Mathematics of Isogeny Based Cryptography

TL;DR

This paper provides an accessible overview of the mathematical foundations of isogeny-based cryptography, aimed at helping students navigate the complex literature and understand key concepts without exhaustive technical detail.

Contribution

It offers a pedagogical introduction to elliptic curves and their application in isogeny-based cryptography, bridging the gap between theory and cryptographic practice.

Findings

Clarifies fundamental elliptic curve concepts

Highlights applications of isogenies in cryptography

Serves as an educational resource for students

Abstract

These lectures notes were written for a summer school on Mathematics for post-quantum cryptography in Thi\`es, Senegal. They try to provide a guide for Masters' students to get through the vast literature on elliptic curves, without getting lost on their way to learning isogeny based cryptography. They are by no means a reference text on the theory of elliptic curves, nor on cryptography; students are encouraged to complement these notes with some of the books recommended in the bibliography. The presentation is divided in three parts, roughly corresponding to the three lectures given. In an effort to keep the reader interested, each part alternates between the fundamental theory of elliptic curves, and applications in cryptography. We often prefer to have the main ideas flow smoothly, rather than having a rigorous presentation as one would have in a more classical book. The reader will excuse us for the inaccuracies and the omissions.