# 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.

The characterization by means of the derivatives below suggests the following definition: a v.B.f. $F$ is said to be strongly-plateuaed if, for every $a$ and every $v$ , the size of the set $\{b\in \mathbb {F} _{2}^{n}:D_{a}D_{b}F(x)=v\}$ does not depend on $x$ , or, equivalently, the size of the set $\{b\in \mathbb {F} _{2}^{n}:D_{a}F(b)=D_{a}F(x)+v\}$ does not depend on $x$ .

## 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. $2^{r+1}$ .

# Characterization of Plateaued Functions 

## Characterization by the Derivatives

Using the fact that a Boolean function $f$ is plateaued if and only if the expression $\sum _{a,b\in \mathbb {F} _{2}^{n}}(-1)^{DaDbf(x)}$ does not depend on $x\in \mathbb {F} _{2}^{n}$ , one can derive the following characterization.

Let $F$ be an $(n,m)$ -function. Then:

• F is plateuaed if and only if, for every $v\in \mathbb {F} _{2}^{m}$ , the size of the set
$\{(a,b)\in (\mathbb {F} _{2}^{n})^{2}:D_{a}D_{b}F(x)=v\}$ does not depend on $x$ ;

• F is plateaued with single amplitude if and only if the size of the set depends neither on $x$ , nor on $v\in \mathbb {F} _{2}^{m}$ for $v\neq 0$ .

Moreover:

• for every $F$ , the value distribution of $D_{a}D_{b}F(x)$ equals that of $D_{a}F(b)+D_{a}F(x)$ when $(a,b)$ ranges over $(\mathbb {F} _{2}^{n})^{2}$ ;
• if two plateaued functions $F,G$ have the same distribution, then all of their component functions $u\cdot F,u\cdot G$ have the same amplitude.

### Power Functions

Let $F(x)=x^{d}$ . Then, for every \$v,x,\lambda \in \mathbb{F}_{2^n}[/itex] with $\lambda \neq 0$ , we have

$|\{(a,b)\in \mathbb {F} _{2^{n}}^{2}:D_{a}F(b)+D_{a}F(x)=v\}|=|\{(a,b)\in \mathbb {F} _{2^{n}}^{2}:D_{a}F(b)+D_{a}F(x/\lambda )=v/\lambda ^{d}\}|.$ Then:

• $F$ is plateaued if and only if, for every $v\in \mathbb {F} _{2^{n}}$ , we have
$|\{(a,b)\in \mathbb {F} _{2^{n}}^{2}:D_{a}F(b)+D_{a}F(1)=v\}|=|\{(a,b)\in \mathbb {F} _{2^{n}}^{2}:D_{a}F(b)+D_{a}F(0)=v\}|;$ • $F$ is plateaued with single amplitude if and only if the size above does not, in addition, depend on $v\neq 0$ .

### Functions with Unbalanced Components

Let $F$ be an $(n,m)$ -function. Then $F$ is plateuaed with all components unbalanced if and only if, for every $v,x\in \mathbb {F} _{2}^{n}$ , we have

$|\{(a,b)\in (\mathbb {F} _{2}^{n})^{2}:D_{a}D_{b}F(x)=v\}|=|\{(a,b)\in (\mathbb {F} _{2}^{n})^{2}:F(a)+F(b)=v\}|.$ Moreover, $F$ is plateaued with single amplitude if and only if this value does not, in addition, depend on $v$ for $v\neq 0$ .

### Strongly-Plateaued Functions

A Boolean function is strongly-plateaued if and only if its partially-bent. A v.B.f. is strongly-plateaued if and only if all of its component functions are partially-bent. In particular, bent and quadratic Boolean and vectorial functions are strongly-plateaued.

The image set ${\rm {Im}}(D_{a}F)$ of any derivative of a strongly-plateaued function $F$ is an affine space.

## Characterization by the Auto-Correlation Functions

Recall that the autocorrelation function of a Boolean function $f$ is defined as ${\Delta _{f}}(a)=\sum _{x\in \mathbb {F} _{2}^{n}}(-1)^{f(x)+f(x+a)}$ .

An $n$ -variable Boolean function $f$ is plateaued if and only if, for every $x\in \mathbb {F} _{2}^{n}$ , we have

$2^{n}\sum _{a\in \mathbb {F} _{2}^{n}}\Delta _{f}(a)\Delta _{f}(a+x)=\left(\sum _{a\in \mathbb {F} _{2}^{n}}\Delta _{f}^{2}(a)\right)\Delta _{f}(x).$ An $(n,m)$ -function $F$ is plateaued if and only if, for every $x\in \mathbb {F} _{2}^{n},u\in \mathbb {F} _{2}^{m}$ , we have

$2^{n}\sum _{a\in \mathbb {F} _{2}^{n}}\Delta _{u\cdot F}(a)\Delta _{u\cdot F}(a+x)=\left(\sum _{a\in \mathbb {F} _{2}^{n}}\Delta _{u\cdot F}^{2}(a)\right)\Delta _{u\cdot F}(x).$ Furthermore, $F$ is plateaued with single amplitude if and only if, for every $x\in \mathbb {F} _{2}^{n},u\in \mathbb {F} _{2}^{m}$ , we have

$\sum _{a\in \mathbb {F} _{2}^{n}}\Delta _{u\cdot F}(a)\Delta _{u\cdot F}(a+x)=\mu ^{2}\Delta _{u\cdot F}(x).$ Alternatively, $F$ is plateuaed if and only if, for every $x,v\in \mathbb {F} _{2}^{n}$ , we have

$2^{n}|\{(a,b,c)\in (\mathbb {F} _{2}^{n})^{3}:F(a)+F(b)+F(c)+F(a+b+c+x)=v\}|=|\{(a,b,c,d)\in (\mathbb {F} _{2}^{n})^{4}:F(a)+F(b)+F(c)+F(a+b+c)+F(d)+F(d+x)=v\}|.$ 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.
2. Carlet C. Boolean and vectorial plateaued functions and APN functions. IEEE Transactions on Information Theory. 2015 Nov;61(11):6272-89.