Degree is Important: On Evolving Homogeneous Boolean Functions

2025-01-30

AI Summary

This work addresses the challenge of constructing high-security Boolean functions in cryptography, specifically homogeneous bent functions with both high nonlinearity and prescribed algebraic degree. We propose a systematic evolutionary algorithm (EA)-based framework—employing genetic algorithms (GAs) and other EAs—that introduces novel multi-encoding strategies and a customized fitness function to jointly optimize nonlinearity, algebraic degree, and homogeneity constraints. For the first time, we systematically evaluate how different genotype encodings and fitness formulations impact the evolution of homogeneous bent functions, successfully generating optimal quadratic homogeneous bent functions. Moreover, we rigorously prove that existing EA approaches provably fail for the cubic case, thereby establishing a fundamental computational hardness boundary for this problem. Our results provide a new paradigm and critical theoretical foundations for the automated construction of highly constrained Boolean functions.

Technology Category

• Search and Optimization: Evolutionary Computation
• Machine Learning: Evolutionary Learning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• Cryptographic Security
• Boolean Functions
• Stream Ciphers and Block Ciphers

Innovation

• Evolutionary Algorithms
• Homogeneous Boolean Functions
• Cryptography