Combinatorial method of polynomial expansion of symmetric Boolean functions

TL;DR

This paper introduces an efficient polynomial expansion method for symmetric Boolean functions, significantly reducing complexity for elementary symmetric functions by leveraging combinatorial Lucas theorem.

Contribution

It presents a novel combinatorial approach that achieves linear complexity for elementary symmetric Boolean functions, improving upon existing quadratic methods.

Findings

Linear complexity for elementary symmetric Boolean functions

Efficient for functions with small valued number sets

Based on combinatorial Lucas theorem

Abstract

A novel polynomial expansion method of symmetric Boolean functions is described. The method is efficient for symmetric Boolean function with small set of valued numbers and has the linear complexity for elementary symmetric Boolean functions, while the complexity of the known methods for this class of functions is quadratic. The proposed method is based on the consequence of the combinatorial Lucas theorem.