Abstract : Computing endomorphism rings of supersingular elliptic curves is an important problem in computational number theory, and it is also closely connected to the security of some of the recently proposed isogeny-based cryptosystems. In this talk we give a new algorithm for computing the endomorphism ring of a supersingular elliptic curve. The algorithm works by first computing two cycles in the l-isogeny graph that create an order in the endomorphism ring of the curve E. Then we determine which maximal order containing this order is the endomorphism ring of E.
This is joint work with Hallgren, Leonardi, Morrison and Park.