Look into the Mirror: Evolving Self-Dual Bent Boolean Functions

Bent Boolean functions are important objects in cryptography and coding theory, and there are several general approaches for constructing such functions. Metaheuristics proved to be a strong choice as they can provide many bent functions, even when the size of the Boolean function is large (e.g., more than 20 inputs). While bent Boolean functions represent only a small part of all Boolean functions, there are several subclasses of bent functions providing specific properties and challenges. One of the most interesting subclasses comprises (anti-)self-dual bent Boolean functions. This paper provides a detailed experimentation with evolutionary algorithms with the goal of evolving (anti-)self-dual bent Boolean functions. We experiment with two encodings and two fitness functions to directly evolve self-dual bent Boolean functions. Our experiments consider Boolean functions with sizes of up to 16 inputs, and we successfully construct self-dual bent functions for each dimension. Moreover, when comparing with the evolution of bent Boolean functions, we notice that the difficulty for evolutionary algorithms is rather similar. Finally, we also tried evolving secondary constructions for self-dual bent functions, but this direction provided no successful results.