Abstract
Matsuo, Chao and Tsujii [16] have proposed an algorithm for counting the number of points on the Jacobian variety of a hyperelliptic curve over a finite field. The Matsuo-Chao-Tsujii algorithm is an improvement of the ‘baby-step-giant-step’ part of the Gaudry-Harley scheme. This scheme consists of two parts: firstly to compute the residue modulo a positive integer m of the order of a given Jacobian variety, and then to search for the actual order by a square-root algorithm. In this paper, following the Matsuo-Chao-Tsujii algorithm, we propose an improvement of the square-root algorithm part in the Gaudry-Harley scheme by optimizing the use of the residue modulo m of the characteristic polynomial of the Frobenius endomorphism of an Abelian variety. It turns out that the computational complexity is \(O \left( q^{\frac{4g -- 2 + i^{2} -- i}{8}} / m^{\frac{i + 1}{2}} \right)\), where i is an integer in the range 1 ≤ i ≤ g. We will show that for each g and each finite field \(\mathbb{F}_q\) of q=p n elements, there exists an i which gives rise to the optimum complexity among all three corresponding algorithms.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adleman, L.M., Huang, M.D.: Counting rational points on curves and Abelian varieties over finite fields. In: Cohen, H. (ed.) ANTS 1996. LNCS, vol. 1122, pp. 1–16. Springer, Heidelberg (1996)
Denef, J., Vercauteren, F.: An extension of Kedlaya’s algorithm to Artin-Schreier curves in characteristic 2. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 308–323. Springer, Heidelberg (2002)
Gaudry, P.: Algorithmique des courbes hyperellitiques et applications á la cryptologie, Ph.D. thesis, École polytechnique (2000)
Gaudry, P.: Algorithms for counting points on curves. In: Talk at ECC 2001, The fifth Workshop on elliptic Curve Cryptography, U. Waterloo (2001)
Gaudry, P., Gürel, N.: An extension of Kedlaya’s point counting algorithm to su-perelliptic curves. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 480–494. Springer, Heidelberg (2001)
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)
Harley, R.: Counting points with the arithmetic-geometric mean. In: Rump talk at EUROCRYPT (2001) (joint work with J.F. Mestre and P. Gaudry)
Huang, M.D., Ierardi, D.: Counting rational point on curves over finite fields. J. Symbolic Computation 25, 1–21 (1998)
Kampkötter, W.: Explizite Gleichungen für Jacobische Varietäten hyperelliptischer Kurven, Ph.D. thesis, GH Essen (1991)
Kedlaya, K.S.: Counting points on hyperelliptic curves using Monsky-Washinitzer cohomology. J. Ramanujan Math. Soc. 16, 323–338 (2001)
Koblitz, N.: Elliptic curve cryptosystems. Math. Comp. 48, 203–209 (1987)
Koblitz, N.: Hyperelliptic curve cryptosystems. J. of Cryptology 1, 139–150 (1989)
Lauder, A., Wan, D.: Computing zeta functions of Artin-Schreier curves over finite fields. LMS J. Comput. Math. 5, 33–55 (2002)
The magma algebraic system, http://www.maths.usyd.edu.au:8000/u/magma/
Manin, J.I.: The Hasse-Witt matrix of an algebraic curve. Transl. Amer. Math. Soc. 45, 245–264 (1965)
Matsuo, K., Chao, J., Tsujii, S.: Baby step giant step algorithms in point counting of hyperelliptic curves. IEICE Trans. Fundamentals E86-A(4) (April 2003)
Miller, V.: Uses of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)
Pila, J.: Frobenius maps of Abelian varieties and finding roots of unity in finite fields. Math. Comp 55, 745–763 (1990)
Weil, A.: Variétés Abéliennes et courbes algébriques. Hermann, Paris (1948)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Izadi, F.A., Murty, V.K. (2003). Counting Points on an Abelian Variety over a Finite Field. In: Johansson, T., Maitra, S. (eds) Progress in Cryptology - INDOCRYPT 2003. INDOCRYPT 2003. Lecture Notes in Computer Science, vol 2904. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24582-7_24
Download citation
DOI: https://doi.org/10.1007/978-3-540-24582-7_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20609-5
Online ISBN: 978-3-540-24582-7
eBook Packages: Springer Book Archive