# 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 ±𝜇 for some positive ineger 𝜇 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⋅𝐹 for 𝑢≠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 𝜑 is injective, resp. takes each value in its image set two times, then $f_{\phi ,h}$ is plateaued of amplitude 2𝑟, resp. 2𝑟+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

$2^{n}|\{(a,b,c)\in (\mathbb {F} _{2}^{n})^{3}:F(a)+F(b)+F(c)+F(a+b+c+x)=v\}|=|\{(a,b,c,d)\in (\mathbb {F} _{2}^{n})^{4}:F(a)+F(b)+F(c)+F(a+b+c)+F(d)+F(d+x)=v\}|.$ ## Characterization by the Means of the Power Moments of the Walsh Transform

### First Characterization

A Boolean function $f:\mathbb {F} _{2^{n}}\rightarrow \mathbb {F} _{2}$ is plateuaed if and only if, for every $0\neq \alpha \in \mathbb {F} _{2}^{n}$ , we have

$\sum _{w\in \mathbb {F} _{2}^{n}}W_{f}(w+\alpha )W_{f}^{3}(w)=0.$ An (𝑛,𝑚)-function 𝐹 is plateuaed if and only if for every $u\in \mathbb {F} _{2}^{m}$ and $0\neq \alpha \in \mathbb {F} _{2}^{n}$ , we have

$\sum _{w\in \mathbb {F} _{2}^{n}}W_{F}(w+\alpha ,u)W_{F}^{3}(w,u)=0.$ Furthermore, 𝐹 is plateaued with single amplitude if and only if, in addition, the sum $\sum _{w\in \mathbb {F} _{2}^{n}}W_{F}^{4}(w,u)$ does not depend on $u$ for $u\neq 0$ .

### Second Characterization

A Boolean function $f:\mathbb {F} _{2^{n}}\rightarrow \mathbb {F} _{2}$ is plateuaed if and only if, for every $b\in \mathbb {F} _{2}$ , we have

$\sum _{a\in \mathbb {F} _{2}}W_{f}^{4}(a)=2^{n}(-1)^{f(b)}\sum _{a\in \mathbb {F} _{2}^{n}}(-1)^{a\cdot b}W_{f}^{3}(a).$ An (𝑛,𝑚)-function 𝐹 is plateuaed if and only if, for every $b\in \mathbb {F} _{2}^{n}$ and every $u\in \mathbb {F} _{m}$ , we have

$\sum _{a\in \mathbb {F} _{2}^{n}}W_{F}^{4}(a,u)=2^{n}(-1)^{u\cdot F(b)}\sum _{a\in \mathbb {F} _{2}^{n}}(-1)^{a\cdot b}W_{F}^{3}(a,u).$ Moreover, 𝐹 is plateaued with single amplitude if and only if the two sums above do not depend on 𝑢 for $u\neq 0$ .

### Third Characterization

Any Boolean function 𝑓 in $n$ variables satisfies

$\left(\sum _{a\in \mathbb {F} _{2}^{n}}W_{f}^{4}(a)\right)^{2}\leq 2^{2n}\left(\sum _{a\in \mathbb {F} _{2}^{n}}W_{f}^{6}(a)\right),$ with equality if and only if 𝑓 is plateuaed.

Any (𝑛,𝑚)-function 𝐹 satisfies

$\sum _{u\in \mathbb {F} _{2}^{m}}\left(\sum _{a\in \mathbb {F} _{2}^{n}}W_{F}^{4}(a,u)\right)^{2}\leq 2^{2n}\sum _{u\in \mathbb {F} _{2}^{m}}\left(\sum _{a\in \mathbb {F} _{2}^{n}}W_{F}^{6}(a,u)\right),$ with equality if and only if 𝐹 is plateuaed.

$\sum _{u\in \mathbb {F} _{2}^{m}}\sum _{a\in \mathbb {F} _{2}^{n}}W_{F}^{4}(a,u)\leq 2^{n}\sum _{u\in \mathbb {F} _{2}^{m}}{\sqrt {\sum _{a\in \mathbb {F} _{2}^{n}}W_{F}^{6}(a,u)}},$ with equality if and only if 𝐹 is plateuaed.

# Characterization of APN among Plateaued Functions

## Characterization by the Derivatives

One very useful property of quadratic functions which extends to plateaued functions is that it suffices to consider the number of solutions to the differential equation $D_{a}F(x)=D_{a}F(0)$ in order to decided the APN-ness of a given function 𝐹. More precisely, a plateuaed (𝑛,𝑛) function 𝐹 is APN if and only if the equation

$F(x)+F(x+a)=F(0)+F(a)$ has at most two solutions for any $0\neq a\in \mathbb {F} _{2}^{n}$ .

## Characterization by the Walsh Transform

Suppose 𝐹 is a plateaued (𝑛,𝑛) function with $F(0)=0$ . Then 𝐹 is APN if and only if

$|\{(x,b)\in \mathbb {F} _{2^{n}}^{2}:F(x)+F(x+b)+F(b)=0\}|=3\cdot 2^{n}-2,$ or, equivalently,

$\sum _{a\in \mathbb {F} _{2^{n}},u\in \mathbb {F} _{2^{n}}^{*}}W_{F}^{3}(a,u)=2^{2n+1}(2^{n}-1).$ Any (𝑛,𝑛)-function satisfies the inequality

$3\cdot 2^{3^{n}}-2^{2n+1}\leq \sum _{u\in \mathbb {F} _{2}^{n}}{\sqrt {\sum _{a\in \mathbb {F} _{2}^{n}}W_{F}^{6}(a,u)}},$ with equality if and only if 𝐹 is APN plateaued.

If we denote by $2^{\lambda _{u}}$ the amplitude of the component function $u\cdot F$ of a given plateuaed function $F$ , then 𝐹 is APN if and only if

$\sum _{0\neq u\in \mathbb {F} _{2}^{n}}2^{2\lambda _{u}}\leq 2^{n+1}(2^{n}-1).$ ## Functions with Unbalanced Components

Let 𝐹 be an (𝑛,𝑛)-plateaued function with all components unbalanced. Then

$|\{(a,b)\in (\mathbb {F} _{2}^{n})^{2}:a\neq b,F(a)=F(b)\}|\geq 2\cdot (2^{n}-1),$ with equality if and only if 𝐹 is APN.

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