Advertisement

Designs, Codes and Cryptography

, Volume 20, Issue 1, pp 5–40 | Cite as

The Xedni Calculus and the Elliptic Curve Discrete Logarithm Problem

  • Joseph H. Silverman
Article

Abstract

Let \(E/{\mathbb{F}}_P\) be an elliptic curve defined over a finite field, and let \(S,T \in E({\mathbb{F}}_P )\) be two points on E. The Elliptic Curve Discrete Logarithm Problem (ECDLP) asks that an integer m be found so that S=mT in \(E({\mathbb{F}}_P )\). In this note we give a new algorithm, termed the Xedni Calculus, which might be used to solve the ECDLP. As remarked by Neal Koblitz, the Xedni method is also applicable to the classical discrete logarithm problem for \({\mathbb{F}}_p^*\) and to the integer factorization problem.

Elliptic curve discrete logarithm Xedni calculus 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. Birch and H. P. F. Swinnerton-Dyer, Elliptic curves and modular functions, Modular Functions of One Variable, Antwerp IV: Springer Lecture Notes 476 (Birch and Kuyk, eds.), Springer-Verlag (1975) pp. 2–32.Google Scholar
  2. 2.
    H. Cohen, A Course in Computational Number Theory, GTM 138, Springer-Verlag (1993).Google Scholar
  3. 3.
    I. Connell, Addendum to a paper of Harada and Lang, J.Algebra, Vol. 145 (1992) pp. 463–467.Google Scholar
  4. 4.
    J. Cremona, Algorithms for Modular Elliptic Curves, (2nd ed.) Cambridge University Press (1997).Google Scholar
  5. 5.
    E. Fouvry, M. Nair, and G. Tenenbaum, L'ensemble exceptionnel dans la conjecture de Szpiro, Bull.Soc.Math.France, Vol. 120 (1992) pp. 485–506.Google Scholar
  6. 6.
    G. Havas, B. S. Majewski, and K. R. Matthews, Extended gcd and Hermite normal form algorithms via lattice basis reduction, Experimental Math., Vol. 7 (1998) pp. 125–136Google Scholar
  7. 7.
    M. Hindry and J. H. Silverman, The canonical height and integral points on elliptic curves, Invent.Math., Vol. 93, pp. 419–450 (1988).Google Scholar
  8. 8.
    M. J. Jacobson, N. Koblitz, J. H. Silverman, A. Stein, and E. Teske, Analysis of the Xedni Calculus Attack, Designs, Codes and Cryptography, Vol. 20 (2000), pp. 41–64.Google Scholar
  9. 9.
    N. Koblitz, Elliptic curve cryptosystems, Mathematics of Computation, Vol. 48, pp. 203–209 (1987).Google Scholar
  10. 10.
    N. Koblitz, Private communications, September 1998.Google Scholar
  11. 11.
    H. Lenstra, Factoring integers with elliptic curves, Annals of Math, Vol. 126 (1987) pp. 649–673.Google Scholar
  12. 12.
    D. W. Masser, Specializations of finitely generated subgroups of abelian varieties, Trans.AMS, Vol. 311 (1989) pp. 413–424.Google Scholar
  13. 13.
    K. Matthews, Short solutions of AX = B using a LLL-based Hermite normal form algorithm, August 18, 1998, preprint.Google Scholar
  14. 14.
    J. F. Mestre, Formules explicites et minoration de conducteurs de variétés algébriques, Compositio Math., Vol. 58 (1986) pp. 209–232.Google Scholar
  15. 15.
    V. S. Miller, Use of elliptic curves in cryptography, Advances in Cryptology CRYPTO '85: Lecture Notes in Computer Science, vol. 218 Springer-Verlag (1986) pp. 417–426.Google Scholar
  16. 16.
    A. Néron, Propriétés arithmétiques et géométriques rattachés à la notion de rang d'une courbe algébrique dans un corps, Bull.Soc.Math.France, Vol. 80 (1952) pp. 101–166.Google Scholar
  17. 17.
    D. Rohrlich, Variation of the root number in families of elliptic curves, Compositio Math, Vol. 87 (1993 ) pp. 119–151Google Scholar
  18. 18.
    D. Rohrlich, An algorithm to compute w3.E/, unpublished.Google Scholar
  19. 19.
    J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Math., Vol. 106, Springer-Verlag, Berlin and New York (1986).Google Scholar
  20. 20.
    J. H. Silverman, Computing heights on elliptic curves, Math.Comp., Vol. 51 (1988) pp. 339–358.Google Scholar
  21. 21.
    J. H. Silverman, Computing canonical heights with little (or no) factorization, Math.Comp., Vol. 66 (1997) pp. 787–805.Google Scholar
  22. 22.
    J. H. Silverman and J. Suzuki, Elliptic curve discrete logarithms and the index calculus, ASIACRYPT '98: Lecture Notes in Computer Science, Springer-Verlag, Berlin, 1998, pp. 110–125.Google Scholar
  23. 23.
    J. H. Silverman and J. Tate, Rational points on elliptic curves, Springer-Verlag, New York (1992).Google Scholar
  24. 24.
    A. Wiles, Modular elliptic curves and Fermat's last theorem, Annals of Math, Vol. 141 (1995) pp. 443–551Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Joseph H. Silverman
    • 1
  1. 1.Mathematics DepartmentBrown UniversityProvidence

Personalised recommendations