A Note on a Conjecture for Balanced Elementary Symmetric Boolean Functions

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.