Counting Points on Hyperelliptic Curves over Finite Fields

  • Pierrick Gaudry
  • Robert Harley
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1838)


We describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm à la Schoof for genus 2 using Cantor’s division polynomials. These are combined with a birthday paradox algorithm to calculate the cardinality. Our methods are practical and we give actual results computed using our current implementation. The Jacobian groups we handle are larger than those previously reported in the literature.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AH92]
    Adleman, L.M., Huang, M.-D.A.: Primality testing and Abelian varieties over finite fields. Lecture Notes in Math, vol. 1512. Springer, Heidelberg (1992)MATHGoogle Scholar
  2. [BC97]
    Bosma, W., Cannon, J.: Handbook of Magma functions, Sydney (1997),
  3. [Can87]
    Cantor, D.G.: Computing in the Jacobian of an hyperelliptic curve. Math. Comp. 48(177), 95–101 (1987)MATHCrossRefMathSciNetGoogle Scholar
  4. [Can94]
    Cantor, D.G.: On the analogue of the division polynomials for hyperelliptic curves. J. Reine Angew. Math. 447, 91–145 (1994)MATHMathSciNetCrossRefGoogle Scholar
  5. [Car57]
    Cartier, P.: Une nouvelle opéeration sur les formes diffiérentielles. C. R. Acad. Sci. Paris Sér. I Math. 244, 426–428 (1957)MATHMathSciNetGoogle Scholar
  6. [Cou96]
    Couveignes, J.-M.: Computing l-isogenies using the p-torsion. In: Cohen, H. (ed.) ANTS 1996. LNCS, vol. 1122, pp. 59–65. Springer, Heidelberg (1996)Google Scholar
  7. [Elk98]
    Elkies, N.: Elliptic and modular curves over finite fields and related computational issues. In: Buell, D.A., Teitelbaum, J.T. (eds.) Computational Perspectives on Number Theory, pp. 21–76. AMS/International Press (1998); Proceedings of a Conference in Honor of A.O.L. AtkinGoogle Scholar
  8. [FR94]
    Frey, G., Rück, H.-G.: A remark concerning m-divisibility and the discretelogarithm in the divisor class group of curves. Math. Comp. 62(206), 865–874 (1994)MATHMathSciNetGoogle Scholar
  9. [Fre83]
    Freitag, E.: SiegelscheModulfunktionen. Grundlehren der mathematischen Wissenschaften, vol. 254. Springer, Heidelberg (1983)Google Scholar
  10. [Har]
    Harley, R.: On modular equations in genus 2 (in preparation)Google Scholar
  11. [HI98]
    Huang, M.-D., Ierardi, D.: Counting points on curves over finite fields. J. Symbolic Comput. 25, 1–21 (1998)CrossRefMathSciNetGoogle Scholar
  12. [IR82]
    Ireland, K.F., Rosen, M.: A classical introduction to modern number theory. Graduate texts in Mathematics, vol. 84. Springer, Heidelberg (1982)MATHGoogle Scholar
  13. [Kam91]
    Kampkötter, W.: Explizite Gleichungen für Jacobische Varietäten hyperelliptischer Kurven. PhD thesis, Univ. Gesamthochschule Essen (August 1991)Google Scholar
  14. [Kli90]
    Klingen, H.: Introductory lectures on Siegel modular forms. Cambridge studies in advanced mathematics, vol. 20. Cambridge University Press, Cambridge (1990)MATHCrossRefGoogle Scholar
  15. [Kob89]
    Koblitz, N.: Hyperelliptic cryptosystems. J. of Cryptology 1, 139–150 (1989)MATHCrossRefMathSciNetGoogle Scholar
  16. [Lec99]
    Lecerf, G.: Kronecker, Polynomial Equation System Solver, Reference manual (1999),
  17. [Ler97]
    Lercier, R.: Algorithmique des courbes elliptiques dans les corps finis. Thése, École polytechnique (June 1997)Google Scholar
  18. [Man65]
    Manin, J.I.: The Hasse-Witt matrix of an algebraic curve. Trans. Amer. Math. Soc. 45, 245–264 (1965)Google Scholar
  19. [Mor95]
    Morain, F.: Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques. J. Théor. Nombres Bordeaux 7, 255–282 (1995)MATHMathSciNetGoogle Scholar
  20. [Mum84]
    Mumford, D.: Tata lectures on theta II. Progr. Math. Birkhauser 43 (1984)Google Scholar
  21. [PH78]
    Pohlig, S., Hellman, M.: An improved algorithm for computing logarithms over GF(p) and its cryptographic significance. IEEE Trans. Inform. Theory, IT 24, 106–110 (1978)MATHCrossRefMathSciNetGoogle Scholar
  22. [Pil90]
    Pila, J.: Frobenius maps of abelian varieties and finding roots of unity in finite fields. Math. Comp. 55(192), 745–763 (1990)MATHCrossRefMathSciNetGoogle Scholar
  23. [Pol78]
    Pollard, J.M.: Monte Carlo methods for index computation mod p. Math. Comp. 32(143), 918–924 (1978)MATHMathSciNetGoogle Scholar
  24. [Rüc99]
    Rück, H.G.: On the discrete logarithm in the divisor class group of curves. Math. Comp. 68(226), 805–806 (1999)MATHCrossRefMathSciNetGoogle Scholar
  25. [Sch]
    Schost, E.: Computing parametric geometric resolutions. Submitted to ISSAC 2000 (2000)Google Scholar
  26. [Sch85]
    Schoof, R.: Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp. 44, 483–494 (1985)MATHMathSciNetGoogle Scholar
  27. [Sch95]
    Schoof, R.: Counting points on elliptic curves over finite fields. J. Théor. Nombres Bordeaux 7, 219–254 (1995)MATHMathSciNetGoogle Scholar
  28. [ST99]
    Stein, A., Teske, E.: Catching kangaroos in function fields (March 1999) (preprint)Google Scholar
  29. [Tat66]
    Tate, J.: Endomorphisms of Abelian varieties over finite fields. Invent. Math. 2, 134–144 (1966)MATHCrossRefMathSciNetGoogle Scholar
  30. [Ver99]
    Vercauteren, F.: #EC(GF(2^1999)). E-mail message to the NMBRTHRY list (October 1999)Google Scholar
  31. [vOW99]
    van Oorschot, P.C., Wiener, M.J.: Parallel collision search with cryptanalytic applications. J. of Cryptology 12, 1–28 (1999)MATHCrossRefGoogle Scholar
  32. [Yui78]
    Yui, N.: On the jacobian varietes of hyperelliptic curves over fields of characteristic p > 2. J. Algebra 52, 378–410 (1978)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Pierrick Gaudry
    • 1
  • Robert Harley
    • 2
  1. 1.LIXÉcole PolytechniquePalaiseau CedexFrance
  2. 2.Projet Cristal, INRIADomaine de Voluceau – RocquencourtLe ChesnayFrance

Personalised recommendations