Computing isogenies in $GF(2^n)$
Résumé
Contrary to what happens over prime fields of large characteristic,
the main cost when counting the number of points of an elliptic
curve $E$ over $GF(2^n)$ is the computation of isogenies of prime
degree $\ell$. The best method so far is due to Couveignes and
needs asymptotically $O(\ell^3)$ field operations. We outline in
this article some nice properties satisfied by these isogenies and
show how we can get from them a new algorithm that seems to perform
better in practice than Couveignes's though of the same complexity.
On a representative problem, we gain a speed-up of 5 for the whole
computation.