Evaluating Large Degree Isogenies between Elliptic Curves

Abstract

An isogeny between elliptic curves is an algebraic morphism which is a group homomorphism. Many applications in cryptography require evaluating large degree isogenies between elliptic curves efficiently. For ordinary curves of the same endomorphism ring, the previous fastest algorithm known has a worst case running time which is exponential in the length of the input. In this thesis we solve this problem in subexponential time under reasonable heuristics. We give two versions of our algorithm, a slower version assuming GRH and a faster version assuming stronger heuristics. Our approach is based on factoring the ideal corresponding to the kernel of the isogeny, modulo principal ideals, into a product of smaller prime ideals for which the isogenies can be computed directly. Combined with previous work of Bostan et al., our algorithm yields equations for large degree isogenies in quasi-optimal time given only the starting curve and the kernel.