# Background and Definition

A Boolean function $f:\mathbb {F} _{2^{n}}\rightarrow \mathbb {F} _{2}$ is said to be plateaued if its Walsh transform takes at most three distinct values, viz. $0$ and $\pm \mu$ for some positive ineger $\mu$ called the amplitude of $f$ .

This notion can be naturally extended to vectorial Boolean functions by applying it to each component. More precisely, if $F$ is an $(n,m)$ -function, we say that $F$ is plateaued if all its component functions $u\cdot F$ for $u\neq 0$ are plateaued. If all of the component functions are plateaued and have the same amplitude, we say that $F$ is plateaued with single amplitude.

## Equivalence relations

The class of functions that are plateaued with single amplitude is CCZ-invariant.

The class of plateaued functions is only EA-invariant.

## Relations to other classes of functions

All bent and semi-bent Boolean functions are plateaued.

Any vectorial AB function is plateaued with single amplitude.

# Constructions of Boolean plateaued functions

## Primary constructons

### Generalization of the Maiorana-MacFarland Functions 

The Maiorana-MacFarland class of bent functions can be generalized into the class of functions $f_{\phi ,h}$ of the form

$f_{\phi ,h}(x,y)=x\cdot \phi (y)+h(y)$ for $x\in \mathbb {F} _{2}^{r},y\in \mathbb {F} _{2}^{s}$ , where $r$ and $s$ are any positive integers, $n=r+s$ , $\phi :\mathbb {F} _{2}^{s}\rightarrow \mathbb {F} _{2}^{r}$ is arbitrary and $h:\mathbb {F} _{2}^{s}\rightarrow \mathbb {F} _{2}$ is any Boolean function.

The Walsh transform of $f_{\phi ,h}$ takes the value

$W_{f_{\phi ,h}}(a,b)=2^{r}\sum _{y\in \phi ^{-1}(a)}(-1)^{b\cdot y+h(y)}$ at $(a,b)$ . If $\phi$ is injective, resp. takes each value in its image set two times, then $f_{\phi ,h}$ is plateaued of amplitude $2^{r}$ , resp. $2^{r+1}$ .

1. Camion P, Carlet C, Charpin P, Sendrier N. On Correlation-immune functions. InAdvances in Cryptology—CRYPTO’91 1992 (pp. 86-100). Springer Berlin/Heidelberg.