Evolving Constructions for Balanced, Highly Nonlinear Boolean Functions

TL;DR

This paper explores the use of genetic programming to evolve constructions for balanced, highly nonlinear Boolean functions, demonstrating GP's ability to find generalizable and novel solutions, including known constructions, for larger function sizes.

Contribution

It introduces a GP-based approach to discover Boolean function constructions with high nonlinearity, capable of generalizing across multiple sizes, including known and novel solutions.

Findings

GP can find constructions that generalize to multiple sizes

GP discovers many equivalent constructions with different representations

The simplest solution is a known indirect sum construction

Abstract

Finding balanced, highly nonlinear Boolean functions is a difficult problem where it is not known what nonlinearity values are possible to be reached in general. At the same time, evolutionary computation is successfully used to evolve specific Boolean function instances, but the approach cannot easily scale for larger Boolean function sizes. Indeed, while evolving smaller Boolean functions is almost trivial, larger sizes become increasingly difficult, and evolutionary algorithms perform suboptimally. In this work, we ask whether genetic programming (GP) can evolve constructions resulting in balanced Boolean functions with high nonlinearity. This question is especially interesting as there are only a few known such constructions. Our results show that GP can find constructions that generalize well, i.e., result in the required functions for multiple tested sizes. Further, we show that GP evolves many equivalent constructions under different syntactic representations. Interestingly, the simplest solution found by GP is a particular case of the well-known indirect sum construction.