Connecting Kani's Lemma and path-finding in the Bruhat-Tits tree to compute supersingular endomorphism rings

TL;DR

This paper presents a deterministic polynomial-time algorithm for computing supersingular elliptic curve endomorphism rings using path-finding in the Bruhat-Tits tree, improving on previous methods with restricted inputs.

Contribution

It introduces a new deterministic algorithm leveraging Bruhat-Tits tree navigation and higher-dimensional isogenies, requiring minimal initial data and surpassing prior subexponential approaches.

Findings

Algorithm runs in polynomial time under given conditions

Locally computes endomorphism rings at various primes

Improves efficiency over previous algorithms

Abstract

We give a deterministic polynomial time algorithm to compute the endomorphism ring of a supersingular elliptic curve in characteristic p, provided that we are given two noncommuting endomorphisms and the factorization of the discriminant of the ring $\mathcal{O}_0$ they generate. At each prime $q$ for which $\mathcal{O}_0$ is not maximal, we compute the endomorphism ring locally by computing a q-maximal order containing it and, when $q \neq p$, recovering a path to $\text{End}(E) \otimes \mathbb{Z}_q$ in the Bruhat-Tits tree. We use techniques of higher-dimensional isogenies to navigate towards the local endomorphism ring. Our algorithm improves on a previous algorithm which requires a restricted input and runs in subexponential time under certain heuristics. Page and Wesolowski give a probabilistic polynomial time algorithm to compute the endomorphism ring on input of a single non-scalar endomorphism. Beyond using techniques of higher-dimensional isogenies to divide endomorphisms by a scalar, our methods are completely different.