On Counts and Densities of Homogeneous Bent Functions: An Evolutionary Approach

TL;DR

This paper explores using Evolutionary Algorithms to generate homogeneous bent Boolean functions with high nonlinearity, introducing the concept of density to guide the search for quadratic and cubic functions across various variable counts.

Contribution

It introduces the concept of density for homogeneous bent functions and demonstrates an evolutionary approach to efficiently find quadratic and cubic bent functions.

Findings

Successful evolution of quadratic and cubic bent functions

Introduction of density as a guiding metric

Effective algorithmic design for different variable counts

Abstract

Boolean functions with strong cryptographic properties, such as high nonlinearity and algebraic degree, are important for the security of stream and block ciphers. These functions can be designed using algebraic constructions or metaheuristics. This paper examines the use of Evolutionary Algorithms (EAs) to evolve homogeneous bent Boolean functions, that is, functions whose algebraic normal form contains only monomials of the same degree and that are maximally nonlinear. We introduce the notion of density of homogeneous bent functions, facilitating the algorithmic design that results in finding quadratic and cubic bent functions in different numbers of variables.