Explicit Isogenies In Quadratic Time In Any Characteristic

Consider two ordinary elliptic curves E, E' defined over a finite field F-q, and suppose that there exists an isogeny phi between E and E'. We propose an algorithm that determines phi, from the knowledge of E, E' and of its degree r, by using the structure of the f-torsion of the curves (where f 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 (O) over tilde (r(2)) p(O(1)) base field operations. On the other hand, the cost of our algorithm is (O) over tilde (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.