Explicit isogenies in quadratic time in any characteristic

Luca de Feo (1) , Cyril Hugounenq (1) , Jérôme Plût (2) , Eric Schost (3)

1 LMV - Laboratoire de Mathématiques de Versailles
2 ANSSI - Agence nationale de la sécurité des systèmes d'information
3 UWO - University of Western Ontario

Consider two ordinary elliptic curves $E,E'$ defined over a finite field $\F_q$, and suppose that there exists an isogeny $\psi$ between $E$ and $E'$. We propose an algorithm that determines $\psi$ from the knowledge of $E$, $E'$ and of its degree $r$, by using the structure of the $ℓ$-torsion of the curves (where $ℓ$~is a prime different from the characteristic~$p$ of the base field). Our approach is inspired by a previous algorithm due to Couveignes, that involved computations using the $p$-torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of~$\tildO(r^2) p^{O(1)}$ base field operations. On the other hand, the cost of our algorithm is $\tildO(r^2) \log(q)^{O(1)}$, for a large class of inputs; this makes it an interesting alternative for the medium- and large-characteristic cases.

Mots clés: Elliptic Curve; isogeny; symbolic computation; isogeny volcanoes

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]; Calcul formel [cs.SC]; Cryptographie et sécurité [cs.CR]

Luca de Feo, Cyril Hugounenq, Jérôme Plût, Eric Schost. Explicit isogenies in quadratic time in any characteristic. 2016. ⟨hal-01320940⟩