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.

翻译：我们提出了一种确定性多项式时间算法，用于计算特征p上超奇异椭圆曲线的自同态环，前提是给定两个非交换的自同态以及它们生成的环$\mathcal{O}_0$的判别式的因式分解。在每一个$\mathcal{O}_0$非极大的素数$q$处，我们通过计算包含它的一个q极大序来局部计算自同态环，并且当$q \neq p$时，在Bruhat-Tits树中恢复一条通往$\text{End}(E) \otimes \mathbb{Z}_q$的路径。我们利用高维同源的技术来导航至局部自同态环。我们的算法改进了先前的一种算法，该算法需要受限的输入并在某些启发式假设下以亚指数时间运行。Page和Wesolowski给出了一种概率多项式时间算法，在输入单个非标量自同态时计算自同态环。除了使用高维同源的技术将自同态除以标量外，我们的方法与之完全不同。