Cryptography with right-angled Artin groups

TL;DR

This paper explores the use of right-angled Artin groups for cryptography, leveraging their efficient word problem and introducing new problems and schemes inspired by graph theory and group theory complexities.

Contribution

It introduces the Subgroup Isomorphism and Group Homomorphism problems for right-angled Artin groups and proposes new authentication schemes based on these problems.

Findings

Group Homomorphism and Graph Homomorphism problems are equivalent for right-angled Artin groups.

The Group Homomorphism problem is NP-complete.

Some right-angled Artin groups have an unsolvable Subgroup Isomorphism problem.

Abstract

In this paper we propose right-angled Artin groups as a platform for secret sharing schemes based on the efficiency (linear time) of the word problem. Inspired by previous work of Grigoriev-Shpilrain in the context of graphs, we define two new problems: Subgroup Isomorphism Problem and Group Homomorphism Problem. Based on them, we also propose two new authentication schemes. For right-angled Artin groups, the Group Homomorphism and Graph Homomorphism problems are equivalent, and the later is known to be NP-complete. In the case of the Subgroup Isomorphism problem, we bring some results due to Bridson who shows there are right-angled Artin groups in which this problem is unsolvable.