Orienting supersingular isogeny graphs

TL;DR

This paper introduces a new category of oriented supersingular elliptic curves and explores their isogeny graphs, leading to a novel cryptographic protocol called OSIDH, which generalizes existing supersingular isogeny-based protocols.

Contribution

It defines $ ext{O}$-oriented supersingular elliptic curves and analyzes their isogeny graphs, culminating in the development of the OSIDH protocol, a new isogeny-based cryptographic scheme.

Findings

Properties of oriented supersingular isogeny graphs derived

Introduction of the OSIDH protocol

Generalization of CSIDH protocol

Abstract

We introduce a category of $\mathcal{O}$-orientedsupersingularellipticcurves and derive properties of the associated oriented and nonoriented $\ell$-isogeny supersingular isogeny graphs. As an application we introduce an oriented super-singular isogeny Diffie-Hellman protocol (OSIDH), analogous to the supersingular isogeny Diffie-Hellman (SIDH) protocol and generalizing the commutative supersingular isogeny Diffie-Hellman (CSIDH) protocol.