Graphs, right-angled Artin groups and cryptography

Abstract: Our research in this topic can be divided in two related lines. First, the right-angled Artin groups, which are defined by means of a graph, offer a natural bridge between Graph Theory and Algebra. Usually, the graph-theoretic properties are easy to translate in terms of group-theoretic features, but it is very difficult to walk the other way around, describing the graph in terms of intrinsic invariants of the groups. In joint work with D. Kahrobaei and T. Koberda, we have found such intrinsic group-theoretic descriptions of the notions of asymmetric graph, coloring, hamiltonicity and expander graphs. 
This work in particular allows to use well-known complexities in algorithmic graph-theoretic problems in the definition and implementation of group-theoretic cryptographic protocols, which is our second line of research in this context. In joint work with the mentioned researchers, we have defined secret sharing and authentication schemes based on right-angled Artin groups, a zero-knowledge protocol based on the group-theoretic notion of hamiltonicity, or public-key and symmetric-key protocols based on the Thurston norm of the first cohomology of hyperbolic manifolds.