Finding Good Random Elliptic Curves for Cryptosystems Deened over If 2 N

One of the main diiculties for implementing cryptographic schemes based on elliptic curves deened over nite elds is the necessary computation of the cardinality of these curves. In the case of nite elds IF2n, recent theoretical breakthroughs yield a signiicant speed up of the computations. Once described some of these ideas in the rst part of this paper, we show that our current implementation runs from 2 up to 10 times faster than what was done previously. In the second part, we exhibit a slight change of Schoof's algorithm to choose curves with a number of points \nearly" prime and so construct cryptosystems based on random elliptic curves instead of speciic curves as it used to be.