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

2025-11-16

AI Summary

This paper addresses the construction of homogeneous Bent Boolean functions—i.e., Boolean functions with maximal nonlinearity whose algebraic normal form (ANF) consists exclusively of monomials of identical degree. Due to the extreme sparsity and combinatorial intractability of the solution space, we introduce the novel concept of *function density* to characterize the distribution of feasible solutions. Leveraging this insight, we design an evolutionary algorithm specifically tailored to the homogeneity constraint: it employs ANF-based encoding, a custom fitness function prioritizing both bentness and homogeneity, and specialized genetic operators preserving degree uniformity. Experiments successfully evolve quadratic and cubic homogeneous Bent functions across multiple input dimensions, demonstrating both effectiveness and scalability. To our knowledge, this is the first systematic application of evolutionary computation to construct high-nonlinearity homogeneous Bent functions. The approach establishes a new paradigm for automated generation of cryptographically significant Boolean functions that are notoriously difficult to construct analytically.

Technology Category

• Search and Optimization: Evolutionary Computation
• Machine Learning: Evolutionary Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Evolving homogeneous bent Boolean functions using Evolutionary Algorithms
• Investigating density of homogeneous bent functions for algorithmic design
• Finding quadratic and cubic bent functions across variable numbers

Innovation

• Evolutionary Algorithms evolve homogeneous bent functions
• Introduces density concept for homogeneous bent functions
• Finds quadratic and cubic bent functions algorithmically