Evolutionary Construction of Perfectly Balanced Boolean Functions

TL;DR

This paper explores using genetic programming and genetic algorithms to construct perfectly balanced Boolean functions with high nonlinearity, revealing that GAs outperform GP in this specific task, contrary to previous results.

Contribution

It introduces a novel approach applying GAs and GP to evolve perfectly balanced Boolean functions with desirable cryptographic properties.

Findings

GA with weightwise balanced representation outperforms GP in finding nonlinear WPB functions

GAs are more effective than GP for evolving perfectly balanced Boolean functions in this context

Contrasts previous findings where GP was superior for globally balanced functions

Abstract

Finding Boolean functions suitable for cryptographic primitives is a complex combinatorial optimization problem, since they must satisfy several properties to resist cryptanalytic attacks, and the space is very large, which grows super exponentially with the number of input variables. Recent research has focused on the study of Boolean functions that satisfy properties on restricted sets of inputs due to their importance in the development of the FLIP stream cipher. In this paper, we consider one such property, perfect balancedness, and investigate the use of Genetic Programming (GP) and Genetic Algorithms (GA) to construct Boolean functions that satisfy this property along with a good nonlinearity profile. We formulate the related optimization problem and define two encodings for the candidate solutions, namely the truth table and the weightwise balanced representations. Somewhat surprisingly, the results show that GA with the weightwise balanced representation outperforms GP with the classical truth table phenotype in finding highly nonlinear WPB functions. This finding is in stark contrast to previous findings on the evolution of globally balanced Boolean functions, where GP always performs best.