NegaBent, No Regrets: Evolving Spectrally Flat Boolean Functions
Claude Carlet, Marko Ðurasevic, Ermes Franch, Domagoj Jakobovic, Luca Mariot, Stjepan Picek
TL;DR
The paper tackles the challenge of constructing negabent and bent-negabent Boolean functions using evolutionary computation, noting the large search space and spectral constraints. It compares bitstring truth-table encoding with symbolic tree-based genetic programming and introduces a fitness function that jointly optimizes nonlinearity under both standard and negaperiodic transforms, while penalizing extreme spectrum values. The results show that while bitstring encoding struggles at larger sizes, symbolic GP consistently evolves highly nonlinear negabent or bent-negabent functions across tested dimensions, highlighting the method's effectiveness for spectral-design problems. This work advances cryptographic function design by validating evolutionary methods as practical tools for exploring specialized transform-domain properties and sets the stage for incorporating additional constraints like balancedness and degree in future studies.
Abstract
Negabent Boolean functions are defined by having a flat magnitude spectrum under the nega-Hadamard transform. They exist in both even and odd dimensions, and the subclass of functions that are simultaneously bent and negabent (bent-negabent) has attracted interest due to the combined optimal periodic and negaperiodic spectral properties. In this work, we investigate how evolutionary algorithms can be used to evolve (bent-)negabent Boolean functions. Our experimental results indicate that evolutionary algorithms, especially genetic programming, are a suitable approach for evolving negabent Boolean functions, and we successfully evolve such functions in all dimensions we consider.