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. (Discret Math 344:112473, 2021) 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 $\mathbf {h}: {\mathbb {F}}_2^t \rightarrow {\mathbb {F}}_{2^t}$ 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.