Supersingular isogeny graphs and orientations in cryptography

Abstract

We discuss the theory of isogeny graphs; we mainly consider supersingular isogeny graphs, where the vertices of the graphs are given by j-invariants of supersingular elliptic curves over some finite field and the edges denote the l-degree isogenies between the elliptic curves that have those j-invariants. We look at some cryptographic protocols, both key exchange protocols and a Sigma-protocol, that use supersingular isogeny graphs. Finally, we introduce orientations, which are injective ring homomorphisms that embed quadratic orders into the endomorphism algebras of (supersingular) elliptic curves. We consider the key exchange protocol OSIDH, which uses orientations and we construct a 3-move protocol that uses orientations and could potentially be a Sigma-protocol.