New bent functions from positive and negative functions of old bent functions

Given a bent function f(x) of n variables we introduce its positive and negative functions as the Boolean functions f⁺(x) and f^-(x) whose supports are M⁺ = {a isin Zopf₂ ⁿ | w(f oplus l_a) = 2^n-1 + 2^n/2-1} and M^- = {a isin Zopf₂ ⁿ | w(f oplus l_a) = 2^n-1 - 2^n/2-1} respectively, where w(f oplus l_a) denotes the Hamming weight of the Boolean function f(x) oplus l_a(x) and l_a(x) is the linear function defined by a isin Zopf₂ ⁿ. We prove that f⁺(x) and f^-(x) are bent functions. Furthermore, combining the 4 minterms of 2 variables with the positive or negative functions of 4 bent functions of n variables we obtain a bent function of n+2 variables.