 
  • inria-00560256, version 1
  • http://hal.archives-ouvertes.fr/inria-00560256
  • oai:hal.archives-ouvertes.fr:inria-00560256
  • From: Gaetan Bisson <>
  • Submitted on: Thursday, 27 January 2011 18:09:09
  • Updated on: Tuesday, 24 May 2011 17:25:12