Degree is Important: On Evolving Homogeneous Boolean Functions

TL;DR

This paper explores using Evolutionary Algorithms to design homogeneous bent Boolean functions with high nonlinearity, finding quadratic solutions but not cubic ones, and compares different encoding and fitness strategies.

Contribution

It evaluates multiple EA configurations for designing homogeneous bent Boolean functions, highlighting the effectiveness of a specific GA approach for quadratic functions.

Findings

EAs successfully find quadratic homogeneous bent functions

The best method is a GA with restricted encoding

No cubic homogeneous bent functions were found

Abstract

Boolean functions with good cryptographic properties like high nonlinearity and algebraic degree play an important in the security of stream and block ciphers. Such functions may be designed, for instance, by algebraic constructions or metaheuristics. This paper investigates the use of Evolutionary Algorithms (EAs) to design homogeneous bent Boolean functions, i.e., functions that are maximally nonlinear and whose algebraic normal form contains only monomials of the same degree. In our work, we evaluate three genotype encodings and four fitness functions. Our results show that while EAs manage to find quadratic homogeneous bent functions (with the best method being a GA leveraging a restricted encoding), none of the approaches result in cubic homogeneous bent functions.