# Computing *l*-isogenies using the *p*-torsion

## Abstract

Computing *l*-isogenies between elliptic curves defined over a finite field F_{q} of small characteristic is of some importance for the computation of the cardinalities of elliptic curves using Schoof-Atkin-Elkies method. In a previous publication [3] we showed that this could be achieved in O(*l*^{3+ε}) multiplications in \(\mathbb{F}_q \)using formal groups. This method has been implemented by Lercier and Morain who obtained spectacular results in this direction [7, 8]. Nevertheless, the use of formal groups seems to be a serious deterent both for man and machine to perform such a work. More recently Lercier proposed an algorithm specific for characteristic 2 that has the same assymptotic complexity but is faster by some significative constant factor. In this paper we propose a general algorithm which does not use formal groups. Instead we take advantage of the elementary Galois properties of the p-torsion. This algorithm has the same complexity as the previous ones if we don't use fast multiplication techniques. But, contrary to the previous methods, it allows the use of fast multiplication for polynomials and then turns out to run in O(*l*^{2+ε}) multiplications in the field \(\mathbb{F}_q \). Our algorithm has also the advantage that it is made exclusively of very classical routines in polynomial and elliptic curve arithmetic. Also one may expect that the implementation of this method should require less work than the previous ones thus bringing new people to this kind of calculation.

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## References

- 1.A. O. L. Atkin. The number of points on an elliptic curve modulo a prime. Preprint, 1988.Google Scholar
- 2.A. O. L. Atkin. The number of points on an elliptic curve modulo a prime (ii). Preprint, 1992.Google Scholar
- 3.J.-M. Couveignes.
*Quelques calculs en théorie des nombres*. Université de Bordeaux, 1994.Google Scholar - 4.N.D. Elkies. Explicit isogenies. 1991.Google Scholar
- 5.Hiroshi Gunji. The hasse invariant and
*p*-division points of an elliptic curve.*Arch. Math. (Basel)*, 27:148–158, 1976.Google Scholar - 6.R. Lercier. Computing isogenies in characteristic 2. Submitted for publication at ANTS 2.Google Scholar
- 7.R. Lercier and F. Morain. Counting points on elliptic curves over \(F_{p^n } \)using Couveignes's algorithm. Research Report LIX/RR/95/09, École Polytechnique-LIX, September 1995.Google Scholar
- 8.R. Lercier and F. Morain. Counting the number of points on elliptic curves over finite fields: strategies and performances. In L.C. Guillou and J.-J. Quisquater, editors,
*Advances in cryptology, EUROCRYPT 95*, volume 921 of*Lecture notes in computer science*, pages 79–94. Springer, 1995.Google Scholar - 9.R. Lidl and H. Niederreiter.
*Introduction to finite fields and their applications*. Cambridge University Press, 1986.Google Scholar - 10.François Morain. Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques. Submitted for publication of the Actes des Journées Arithmétiques 1993, March 1994.Google Scholar
- 11.R. Schoof. Elliptic curves over finite fields and the computation of square roots mod
*p. Math. of Comp.*, 44:483–494, 1985.Google Scholar - 12.René Schoof. Counting points on elliptic curves over finite fields. to appear in the Journal de Théorie des nombres de Bordeaux.Google Scholar
- 13.J. F. Voloch. Explicit p-descent in characteristic
*p*. Comp. Math., 74:247–258, 1990.Google Scholar