Finding Good Random Elliptic Curves for Cryptosystems Defined over F2n

One of the main difficulties for implementing cryptographic schemes based on elliptic curves defined over finite fields is the necessary computation of the cardinality of these curves. In the case of finite fields F2n, recent theoretical breakthroughs yield a significant speed up of the computations. Once described some of these ideas in the first 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 specific curves as it used to be.

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