# Background and Definition

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

This notion can be naturally extended to vectorial Boolean functions by applying it to each component. More precisely, if ${\displaystyle F}$ is an ${\displaystyle (n,m)}$-function, we say that ${\displaystyle F}$ is plateaued if all its component functions ${\displaystyle u\cdot F}$ for ${\displaystyle u\neq 0}$ are plateaued. If all of the component functions are plateaued and have the same amplitude, we say that ${\displaystyle F}$ 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 [1]

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

${\displaystyle f_{\phi ,h}(x,y)=x\cdot \phi (y)+h(y)}$

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

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

${\displaystyle W_{f_{\phi ,h}}(a,b)=2^{r}\sum _{y\in \phi ^{-1}(a)}(-1)^{b\cdot y+h(y)}}$

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

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.