A Cryptographic Application of Weil Descent

  • Steven D. Galbraith⋆
  • Nigel P. Smart
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1746)


This paper gives some details about howWeil descent can be used to solve the discrete logarithm problem on elliptic curves which are defined over finite fields of small characteristic. The original ideas were first introduced into cryptography by Frey. We discuss whether these ideas are a threat to existing public key systems based on elliptic curves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Adleman, J. De Marrais, and M.-D. Huang. A subexponential algorithm for discrete logarithms over the rational subgroup of the Jacobians of large genus hyperelliptic curves over finite fields. In ANTS-1: Algorithmic Number Theory, L.M. Adleman and M-D. Huang, editors. Springer-Verlag, LNCS 877, 28–40, 1994.Google Scholar
  2. 2.
    L.M. Adleman, M.-D. Huang. Primality testing and abelian varieties over finite fields. Springer LNM 1512, 1992.Google Scholar
  3. 3.
    M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger. KANT V4. J. Symbolic Computation, 24, 267–283, 1997.zbMATHCrossRefGoogle Scholar
  4. 4.
    G. Frey. Weil descent. Talk at Waterloo workshop on the ECDLP, 1998.
  5. 5.
    S.D. Galbraith, S. Paulus and N.P. Smart. Arithmetic on super-elliptic curves. Preprint, 1998.Google Scholar
  6. 6.
    S.D. Galbraith and N.P. Smart. A cryptographic application of Weil descent. HPLabs Technical Report, HPL-1999-70.Google Scholar
  7. 7.
    P. Gaudry. A variant of the Adleman-DeMarrais-Huang algorithm and its application to small genera. Preprint, 1999.Google Scholar
  8. 8.
    J.L. Hafner and K.S. McCurley. A rigorous subexponential algorithm for computation of class groups. J. AMS, 2, 837–850, 1989.zbMATHMathSciNetGoogle Scholar
  9. 9.
    R. Hartshorne. Algebraic geometry. Springer GTM 52, 1977.Google Scholar
  10. 10.
    M.-D. Huang, D. Ierardi. Counting points on curves over finite fields. J. Symbolic Computation, 25, 1–21, 1998.CrossRefMathSciNetGoogle Scholar
  11. 11.
    J.S. Milne. Jacobian Varieties. In Arithmetic Geometry, G. Cornell and J.H. Silverman, editors. Springer-Verlag, 167–212, 1986.Google Scholar
  12. 12.
    D. Mumford. Abelian varieties. Oxford, 1970.Google Scholar
  13. 13.
    J. Pila. Frobenius maps of abelian varieties and finding roots of unity in finite fields. Math. Comp., 55, 745–763, 1990.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Steven D. Galbraith⋆
    • 1
  • Nigel P. Smart
    • 2
  1. 1.Mathematics DepartmentRoyal Holloway University of LondonSurreyUK
  2. 2.Hewlett-Packard LaboratoriesBristolUK

Personalised recommendations