A Note on a Conjecture for Balanced Elementary Symmetric Boolean Functions

TL;DR

This paper investigates the weight of elementary symmetric Boolean functions, providing partial proofs for a longstanding conjecture about their balancedness and offering new insights based on modular analysis.

Contribution

The paper proves that most elementary symmetric Boolean functions have weight less than half of the total, supporting the conjecture and simplifying its conditions through modular analysis.

Findings

Most functions satisfy wt(σ_{n,d})<2^{n-1}

Results cover known cases of the conjecture

Experimental results on σ_{n, 2^t+2^s}

Abstract

In 2008, Cusick {\it et al.} conjectured that certain elementary symmetric Boolean functions of the form $\sigma_{2^{t+1}l-1, 2^t}$ are the only nonlinear balanced ones, where $t$, $l$ are any positive integers, and $\sigma_{n,d}=\bigoplus_{1\le i_1<...<i_d\le n}x_{i_1}x_{i_2}...x_{i_d}$ for positive integers $n$, $1\le d\le n$. In this note, by analyzing the weight of $\sigma_{n, 2^t}$ and $\sigma_{n, d}$, we prove that ${\rm wt}(\sigma_{n, d})<2^{n-1}$ holds in most cases, and so does the conjecture. According to the remainder of modulo 4, we also consider the weight of $\sigma_{n, d}$ from two aspects: $n\equiv 3({\rm mod\}4)$ and $n\not\equiv 3({\rm mod\}4)$. Thus, we can simplify the conjecture. In particular, our results cover the most known results. In order to fully solve the conjecture, we also consider the weight of $\sigma_{n, 2^t+2^s}$ and give some experiment results on it.