Improved Sieving on Algebraic Curves

Conference paper

DOI: 10.1007/978-3-319-22174-8_16

Volume 9230 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Vitse V., Wallet A. (2015) Improved Sieving on Algebraic Curves. In: Lauter K., Rodríguez-Henríquez F. (eds) Progress in Cryptology -- LATINCRYPT 2015. LATINCRYPT 2015. Lecture Notes in Computer Science, vol 9230. Springer, Cham

Abstract

The best algorithms for discrete logarithms in Jacobians of algebraic curves of small genus are based on index calculus methods coupled with large prime variations. For hyperelliptic curves, relations are obtained by looking for reduced divisors with smooth Mumford representation (Gaudry); for non-hyperelliptic curves it is faster to obtain relations using special linear systems of divisors (Diem, Kochinke). Recently, Sarkar and Singh have proposed a sieving technique, inspired by an earlier work of Joux and Vitse, to speed up the relation search in the hyperelliptic case. We give a new description of this technique, and show that this new formulation applies naturally to the non-hyperelliptic case with or without large prime variations. In particular, we obtain a speed-up by a factor approximately 3 for the relation search in Diem and Kochinke’s methods.

Keywords

Discrete logarithm Index calculus Algebraic curves Curve-based cryptography 

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut Fourier, UJF-CNRS, UMR 5582Saint-martin d’hèresFrance
  2. 2.Sorbonnes Universités, UPMC Univ Paris 06, CNRS, INRIA, LIP6 UMR 7606ParisFrance
  3. 3.Projet POLSYS, INRIA RocquencourtLe Chesnay CedexFrance