Evolutionary algorithm solution of the multiple conjugacy search problem in groups, and its applications to cryptography

Abstract.

We consider the multiple conjugacy search problem over a subclass of partially commutative groups and experimentally attack it with a genetic algorithm hybridised with a “length attack”. We detail symbolic computation of words over the groups, constructing functions which measure certain statistics of those words. By experimentation, the hybrid algorithm is shown to be effective, showing that the standard conjugacy search problem is harder than the multiple conjugacy search problem for our groups. Moreover, some intuitive methods of increasing problem difficulty are overcome by the algorithm, and in fact make the problem easier to solve. We show our algorithm is efficient, comparing well with traditional approaches in groups that are statistically similar. Finally, via “approximation” of braid groups by our subclass, we consider implications of the attack on certain cryptosystems, pointing to further work in the discipline of group-theoretic cryptography.