A Discrete Particle Swarm Optimizer for the Design of Cryptographic Boolean Functions
Luca Mariot, Alberto Leporati, Luca Manzoni
TL;DR
We tackle the challenge of designing balanced Boolean functions with strong cryptographic properties by introducing a discrete PSO with a Hamming weight–preserving swap update and Hill Climbing, complemented by meta-optimization of velocity parameters. The CGA-tuned PSO is evaluated on $n$ from 7 to 12, yielding functions with competitive nonlinearity, correlation immunity, and propagation criteria relative to prior heuristics, particularly for smaller $n$. However, performance declines for larger $n$ and CI(2)/AC$_{max}$ metrics, indicating the need for further parameter and update refinements; CGA incurs substantial computational cost compared to LUS. The work contributes a scalable, competitive heuristic framework for cryptographic Boolean-function design and points toward future improvements via alternative fitness designs and refined balance-preserving updates.
Abstract
A Particle Swarm Optimizer for the search of balanced Boolean functions with good cryptographic properties is proposed in this paper. The algorithm is a modified version of the permutation PSO by Hu, Eberhart and Shi which preserves the Hamming weight of the particles positions, coupled with the Hill Climbing method devised by Millan, Clark and Dawson to improve the nonlinearity and deviation from correlation immunity of Boolean functions. The parameters for the PSO velocity equation are tuned by means of two meta-optimization techniques, namely Local Unimodal Sampling (LUS) and Continuous Genetic Algorithms (CGA), finding that CGA produces better results. Using the CGA-evolved parameters, the PSO algorithm is then run on the spaces of Boolean functions from $n=7$ to $n=12$ variables. The results of the experiments are reported, observing that this new PSO algorithm generates Boolean functions featuring similar or better combinations of nonlinearity, correlation immunity and propagation criterion with respect to the ones obtained by other optimization methods.