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 $\mathbf{H}(x)=\mathbf{h}(T{r}_1^n(u_{1}x),\dots ,T{r}_1^n(u_{t}x))$, 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.

dr. Amar Bapić

Young Researcher

dr. Enes Pasalic

Full Professor, Head of the center