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：我们提出一种确定性多项式时间算法，用于计算特征p下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$\mathcal{O}_0$的判别式的因子分解。对于每个使得$\mathcal{O}_0$非极大的素数$q$，我们通过计算包含该环的q-极大阶来局部计算自同态环，并在$q \neq p$时，在Bruhat-Tits树中恢复通往$\text{End}(E) \otimes \mathbb{Z}_q$的路径。我们利用高维同源的技术来导航至局部自同态环。该算法改进了此前需要受限输入且在某些启发式假设下运行于亚指数时间的算法。Page和Wesolowski给出了一种概率多项式时间算法，该算法在输入单个非标量自同态时即可计算自同态环。我们的方法与之完全不同的，不仅采用了利用高维同源技术将自同态除以标量的方法。