Abstract
In this paper we present an algorithm for counting points on elliptic curves over a finite field \( \mathbb{F}_{p^n } \) of small characteristic, based on Satoh's algorithm. The memory requirement of our algorithm is O(n2), where Satoh's original algorithm needs O(n 3) memory. Furthermore, our version has the same run time complexity of O(n n+ε) bit operations, but is faster by a constant factor. We give a detailed description of the algorithm in characteristic 2 and show that the amount of memory needed for the generation of a secure 200-bit elliptic curve is within the range of current smart card technology.
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F.W.O. research assistant, sponsored by the Fund for Scientific Research - Flanders (Belgium).
References
A.O.L. Atkin. The number of points on an elliptic curve modulo a prime. Series of e-mails to the NMBRTHRY mailing list, 1992.
J.M. Couveignes. Quelques calculs en théorie des nombres. PhD thesis, Université de Bordeaux, 1994.
J.M. Couveignes. Computing l-isogenies with the p-torsion. ANTS-II, Lecture Notes in Comp. Sci., 1122:59–65, 1996.
J.A. Csirik. Counting the number of points on an elliptic curve on a low-memory device. 1998. Preprint.
M. Deuring. Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Univ. Hamburg, 14:197–272, 1941.
N. Elkies. Elliptic and modular curves over finite fields and related computational issues. Computational Perspectives on Number Theory, pages 21–76, 1998.
M. Fouquet, P. Gaudry, and R. Harley. On Satoh’s algorithm and its implementation. J. Ramanujan Math. Soc., 15:281–318, 2000.
H. Hasse. Beweis des Analogons der Riemannschen Vermutung für die Artinschen und F. K. Smidtschen Kongruenzzetafunctionen in gewissen elliptischen Fällen. Ges. d. Wiss. Nachrichten. Math.-Phys. Klasse, pages 253–262, 1933.
R. Lercier. Algorithmique des courbes elliptiques dans les corps finis. PhD thesis, L'École Polytechnique, Laboratoire D'Informatique, CNRS, Paris, June 1997.
J. Lubin, J.P. Serre, and J. Tate. Elliptic curves and formal groups. Lecture notes prepared in connection with the seminars held at the Summer Institute on Algebraic Geometry, Whitney Estate, Woods Hole, Massachusetts, 1964.
T. Satoh. The canonical lift of an ordinary elliptic curve over a finite field and its point counting. J. Ramanujan Math. Soc., 15:247–270, 2000.
R. Schoof. Elliptic curves over finite fields and the computation of square roots mod p. Math. Comput., 44:483–494, 1985.
J.P. Serre. Local Fields, volume 67 of Graduate Texts in Mathematics. Springer-Verlag, 1979.
B. Skjernaa. Satoh’s algorithm in characteristic 2. Preprint, 2000.
J. Vélu. Isogénies entre courbes elliptiques. C.R. Acad. Sc. Paris, 273:238–241, 1971.
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Vercauteren, F., Preneel, B., Vandewalle, J. (2001). A Memory Efficient Version of Satoh’s Algorithm. In: Pfitzmann, B. (eds) Advances in Cryptology — EUROCRYPT 2001. EUROCRYPT 2001. Lecture Notes in Computer Science, vol 2045. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44987-6_1
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DOI: https://doi.org/10.1007/3-540-44987-6_1
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