In this talk, we discuss the problem of computing the endomorphism ring of a supersingular elliptic curve from a full-rank suborder. We begin with a quantum algorithm that treats a special family of such suborders and then explain how to extend this approach to arbitrary full-rank suborders. Beyond its intrinsic number-theoretic interest, this problem is also of great importance in isogeny-based cryptography. It underlies the security of the key exchange protocol pSIDH and is closely related to the fundamental hard problem of computing endomorphism rings of supersingular elliptic curves. This talk is based on two joint works: the first with Muhammad Imran, Gábor Ivanyos, Péter Kutas, Antonin Leroux, and Christophe Petit, and the second with Christophe Petit.
