Degree is Important: On Evolving Homogeneous Boolean Functions

TL;DR

The paper tackles the design of homogeneous bent Boolean functions via Evolutionary Algorithms, addressing a sparsely explored problem in cryptographic function design. By comparing bitstring, restricted, and symbolic encodings across multiple fitness objectives (homogeneity, bentness, and their combination), the authors reveal that quadratic homogeneous bent functions can be reliably obtained, especially using a restricted encoding with a bent-specific objective, while cubic cases remain elusive. The study highlights the influence of search space structure and fitness landscape on success, and suggests directions to improve higher-degree results and deepen understanding of homogeneous bent function search. These insights have implications for efficient evaluation and synthesis of cryptographic primitives where fixed algebraic degree and maximal nonlinearity are desirable.

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.