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We give an algorithm for computing inseparable endomorphisms of asupersingular elliptic curve $E$ defined over $\mathbb F_{p^2}$, which,conditional on GRH, runs in expected $O(\sqrt{p}(\log p)^2(\log\log p)^3)$time. With two calls to this algorithm, we compute a Bass suborder of$\text{End}(E)$, improving on the results of Eisentr\"{a}ger, Hallgren,Leonardi, Morrison, and Park (ANTSXIV) who only gave a heuristic algorithm forcomputing a Bass suborder. We further improve on the results of Eisentr\"{a}geret al. by removing the heuristics involved in an algorithm for recovering$\text{End}(E)$ from a Bass suborder. We conclude with an argument that $O(1)$endomorphisms generated by our algorithm along with negligible overhead sufficeto compute $\text{End}(E)$, conditional on a heuristic assumption about thedistribution of the discriminants of these endomorphisms.

Computing supersingular endomorphism rings using inseparable endomorphisms