Curves, Jacobians, and Cryptography

TL;DR

This paper provides a comprehensive overview of abelian varieties, especially Jacobian varieties, covering their mathematical theory and applications in cryptography, including elliptic and hyperelliptic curve cryptography, with recent developments and open problems.

Contribution

It offers an extensive synthesis of the mathematical background and recent cryptographic applications of Jacobian varieties, highlighting new directions and open challenges.

Findings

Overview of abelian varieties and Jacobians in cryptography

Applications to elliptic and hyperelliptic curve cryptography

Discussion of open problems and future research directions

Abstract

The main purpose of this paper is to give an overview over the theory of abelian varieties, with main focus on Jacobian varieties of curves reaching from well-known results till to latest developments and their usage in cryptography. In the first part we provide the necessary mathematical background on abelian varieties, their torsion points, Honda-Tate theory, Galois representations, with emphasis on Jacobian varieties and hyperelliptic Jacobians. In the second part we focus on applications of abelian varieties on cryptography and treating separately, elliptic curve cryptography, genus 2 and 3 cryptography, including Diffie-Hellman Key Exchange, index calculus in Picard groups, isogenies of Jacobians via correspondences and applications to discrete logarithms. Several open problems and new directions are suggested.