Graphs of Vectorial Plateaued Functions as Difference Sets

A function $F:{\mathbb{F}}_{p^n}\to {\mathbb{F}}_{p^m},$ is a vectorial $s$-plateaued function if for each component function $F_{b}(\mu)=Tr_n(\alpha F(x)), b\in $\mathbb{F}${p^m}^*$and$\mu \in $\mathbb{F}${p^n}$, the Walsh transform value $\mathrm{\mid }\widehat{F_b}(\mu )\mathrm{\mid }$ is either $0$ or $p^{\frac{n+s}{2}}$. In this paper, we explore the relation between (vectorial) $s$-plateaued functions and partial geometric difference sets. Moreover, we establish the link between three-valued cross-correlation of $p$-ary sequences and vectorial $s$-plateaued functions. Using this link, we provide a partition of ${\mathbb{F}}_{3^n}$ into partial geometric difference sets. Conversely, using a partition of ${\mathbb{F}}_{3^n}$ into partial geometric difference sets, we constructed ternary plateaued functions $f:$\mathbb{F}$_{3^n}\rightarrow $\mathbb{F}$_3$\mathrm{.}Wealsogiveacharacterizationof$p$-ary plateaued functions in terms of special matrices which enables us to give the link between such functions and second-order derivatives using a different approach.