an:01355981
Zbl 0963.11032
Lercier, R.; Morain, F.
Computing isogenies between elliptic curves over \(F_{p^n}\) using Couveignes's algorithm
EN
Math. Comput. 69, No. 229, 351-370 (2000).
00061298
2000
j
11G20 11T71 94A60 11Y16
elliptic curves over finite fields; isogenies; formal groups; Schoof's algorithm
Elliptic curves over finite fields have been used to factor integers [cf. \textit{H. W. Lenstra jun.}, Ann. Math. (2) 126, 649-673 (1987; Zbl 0629.10006) and \textit{P. L. Montgomery}, Math. Comput. 48, 243-264 (1987; Zbl 0608.10005)]. One of the important steps in the solution of this problem is to compute the number of elliptic curves over a given finite field. Schoof's polynomial time algorithm to solve this problem was efficiently implemented due to the work of \textit{A. O. L. Atkin} [The number of points on an elliptic curve modulo a prime, draft (1988), \url{http://listserv.nodak.edu./archives/nmbry.html}] and \textit{N. D. Elkies} [Explicit isogenies, draft (1991) and Elliptic and modular curves over finite fields and related computational issues, in Computational Perspectives in Number Theory, AMS/IP Stud. Adv. Math. 7, 21-76 (1998; Zbl 0915.11036)]. The methods used were good for large characteristics, however could not be used when the characteristic is small. The first answer to this situation appeared in Couveigne's thesis [\textit{J.-M. Couveignes}, Quelques calculs en th??orie de nombres, Th??se, Universit?? Bordeaux I, July 1994].
The aim of the paper under review is to explain how Couveigne's algorithm can be implemented in an efficient way.
Am??lcar Pacheco (Rio de Janeiro)
Zbl 0629.10006; Zbl 0608.10005; Zbl 0915.11036