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

TL;DR

This work addresses evolving homogeneous bent Boolean functions with Evolutionary Algorithms, introducing a density-based framework to quantify the space of such functions by degree and variable count. It compares multiple encodings (GP symbolic, TT, rANF, wANF) and fitness criteria to search for quadratic and cubic homogeneous bent functions across $n=6$–$12$ variables. The authors derive exact counts and densities for quadratic and cubic cases (e.g., $|\\mathcal{HB}_{6,3}|=30$, $|\\mathcal{HB}_{8,3}|=293{,}760$) and show cubic bent functions are extremely sparse, existing only for restricted monomial counts; they demonstrate cubic functions can be found with carefully constrained encodings in low dimensions. The results highlight that encoding selection and explicit structural constraints are crucial for navigating the sparse cubic search space, offering novel cubic homogeneous bent functions discovered via evolving strategies and guiding future design of cryptographic primitives.

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.

Figures (3)

• Figure 1: Non-zero density values $\delta_{n,2,k}$.
• Figure 2: Non-zero density values $\delta_{n,3,k}$.
• Figure 3: Accumulated fitness values for all encodings, sizes 8-12.