Supersingular elliptic curves, quaternion algebras and applications to cryptography

TL;DR

This survey explores supersingular isogeny graphs and their cryptographic applications, introduces recent results, and discusses generalizations to superspecial abelian varieties, aiming to be accessible to both novices and experts.

Contribution

It provides a broad overview of supersingular isogeny graphs, presents three recent results with sketches of proofs, and discusses generalizations to superspecial abelian varieties with real multiplication.

Findings

Presentation of three recent results on supersingular isogeny graphs

A general theorem on lattice generation over totally real fields

Clarification of folklore and vague statements in existing literature

Abstract

This paper contains a survey of supersingular isogeny graphs associated to supersingular elliptic curves and their various applications to cryptography. Within limitation of space, we attempt to address a broad audience and make this part widely accessible. For those graphs we also present three recent results and sketch their proofs. We then discuss a generalization to superspecial isogeny graphs associated to superspecial abelian varieties with real multiplication. These graphs were introduced by Charles, Goren and Lauter and so our discussion is brief. Motivated by their cryptographic applications, we prove a general theorem concerning generation of lattices over totally real fields by elements of specified norm. Throughout the paper we have attempted to clarify certain considerations that are either vaguely stated in the literature, or are folklore. We hope this paper will be useful both to a novice wishing to familiarize themselves with this very active area, and to the expert who may enjoy some vignettes and an overview of some new results.