Secondary constructions of vectorial pp-ary weakly regular bent functions

In \cite{Bapic, Tang, Zheng} a new method for the secondary construction of vectorial/Boolean bent functions via the so-called $(P_U)$ property was introduced. In 2018, Qi et al. generalized the methods in \cite{Tang} for the construction of $p$-ary weakly regular bent functions. The objective of this paper is to further generalize these constructions, following the ideas in \cite{Bapic, Zheng}, for secondary constructions of vectorial $p$-ary weakly regular bent and plateaued functions. We also present some infinite families of such functions via the $p$-ary Maiorana-McFarland class. Additionally, we give another characterization of the $(P_U)$ property for the $p$-ary case via second-order derivatives, as it was done for the Boolean case in \cite{Zheng}

Secondary constructions of (non)weakly regular plateaued functions over finite fields

Plateaued (vectorial) functions over finite fields have diverse applications in symmetric cryptography, coding theory, and sequence theory. Constructing these functions is an attractive research topic in the literature. We can distinguish two kinds of constructions of plateaued functions: secondary constructions and primary constructions. The first method uses already known functions to obtain new functions while the latter do not need to use previously constructed functions to obtain new functions. In this work, the first secondary constructions of (non)weakly regular plateaued (vectorial) functions are presented over the finite fields of odd characteristics. We also introduce some recursive constructions of (non)weakly regular plateaued p-ary functions by using already known such functions. We obtain nontrivial plateaued functions from the previously known trivial plateaued (partially bent) functions in the proposed construction methods

A new method for secondary constructions of vectorial bent functions

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 $\mathbf{H } (x)=\mathbf{h} (Tr_1^n(u_1x),\ldots,Tr_1^n(u_tx))$, the function $\mathbf{h} :\mathbb{F}_2^t \rightarrow \mathbb{F}_{2^t}$ can be chosen arbitrary 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