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    Secondary constructions of vectorial pp-ary weakly regular bent functions

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    In \cite{Bapic, Tang, Zheng} a new method for the secondary construction of vectorial/Boolean bent functions via the so-called (PU)(P_U) property was introduced. In 2018, Qi et al. generalized the methods in \cite{Tang} for the construction of pp-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 pp-ary weakly regular bent and plateaued functions. We also present some infinite families of such functions via the pp-ary Maiorana-McFarland class. Additionally, we give another characterization of the (PU)(P_U) property for the pp-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

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    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

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    In 2017, Tang et al. have introduced a generic construction for bent functions of the form f(x)=g(x)+h(x)f(x)=g(x)+h(x), where gg is a bent function satisfying some conditions and hh 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)F(x)=G(x)+h(X), where GG is a bent vectorial and hh a Boolean function. In this paper we further generalize this construction to obtain vectorial bent functions of the form F(x)=G(x)+H(X)F(x)=G(x)+\mathbf{H}(X), where H\mathbf{H} is also a vectorial Boolean function. This allows us to construct new infinite families of vectorial bent functions, EA-inequivalent to GG, which was used in the construction. Most notably, specifying H(x)=h(Tr1n(u1x),…,Tr1n(utx))\mathbf{H } (x)=\mathbf{h} (Tr_1^n(u_1x),\ldots,Tr_1^n(u_tx)), the function h:F2t→F2t\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 GG. We also propose a method of constructing vectorial (n,n)(n,n)-functions having maximal number of bent components
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