Date: 02 Jun 2005

Computing isogenies in \(\mathbb{F}_{2^n } \)

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Abstract

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 \(\mathbb{F}_{2^n } \)is the computation of isogenies of prime degree ℓ. The best method so far is due to Couveignes and needs asymptotically O(ℓ3) 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.