Faster Algorithms for Isogeny Problems Using Torsion Point Images

There is a recent trend in cryptography to construct protocols based on the hardness of computing isogenies between supersingular elliptic curves. Two prominent examples are Jao-De Feo’s key exchange protocol and the resulting encryption scheme by De Feo-Jao-Plût. One particularity of the isogeny problems underlying these protocols is that some additional information is given as input, namely the image of some torsion points with order coprime to the isogeny. This additional information was used in several active attacks against the protocols but the current best passive attacks make no use of it at all. In this paper, we provide new algorithms that exploit the additional information provided in isogeny protocols to speed up the resolution of the underlying problems. Our techniques lead to heuristic polynomial-time key recovery on two nonstandard variants of De Feo-Jao-Plût’s protocols in plausible attack models. This shows that at least some isogeny problems are easier to solve when additional information is leaked.