Improved Sieving on Algebraic Curves

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9230)

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 

References

  1. 1.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3–4), 235–265 (1997). Computational algebra and number theory (London, 1993)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Diem, C.: An index calculus algorithm for plane curves of small degree. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 543–557. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  3. 3.
    Diem, C., Kochinke, S.: Computing discrete logarithms with special linear systems (2013). http://www.math.uni-leipzig.de/diem/preprints/dlp-linear-systems.pdf
  4. 4.
    Gaudry, P.: An algorithm for solving the discrete log problem on hyperelliptic curves. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 19–34. Springer, Heidelberg (2000) CrossRefGoogle Scholar
  5. 5.
    Gaudry, P., Hess, F., Smart, N.P.: Constructive and destructive facets of Weil descent on elliptic curves. J. Cryptol. 15(1), 19–46 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gaudry, P., Thomé, E., Thériault, N., Diem, C.: A double large prime variation for small genus hyperelliptic index calculus. Math. Comput. 76(257), 475–492 (2007)MATHCrossRefGoogle Scholar
  7. 7.
    Joux, A., Vitse, V.: Cover and decomposition index calculus on elliptic curves made practical. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 9–26. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  8. 8.
    Laine, K., Lauter, K.: Time-memory trade-offs for index calculus in genus 3. J. Math. Cryptol. 9(2), 95–114 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    LaMacchia, B.A., Odlyzko, A.M.: Computation of discrete logarithms in prime fields. Des. Codes Crypt. 1(1), 47–62 (1991)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sarkar, P., Singh, S.: A new method for decomposition in the Jacobian of small genus hyperelliptic curves. Cryptology ePrint Archive, Report 2014/815 (2014)Google Scholar
  11. 11.
    Thériault, N.: Index calculus attack for hyperelliptic curves of small genus. In: Laih, C.-S. (ed.) ASIACRYPT 2003. LNCS, vol. 2894, pp. 75–92. Springer, Heidelberg (2003) CrossRefGoogle Scholar

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

Personalised recommendations