We present necessary and sufficient conditions for a Boolean function to be a negabent function for both an even and an odd number of variables, which demonstrates the relationship between negabent functions and bent functions. By using these necessary and sufficient conditions for Boolean functions to be negabent, we obtain that the nega spectrum of a negabent function has at most four values. We determine the nega spectrum distribution of negabent functions. Further, we provide a method to construct bent-negabent functions innvariables (neven) of algebraic degree ranging from 2 to [(n)/2], which implies that the maximum algebraic degree of ann-variable bent-negabent function is equal to [(n)/2]. Thus, we answer two open problems proposed by Parker and Pott and by Stănică et al.

In 2008, Cusick et al conjectured that certain elementary symmetric Boolean functions of the form σ(2t+1)l-1, (2t) are the only nonlinear balanced ones, wheret,lare any positive integers, and σn,d=⊕1≤( i1)d≤ n xi1xi2⋯xid) for positive integersn, 1 ≤d≤n. In this paper, by analyzing the weight of σn,(2t) and σn, d, we prove that wt σn,dn-1holds in most cases, and so does the conjecture. According to the remainder modulo 4, we also consider the weight of σn, dfrom 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 σn, (2t+2s) and also give some experimental results on it.