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Computing endomorphism rings of supersingular elliptic curves is an importantproblem in computational number theory, and it is also closely connected to thesecurity of some of the recently proposed isogeny-based cryptosystems. In thispaper we give a new algorithm for computing the endomorphism ring of asupersingular elliptic curve $E$ that runs, under certain heuristics, in time$O((\log p)^2p^{1/2})$. The algorithm works by first finding two cycles of acertain form in the supersingular $\ell$-isogeny graph $G(p,\ell)$, generatingan order $\Lambda \subseteq \operatorname{End}(E)$. Then all maximal orderscontaining $\Lambda$ are computed, extending work of Voight. The final step isto determine which of these maximal orders is the endomorphism ring. As part ofthe cycle finding algorithm, we give a lower bound on the set of all$j$-invariants $j$ that are adjacent to $j^p$ in $G(p,\ell)$, answering aquestion in arXiv:1909.07779.

Computing endomorphism rings of supersingular elliptic curves and connections to pathfinding in isogeny graphs