The Diffie-Hellman scheme is the most well-known key exchange protocol. However, its security relies on the Discrete Logarithm problem which is not a difficult problem for quantum computers. In this presentation, we will first focus on the task of adapting Diffie-Hellman to a post-quantum context by replacing the cyclic groups in Diffie-Hellman by group actions. We will see how elliptic curves and isogenies can be used as a framework for realizing this task. In particular, we will introduce the notion of orientations — embeddings from quadratic number fields into endomorphism rings of supersingular elliptic curves — and see how they induce suitable group actions for this objective. Finally, we will study the computational complexity of computing the endomorphism ring of supersingular elliptic curves given an orientation, which is an important problem that naturally arises from these constructions.
