Paper 2020/1609

A new method for secondary constructions of vectorial bent functions

Amar Bapić and Enes Pasalic

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 function from known ones in the form $F(x)=G(x)+h(X)$, where $G$ is a bent vectorial and $h$ 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 $\mathbf{h}:{\mathbb{F}}_2^t\to {\mathbb{F}}_{2^t}$ can be chosen arbitrary which gives a relatively large class of different functions for a fixed function $G$. We also propose a method of constructing vectorial $$-functions having maximal number of bent components.