In recent years, isogeny-based cryptosystems have captured the attention of the math/crypto community for their conjectured resistance to quantum attacks. In this context, the most promising protocols have supersingular elliptic curves defined over finite fields as central objects, and their security is based on the mathematical problem of computing an isogeny between two supersingular elliptic curves E and E'. It has been shown that this problem can be reduced to the computation of the endomorphism rings of E and E'. In this talk, after reviewing the mathematical and cryptographic context, we will present an improved algorithm for computing the endomorphism ring of a supersingular elliptic curve over a finite field. This is a joint work with Jenny G. Fuselier, Mark Kozek, Travis Morrison and Changningphaabi Namoijam.