Several new infinite families of bent functions via second order derivatives

Abstract

Inspired by a recent work of Tang et al. on constructing bent functions [14, IEEE TIT, 63(1): 6149-6157, 2017], we introduce a property (P_τ) of any Boolean function that its second order derivatives vanish at any direction (u_i,u_j) for some τ-subset {u₁,…,u_τ} of $\mathbb {F}_{2^{n}}$, and then establish a link between this property and the construction of Tang et al. (IEEE Trans. Inf. Theory 63(10), 6149–6157 2017). It enables us to find more bent functions efficiently. We construct (at least) five new infinite families of bent functions from some known functions: the Gold’s bent functions and some quadratic non-monomial bent functions, Leander’s monomial bent functions, Canteaut-Charpin-Kyureghyan’s monomial bent functions, and the Maiorana-McFarland class of bent functions, respectively. Our result generalizes some recent works on bent functions. We also provide the corresponding dual functions in all our constructions except the quadratic non-monomial one. It also turns out that we can get new bent functions outside the Maiorana-McFarland completed class.