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Computing isogenies between elliptic curves over \(F_{p^n}\) using Couveignes’s algorithm. (English) Zbl 0963.11032
Elliptic curves over finite fields have been used to factor integers [cf. H. W. Lenstra jun., Ann. Math. (2) 126, 649-673 (1987; Zbl 0629.10006) and 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 A. O. L. Atkin [The number of points on an elliptic curve modulo a prime, draft (1988), http://listserv.nodak.edu./archives/nmbry.html] and 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 [J.-M. Couveignes, Quelques calculs en théorie de nombres, Thèse, Université 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.

MSC:
11G20 Curves over finite and local fields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94A60 Cryptography
11Y16 Number-theoretic algorithms; complexity
Software:
ECPP; gfun
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References:
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