A Note on a Conjecture for Balanced Elementary Symmetric Boolean Functions

In 2008, Cusick et al conjectured that certain elementary symmetric Boolean functions of the form σ(2^t+1)_l-1, (2^t) are the only nonlinear balanced ones, where t, l are any positive integers, and σ_n,d=⊕₁ ≤( i₁) <; ⋯ <;(i_d ≤ n x_i1 x_i2⋯x_id) for positive integers n, 1 ≤ d ≤ n. In this paper, by analyzing the weight of σ_n,(2^t) and σ_{n, d}, we prove that wt σ_n,d <; 2^n-1 holds in most cases, and so does the conjecture. According to the remainder modulo 4, we also consider the weight of σ_{n, d} from two aspects: n ≠ 3(mod 4) and n not ≡ 3(mod 4). In particular, our results not only cover the most known results, but also contain some new cases. Thus, we can reduce the conjecture to few remaining cases. We do not fully solve the conjecture, but we also consider the weight of σ_n, (2^t+2^s) and also give some experimental results on it.