The choice of an elliptic curve for the implementation of an elliptic curve cryptosystem requires counting the number of points on such a curve over a finite field. An improvement of Schoof's algorithm for counting the number of rational points on an elliptic curve defined over a finite field takes advantage of some factor of the division polynomials. In this paper, we study the possible factorisations of such division polynomials.
References
Elkies, N. D.: Elliptic and modular curves over finite fields and related computational issues. Computational perspectives on number theory (Chicago, IL, 1995), AMS/IP Stud. Adv. Math., vol. 7, Amer. Math. Soc., Providence, RI, pp. 21–76 (1998). MR1486831 Elkies, N. D.: Elliptic and modular curves over finite fields and related computational issues. Computational perspectives on number theory (Chicago, IL, 1995), AMS/IP Stud. Adv. Math., vol. 7, Amer. Math. Soc., Providence, RI, pp. 21–76 (1998). MR1486831
