Skip to main content
SpringerLink
Log in

Point Counting in Families of Hyperelliptic Curves

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

Abstract

Let E Γ be a family of hyperelliptic curves defined by \(Y^{2}=\bar {Q}(X,\ensuremath {\Gamma })\) , where \(\bar{Q}\) is defined over a small finite field of odd characteristic. Then with \(\ensuremath {\bar {\gamma }}\) in an extension degree n field over this small field, we present a deterministic algorithm for computing the zeta function of the curve \(E_{\ensuremath {\bar {\gamma }}}\) by using Dwork deformation in rigid cohomology. The time complexity of the algorithm is \(\ensuremath {\mathcal {O}}(n^{2.667})\) and it needs \(\ensuremath {\mathcal {O}}(n^{2.5})\) bits of memory. A slight adaptation requires only \(\ensuremath {\mathcal {O}}(n^{2})\) space, but costs time \(\ensuremath {\widetilde {\mathcal {O}}}(n^{3})\) . An implementation of this last result turns out to be quite efficient for n big enough.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Anshel and D. Goldfeld, Zeta functions, one-way functions and pseudorandom number generators, Duke Math. J. 88(2) (1997), 371–390.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. J. Bernstein, Fast multiplication and its applications, URL: http://cr.yp.to/papers.html#multapps, to appear in Buhler–Stevenhagen Algorithmic Number Theory.

  3. P. Berthelot, Géométrie rigide et cohomologie des variétés algébriques de caractéristique p, Mém. Soc. Math. France (N.S.), 23(3) (1986), 7–32, Introductions aux cohomologies p-adiques (Luminy, 1984).

    MATH  MathSciNet  Google Scholar 

  4. S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean Analysis, Springer, Berlin, 1984.

    MATH  Google Scholar 

  5. H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, and F. Vercauteren, Eds., Handbook of Elliptic and Hyperelliptic Curve Cryptography, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, 2006.

    MATH  Google Scholar 

  6. J. Denef and F. Vercauteren, Errata for “An extension of Kedlaya’s algorithm to hyperelliptic curves in characteristic 2”, and related papers, available on http://wis.kuleuven.be/algebra/denef_papers/ErrataPointCounting.pdf.

  7. J. Denef and F. Vercauteren, An extension of Kedlaya’s algorithm to hyperelliptic curves in characteristic 2, J. Cryptol. 19(1) (2006), 1–25, Erratum available as [6].

    Article  MATH  MathSciNet  Google Scholar 

  8. B. Dwork, A deformation theory for the zeta function of a hypersurface, in Proc. Internat. Congr. Mathematicians (Stockholm, 1962), pp. 247–259, Inst. Mittag–Leffler, Djursholm, 1963.

    Google Scholar 

  9. B. Edixhoven, Point Counting after Kedlaya, EIDMA-Stieltjes Graduate Course, Leiden.

  10. R. Gerkmann, Relative rigid cohomology and point counting on families of elliptic curves, Preprint, available on http://joguinf.informatik.uni-mainz.de/~gerkmann/.

  11. R. Harley, Asymptotically optimal p-adic point-counting, E-mail to NMBRTHRY list.

  12. X. Huang and V. Y. Pan, Fast rectangular matrix multiplication and applications, J. Complex. 14 (1998), 257–299.

    Article  MATH  MathSciNet  Google Scholar 

  13. H. Hubrechts, Elliptic and Hyperelliptic Curve Point Counting Through Deformation, PhD thesis, KU Leuven, Belgium.

  14. H. Hubrechts, Point counting in families of hyperelliptic curves in characteristic 2, LMS J. Comput. Math. 10 (2007), 207–234.

    MathSciNet  Google Scholar 

  15. H. Hubrechts, Quasi-quadratic elliptic curve point counting using rigid cohomology, submitted, available on http://wis.kuleuven.be/algebra/hubrechts/.

  16. K. S. Kedlaya, Counting points on hyperelliptic curves using Monsky–Washnitzer cohomology, J. Ramanujan Math. Soc. 16(4) (2001), 323–338.

    MATH  MathSciNet  Google Scholar 

  17. K. S. Kedlaya, Errata for: “Counting points on hyperelliptic curves using Monsky–Washnitzer cohomology” [J. Ramanujan Math. Soc. 16(4) (2001), 323–338], J. Ramanujan Math. Soc. 18(4) (2003), 417–418, dedicated to Professor K.S. Padmanabhan.

    MATH  MathSciNet  Google Scholar 

  18. K. S. Kedlaya, Computing zeta functions via p-adic cohomology, in Algorithmic Number Theory, Lecture Notes in Computer Science, Vol. 3076, Springer, Berlin, 2004, pp. 1–17.

    Google Scholar 

  19. N. Koblitz, Elliptic curve cryptosystems, Math. Comp. 48(177) (1987), 203–209.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. G. Lauder, Rigid cohomology and p-adic point counting, to appear in a special issue of J. Théor. Nombres Bordeaux.

  21. A. G. Lauder and D. Wan, Counting points of varieties over finite fields of small characteristic, to appear in MSRI Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography.

  22. A. G. B. Lauder, Deformation theory and the computation of zeta functions, Proc. London Math. Soc. (3) 88(3) (2004), 565–602.

    Article  MATH  MathSciNet  Google Scholar 

  23. A. G. B. Lauder, A recursive method for computing zeta functions of varieties, LMS J. Comput. Math. 9 (2006), 222–269 (electronic).

    MATH  MathSciNet  Google Scholar 

  24. M. Madsen, A general framework for p-adic point counting and application to elliptic curves on Legendre form, Preprint, 2004.

  25. V. S. Miller, Use of elliptic curves in cryptography, in Advances in Cryptology—CRYPTO ’85 (Santa Barbara, CA, 1985), Lecture Notes in Computer Science, Vol. 218, pp. 417–426, Springer, Berlin, 1986.

    Google Scholar 

  26. M. Monagan, S. Tse, A. Wittkopf, Modular algorithms for resultants, in Maple Summer Workshop (2002).

  27. T. Satoh, The canonical lift of an ordinary elliptic curve over a finite field and its point counting, J. Ramanujan Math. Soc. 15(4) (2000), 247–270.

    MATH  MathSciNet  Google Scholar 

  28. R. Schoof, Counting points on elliptic curves over finite fields, J. Théor. Nombres Bordeaux 7(1) (1995), 219–254, Les Dix-Huitièmes Journées Arithmétiques (Bordeaux, 1993).

    MATH  MathSciNet  Google Scholar 

  29. V. Shoup, Efficient computation of minimal polynomials in algebraic extension of finite fields, in Proc. 1999 International Symposium on Symbolic and Algebraic Computation.

  30. J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Vol. 106, Springer, New York, 1992, corrected reprint of the 1986 original.

    Google Scholar 

  31. M. A. Tsfasman, Algebraic geometry lattices and codes, in Algorithmic Number Theory (Talence, 1996), Lecture Notes in Computer Science, Vol. 1122, pp. 385–389, Springer, Berlin, 1996.

    Google Scholar 

  32. N. Tsuzuki, Bessel F-isocrystals and an algorithm of computing Kloosterman sums, Preprint, 2003.

  33. B. L. van der Waerden, Modern Algebra, Ungar, New York, 1953.

    Google Scholar 

  34. F. Vercauteren, Computing Zeta Functions of Curves over Finite Fields, PhD thesis, KU Leuven, Belgium, 2003.

  35. J. von zur Gathen, J. Gerhard, Modern Computer Algebra, Cambridge University Press, Cambridge, 2003.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B–bus 2400, 3001, Leuven, Belgium

    Hendrik Hubrechts

Authors
  1. Hendrik Hubrechts
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hendrik Hubrechts.

Additional information

Communicated by Hendrik Lenstra

H. Hubrechts is a Research Assistant of the Research Foundation–Flanders (FWO–Vlaanderen).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hubrechts, H. Point Counting in Families of Hyperelliptic Curves. Found Comput Math 8, 137–169 (2008). https://doi.org/10.1007/s10208-007-9000-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-007-9000-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation