Applications of Finite non-Abelian Simple Groups to Cryptography in the Quantum Era

TL;DR

This paper explores how finite non-abelian simple groups can be applied to cryptography, especially in the quantum era, by examining group-theoretic problems, hash functions, and homomorphic encryption.

Contribution

It provides a comprehensive review of applications of finite simple groups in cryptography, connecting group theory with cryptographic constructions and quantum considerations.

Findings

Group-theoretic factorization problems are central to cryptographic schemes.

Group theoretical hash functions are reviewed as potential cryptographic primitives.

Discussion on fully homomorphic encryption using simple groups in the quantum context.

Abstract

The theory of finite simple groups is a (rather unexplored) area likely to provide interesting computational problems and modelling tools useful in a cryptographic context. In this note, we review some applications of finite non-abelian simple groups to cryptography and discuss different scenarios in which this theory is clearly central, providing the relevant definitions to make the material accessible to both cryptographers and group theorists, in the hope of stimulating further interaction between these two (non-disjoint) communities. In particular, we look at constructions based on various group-theoretic factorization problems, review group theoretical hash functions, and discuss fully homomorphic encryption using simple groups. The Hidden Subgroup Problem is also briefly discussed in this context.