Chapter

Public Key Cryptography – PKC 2010

Volume 6056 of the series Lecture Notes in Computer Science pp 368-383

Using Equivalence Classes to Accelerate Solving the Discrete Logarithm Problem in a Short Interval

  • Steven D. GalbraithAffiliated withMathematics Department, Auckland University
  • , Raminder S. RupraiAffiliated withMathematics Department, Royal Holloway University of London

Abstract

The Pollard kangaroo method solves the discrete logarithm problem (DLP) in an interval of size N with heuristic average case expected running time approximately \(2 \sqrt{N}\) group operations. It is well-known that the Pollard rho method can be sped-up by using equivalence classes (such as orbits of points under an efficiently computed group homomorphism), but such ideas have not been used for the DLP in an interval. Indeed, it seems impossible to implement the standard kangaroo method with equivalence classes.

The main result of the paper is to give an algorithm, building on work of Gaudry and Schost, to solve the DLP in an interval of size N with heuristic average case expected running time of close to \(1.36\sqrt{N}\) group operations for groups with fast inversion. In practice the algorithm is not quite this fast, due to the usual problems with pseudorandom walks such as fruitless cycles. In addition, we present experimental results.

Keywords

discrete logarithm problem (DLP) elliptic curves negation map efficiently computable group homomorphisms