Counting Points for Hyperelliptic Curves of Type y2=x5+ax over Finite Prime Fields

  • Eisaku Furukawa
  • Mitsuru Kawazoe
  • Tetsuya Takahashi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3006)


Counting rational points on Jacobian varieties of hyperelliptic curves over finite fields is very important for constructing hyperelliptic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running time. In this article, we propose an extremely fast point counting algorithm for hyperelliptic curves of type y 2=x 5+ax over given large prime fields \(\mathbb{F}_{p}\), e.g. 80-bit fields. For these curves, we also determine the necessary condition to be suitable for HCC, that is, to satisfy that the order of the Jacobian group is of the form l· c where l is a prime number greater than about 2160 and c is a very small integer. We show some examples of suitable curves for HCC obtained by using our algorithm. We also treat curves of type y 2=x 5+a where a is not square in \(\mathbb{F}_{p}\).


Prime Number Characteristic Polynomial Elliptic Curf Abelian Variety Hyperelliptic Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi Sums. Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 21. A Wiley-Interscience Publication, Chichester (1998)zbMATHGoogle Scholar
  2. 2.
    Buhler, J., Koblitz, N.: Lattice Basis Reduction, Jacobi Sums and Hyperelliptic Cryptosystems. Bull. Austral. Math. Soc. 58, 147–154 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cantor, D.G.: Computing in the Jacobian of hyperelliptic curve. Math. Comp. 48, 95–101 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Choie, Y., Jeong, E., Lee, E.: Supersingular Hyperelliptic Curves of Genus 2 over Finite Fields, Cryptology ePrint Archive: Report 2002/032 (2002),
  5. 5.
    Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138. Springer, Heidelberg (1996)Google Scholar
  6. 6.
    Duursma, I., Gaudry, P., Morain, F.: Speeding up the Discrete Log Computation on Curves with Automorphisms. In: Lam, K.-Y., Okamoto, E., Xing, C. (eds.) ASIACRYPT 1999. LNCS, vol. 1716, pp. 103–121. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  7. 7.
    Frey, G., Rück, H.-G.: A Remark Concerning m-divisibility and the Discrete Logarithm in the Divisor Class Group of Curves. Math. Comp. 62(206), 865–874 (1994)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Galbraith, S.G.: Supersingular Curves in Cryptography. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 495–513. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Gaudry, P., Harley, R.: Counting Points on Hyperelliptic Curves over Finite Fields. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 297–312. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  10. 10.
    Hudson, R.H., Williams, K.S.: Binomial Coefficients and Jacobi Sums. Trans. Amer. Math. Soc. 281, 431–505 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Koblitz, N.: Algebraic Aspects of Cryptography. Algorithms and Computation in Mathematics, vol. 3. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  12. 12.
    Lang, S.: Abelian Varieties. Springer, Heidelberg (1983)zbMATHCrossRefGoogle Scholar
  13. 13.
    Leprévost, F., Morain, F.: Revêtements de courbes elliptiques à multiplication complexe par des courbes hyperelliptiques et sommes de caractères. J. Number Theory 64, 165–182 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Manin, J.I.: The Hasse-Witt Matrix of an Algebraic Curve. Amer. Math. Soc. Transl. Ser. 45, 245–264 (1965)Google Scholar
  15. 15.
    Matsuo, K., Chao, J., Tsujii, S.: An improved baby step giant step algorithm for point counting of hyperelliptic curves over finite fields. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 461–474. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  16. 16.
    Mumford, D.: Tata Lectures on Theta II. In: Progress in Mathematics 43, Birkhäuser, Basel (1984)Google Scholar
  17. 17.
    Rück, H.-G.: Abelian surfaces and Jacobian varieties over finite fields. Compositio Math. 76, 351–366 (1990)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Tate, J.: Endomorphisms of abelian varieties over finite fields. Invent. Math. 2, 134–144 (1996)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Waterhouse, W.C.: Abelian varieties over finite fields. Ann. Sci. École Nor. Sup. 2(4), 521–560 (1969)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Wolfram, S.: The Mathematica Book, 4th edn. Wolfram Media/Cambridge University Press (1999)Google Scholar
  21. 21.
    Xing, C.: On supersingular abelian varieties of dimension two over finite fields. Finite Fields and Their Appl. 2, 407–421 (1996)zbMATHCrossRefGoogle Scholar
  22. 22.
    Yui, N.: On the Jacobian Varieties of Hyperelliptic Curves over Fields of Characteristic p > 2. J. Alg. 52, 378–410 (1978)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Eisaku Furukawa
    • 1
  • Mitsuru Kawazoe
    • 2
  • Tetsuya Takahashi
    • 2
  1. 1.Fujitsu Kansai-Chubu Net-Tech Limited 
  2. 2.Department of Mathematics and Information Sciences College of Integrated Arts and SciencesOsaka Prefecture UniversityOsakaJapan

Personalised recommendations