Improved algorithms for finding fixed-degree isogenies between supersingular elliptic curves

Abstract

Finding isogenies between supersingular elliptic curves is a natural algorithmic problem which is known to be equivalent to computing the curves\u27 endomorphism rings. When the isogeny is additionally required to have a specific degree $d$, the problem appears to be somewhat different in nature, yet it is also considered a hard problem in isogeny-based cryptography. Let $E_1,E_2$ be supersingular elliptic curves over $\mathbb{F}_{p^2}$. We present improved classical and quantum algorithms that compute an isogeny of degree $d$ between $E_1$ and $E_2$ if it exists. Let the sought-after degree be $d = p^{1/2+ \epsilon}$ for some $\epsilon>0$. Our essentially memory-free algorithms have better time complexity than meet-in-the-middle algorithms, which require exponential memory storage, in the range $1/2\leq\epsilon\leq 3/4$ on a classical computer and quantum improvements in the range $0<\epsilon<5/2$