Explicit isogenies in quadratic time in any characteristic

Abstract

Consider two ordinary elliptic curves$E,E^{\prime }$defined over a finite field$\mathbb{F}_{q}$, and suppose that there exists an isogeny$\unicode[STIX]{x1D713}$between$E$and$E^{\prime }$. We propose an algorithm that determines$\unicode[STIX]{x1D713}$from the knowledge of$E$,$E^{\prime }$and of its degree$r$, by using the structure of the$\ell$-torsion of the curves (where $\ell$ is a prime different from the characteristic $p$of the base field). Our approach is inspired by a previous algorithm due to Couveignes, which involved computations using the$p$-torsion on the curves. The most refined version of that algorithm, due to De Feo, has a complexity of $\tilde{O} (r^{2})p^{O(1)}$base field operations. On the other hand, the cost of our algorithm is$\tilde{O} (r^{2})\log (q)^{O(1)}$, for a large class of inputs; this makes it an interesting alternative for the medium- and large-characteristic cases.