Algebraic groups at the heart of post-quantum cryptography

Introduction

Contemporary public-key cryptography builds its foundations on a handful of computational problems rooted in arithmetic and geometry. The vast majority of deployed cryptosystems rely on two classical problems: computing discrete logarithms and factoring integers. These problems would not resist a large-scale quantum computer. Research on quantum technology is accelerating, endangering the world's information systems. New foundations are being proposed by the cryptologic community, promising post-quantum security, but suffering in many aspects from the lack of adequate scrutiny.

Emerging Post-Quantum Candidates

Emerging post-quantum candidates can be naturally embedded into rich and modern mathematical theories. This is the case for:

1. Lattice-based cryptography
2. Isogeny-based cryptography

These two areas share surprising connections once recast in the world of algebraic groups. Algebraic groups are at the forefront of modern mathematics. Their study across the past century has blossomed with the development of powerful theories, such as representation theory and automorphic forms.

The Dialogue Between Disciplines

Yet, the dialogue between arithmeticians and cryptologists has been sparse. The link between algebraic groups and the objects of post-quantum cryptography has been mostly anecdotal.

Project Goals

This project brings this connection to the forefront, observing that the theory of algebraic groups shines a powerful light on problems raised by lattice-based and isogeny-based cryptography.

Unique Abilities of Algebraic Groups

It has the unique ability to turn the set of all instances of a computational problem into one meaningful object in itself — a 'moduli space' — with:

• An arithmetic structure
• A geometry
• A topology
• A harmonic theory

Exposing these problems to the powerful artillery of modern arithmetic will lead to cryptanalytic breakthroughs, security proofs, and the construction of cutting-edge cryptosystems.