Pairing the Volcano
Résumé
Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges are $l$-isogenies. Algorithms allowing to travel on these graphs were developed by Kohel in his thesis (1996) and later on, by Fouquet and Morain (2001). However, up to now, no method was known, to predict, before taking a step on the volcano, the direction of this step. Consequently, in Kohel's and Fouquet-Morain's algorithms, we always take more steps than necessary, before choosing the right direction. Suppose we know the cardinality of the curve. Given a point $P$ of order $l$ on the elliptic curve, we develop a method to decide whether the subgroup generated by $P$ is the kernel of a horizontal isogeny, a descending or an ascending one. In most cases, our method is very efficient and gives simple algorithms, which are more efficient than previous ones. In the other cases, we show that the two methods should be combined in order to obtain efficient algorithms.
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