A new method for secondary constructions of vectorial bent functions

Abstract: In 2017, Tang et al. have introduced a generic construction for bent functions of the form $f(x)=g(x)+h(x)$, where $g$ is a bent function satisfying some conditions and $h$ is a Boolean function. Recently, Zheng et al. generalized this result to construct large classes of bent vectorial Boolean functions from known ones in the form $F(x)=G(x)+h(X)$, where $G$ is a vectorial bent and $h$ is a Boolean function. In this paper, we further generalize this construction to obtain vectorial bent functions of the form $F(x)=G(x)+\mathbf{H}(X)$, where $\mathbf{H}$ is also a vectorial Boolean function. This allows us to construct new infinite families of vectorial bent functions, EA-inequivalent to $G$, which was used in the construction. Most notably, specifying $\mathbf{H} (x)=\mathbf{h} (Tr_1^n(u_1x),\ldots,Tr_1^n(u_tx))$, the function $\mathbf{h}$ can be chosen arbitrarily, which gives a relatively large class of different functions for a fixed function $G$. We also propose a method of constructing vectorial $(n,n)$-functions having maximal number of bent components.