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Computing Zeta Functions of Artin–schreier Curves over Finite Fields

Published online by Cambridge University Press:  01 February 2010

Alan G. B. Lauder  and
Daqing Wan
Alan G. B. Lauder
Affiliation:
Computing Laboratory, Oxford University, Oxford OX1 3QD, alan.lauder@comlab.ox.ac.uk, http://web.comlab.ox.ac.uk/oucl/work/alan.lauder/
Daqing Wan
Affiliation:
Department of Mathematics, University of California, Irvine, CA 92697, USA, dwan@math.uci.edu, http://www.math.uci.edu/~dwan/Overview.html

Abstract

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The authors present a practical polynomial-time algorithm for computing the zeta function of certain Artin–Schreier curves over finite fields. This yields a method for computing the order of the Jacobian of an elliptic curve in characteristic 2, and more generally, any hyperelliptic curve in characteristic 2 whose affine equation is of a particular form. The algorithm is based upon an efficient reduction method for the Dwork cohomology of one-variable exponential sums.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 5 , 2002 , pp. 34 - 55
Copyright
Copyright © London Mathematical Society 2002

References

1Adolphson, A. and Sperber, S., ‘Exponential sums and Newton polyhedra: cohomology and estimates’, Ann.of Math. 130 (1989) 367–406.CrossRefGoogle Scholar
2Blake, I., Seroussi, G. and Smart, N., Elliptic curves in cryptography, LMS Lecture Note Ser. 265 (Cambridge Univ. Press, 1999).CrossRefGoogle Scholar
3Cantor, D. and Kaltofen, E., ‘On fast multiplication of polynomials over arbitrary algebras’, Acta Inform.28 (1991) 693–701.CrossRefGoogle Scholar
4Coleman, R. F., ‘P-adic Banach spaces and families of modular forms’, Invent.Math.127 (1997) 417–479.CrossRefGoogle Scholar
5Dwork, B., ‘On the rationality of the zeta function of an algebraic variety’, Amer.J.Math.82 (1960) 631–648.CrossRefGoogle Scholar
6Dwork, B., ‘On the zeta function of a hypersurface’, Inst.Hautes Études Sci Publ.Math.12 (1962).CrossRefGoogle Scholar
7Elkies, N., ‘Elliptic and modular curves over finite fields and related computational issues’, Computational perspectives in number theory: Proceedings of a conference n honour of A.O.L.Atkin, AMS/IP Stud. Adv. Math. 7 (ed. Buell, D. A. and Teitelbaum, J. T., Amer. Math. Soc, Providence, RI, 1998) 21–76.Google Scholar
8Fouquet, M., Gaudry, P. and Harley, R., ‘An extension of Satoh's algorithm and its implementation’, J.Ramanujan Math. Soc.15 (2000 )281–318.Google Scholar
9Fouquet, M., Gaudry, P. and Harley, R., ‘Finding secure curves with the Satoh-FGHalgorithm and an early abort strategy’, Advances in Cryptology - EUROCRYPT2001, Lecture Notes in Comput. Sci. 2045 (ed. Pfitzmann, B., Springer, 2001) 14–29.Google Scholar
10Gaudry, P. and Gürel, N., ‘An extension of Kedlaya's algorithm for counting points on superelliptic curves’, preprint, 2001.CrossRefGoogle Scholar
11Gaudry, P. and Harley, R., ‘Counting points on hyperelliptic curves over finite fields’, Advances in Cryptology - EUROCRYPT 2000, Lecture Notes in Comput. Sci. 1807 (ed. B. Preneel, Springer, 2000) 19–34.CrossRefGoogle Scholar
12Kedlaya, K. S., ‘Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology’, preprint, 2001.Google Scholar
13Koblitz, N., ‘Hyperelliptic cryptosystems, J. Cryptology 1 (1989) 139–150.CrossRefGoogle Scholar
14Lauder, A.G.B. and Wan, D., ‘Counting points on varieties over finite fields of small characteristic’, preprint, 2001.Google Scholar
15Lidl, R. and Niederreiter, H., Introduction to finite fields and their applications (Cambridge Univ. Press, 1986).Google Scholar
16Poonen, B., ‘Computational aspects of curves of genus at least 2’, Algorithmic number theory II, Lecture Notes in Comput. Sci. 1122, (ed. Cohen, H., Springer, 1996) 283–306.CrossRefGoogle Scholar
17Satoh, T., ‘The canonical lift of an ordinary elliptic curve over a finite fields and its points counting’, J.Ramanujan Math.Soc. 15 (2000) 247–270.Google Scholar
18Schoof, R., ‘Elliptic curves over finite fields and the computation of square roots modp’ Math.Comp. 44 (1985) 483–494.Google Scholar
19Schoof, R., ‘Counting points on elliptic curves over finite fields’, J.Théor. Nombres Bordeaux 7 (1998) 219–254.CrossRefGoogle Scholar
20Serre, J. P., ‘Endomorphisms complètement continus des espaces de Banach p-adique’, Inst. Hautes Études Sci. Publ. Math 12 (1962) 69–85.CrossRefGoogle Scholar
21Sperber, S., ‘On the p-adic theory of exponential sums’, Amer.J.Math 108 (1983) 255–296.CrossRefGoogle Scholar
22Vercauteren, F., Preneel, B. and Vandewalle, J., ‘A memory efficient version of Satoh's algorithm’, Advances in Cryptology - EUROCRYPT 2001, Lecture Notes in Comput. Sci. 2045 (ed. Pfitzmann, B., Springer, 2001) 1–13.Google Scholar
23Wan, D., ‘Computing zeta functions over finite fields’, Contemp.Math 225 (1999) 131–141.CrossRefGoogle Scholar
24Wan, D., ‘Pure L-functions from algebraic geometry over finite fields’, Finite fields and applications (ed. Jungnickel, D. and Niederreiter, H., Springer, 2000) 437–461.Google Scholar
25Wan, D., ‘Algorithmic theory of zeta functions over finite fields’, preprint, 2001.Google Scholar