Identification of Disjoint Bi-decompositions in Boolean Functions through Walsh Spectrum

Abstract

A new algorithm for the identification of disjoint bi-decomposition in Boolean functions from its Walsh spectrum is proposed. The type of bi-decomposition and its existence is derived from the knowledge of a subset of Walsh spectrum for a Boolean function. All three types of bi-decomposition are considered including OR, AND and EXOR type. A filtering procedure that uses just few Walsh spectral coefficients (SC) is applied to quickly eliminate the functions that are not bi-decomposable and hence the algorithm is very efficient. The type of bi-decomposition and affirmation/negation of variables in its logic sub-functions are directly identified by manipulation on the reduced cubical representation of Boolean functions and their corresponding Walsh spectra. The presented algorithm has been implemented in C and tested on the standard benchmark functions. The number of Boolean functions having various disjoint bi-decompositions has also been enumerated.