Abstract
We present a variant of an algorithm of Oliver Atkin for counting the number of points on an elliptic curve over a finite field. We describe an implementation of this algorithm for prime fields. We report on the use of this implementation to count the number of points on a curve over \(\mathbb{F}\) p, where p is a 375-digit prime.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A.O.L. Atkin: The number of points on an elliptic curve modulo a prime, unpublished manuscript
J. Buchmann, V. Müller: Computing the number of points on an elliptic curve over finite fields, Proceedings of ISSAC 1991, 179–182
J. M. Couveignes, F. Morain: Schoof's algorithm and isogeny cycles, Proceedings of ANTS 1994
N. Koblitz: Elliptic curve cryptosystems, Mathematics of Computation, 48 (1987), 203–209
V. Miller: Uses of elliptic curves in cryptography, Advances in Cryptology: Proceedings of Crypto '85, Lecture Notes in Computer Science, 218 (1986), Springer-Verlag, 417–426
V. Müller: Die Berechnung der Punktanzahl elliptischer Kurven über endlichen Körpern der Charakteristik gröβer 3, Thesis, to be published
P. L. Montgomery: Speeding the Pollard and Elliptic Curve Methods for Factorization, Mathematics of Computation, 48 (1987), 243–264
A. Odlyzko: Discrete logarithms and their cryptographic significance, Advances in Cryptology: Proceedings of Eurocrypt '84, Lecture Notes in Computer Science, 209 (1985), Springer-Verlag, 224–314
R. Roth, Th. Setz: LiPS: a system for distributed processing on workstations, University of Saarland, 1993
R. Schoof: Elliptic curves over finite fields and the computation of square roots mod p, Mathematics of Computation, 44 (1985), 483–494
V. Shoup: Practical computations with large polynomials over finite fields, in preparation (1994).
J. Silverman: The arithmetic of elliptic curves, Springer-Verlag, 1986
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lehmann, F., Maurer, M., Müller, V., Shoup, V. (1994). Counting the number of points on elliptic curves over finite fields of characteristic greater than three. In: Adleman, L.M., Huang, MD. (eds) Algorithmic Number Theory. ANTS 1994. Lecture Notes in Computer Science, vol 877. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58691-1_44
Download citation
DOI: https://doi.org/10.1007/3-540-58691-1_44
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58691-3
Online ISBN: 978-3-540-49044-9
eBook Packages: Springer Book Archive