Explicit isogenies in quadratic time in any characteristic

TL;DR

The paper introduces a new algorithm for computing isogenies between elliptic curves over finite fields, which operates in quadratic time regardless of the characteristic, offering efficiency improvements especially in medium to large characteristic cases.

Contribution

It presents a novel isogeny computation algorithm that leverages the structure of the $ ext{l}$-torsion, improving over previous methods in certain characteristic regimes.

Findings

Algorithm runs in $ ilde{O}(r^2 + \sqrt{r} ext{log}(q))$ operations.

Outperforms previous algorithms in medium and large characteristic cases.

Provides a practical alternative for isogeny computations in cryptographic applications.

Abstract

Consider two elliptic curves $E,E'$ defined over the finite field $\mathbb{F}_q$, and suppose that there exists an isogeny $\psi$ between $E$ and $E'$. We propose an algorithm that determines $\psi$ from the knowledge of $E$, $E'$ 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, that 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 $\tilde{O}(r^2 + \sqrt{r} \log(q))$; this makes it an interesting alternative for the medium- and large-characteristic cases.