# Plateaued Functions

## Contents

- 1 Background and Definition
- 2 Constructions of Boolean plateaued functions
- 3 Characterization of Plateaued Functions
^{[2]} - 4 Characterization of APN among Plateaued Functions

# Background and Definition

A Boolean function is said to be *plateaued* if its Walsh transform takes at most three distinct values, viz. and for some positive ineger called the *amplitude* of .

This notion can be naturally extended to vectorial Boolean functions by applying it to each component. More precisely, if is an -function, we say that is *plateaued* if all its component functions for are plateaued. If all of the component functions are plateaued and have the same amplitude, we say that is *plateaued with single amplitude*.

The characterization by means of the derivatives below suggests the following definition: a v.B.f. is said to be *strongly-plateuaed* if, for every and every , the size of the set does not depend on , or, equivalently, the size of the set does not depend on .

## 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 of the form

for , where and are any positive integers, , is arbitrary and is any Boolean function.

The Walsh transform of takes the value

at . If is injective, resp. takes each value in its image set two times, then is plateaued of amplitude , resp. .

# Characterization of Plateaued Functions ^{[2]}

## Characterization by the Derivatives

Using the fact that a Boolean function is plateaued if and only if the expression does not depend on , one can derive the following characterization.

Let be an -function. Then:

- F is plateuaed if and only if, for every , the size of the set

does not depend on ;

- F is plateaued with single amplitude if and only if the size of the set depends neither on , nor on for .

Moreover:

- for every , the value distribution of equals that of when ranges over ;

- if two plateaued functions have the same distribution, then all of their component functions have the same amplitude.

### Power Functions

Let . Then, for every $v,x,\lambda \in \mathbb{F}_{2^n}</math> with , we have

Then:

- is plateaued if and only if, for every , we have

- is plateaued with single amplitude if and only if the size above does not, in addition, depend on .

### Functions with Unbalanced Components

Let be an -function. Then is plateuaed with all components unbalanced if and only if, for every , we have

Moreover, is plateaued with single amplitude if and only if this value does not, in addition, depend on for .

### 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 of any derivative of a strongly-plateaued function is an affine space.

## Characterization by the Auto-Correlation Functions

Recall that the autocorrelation function of a Boolean function is defined as .

An -variable Boolean function is plateaued if and only if, for every , we have

An -function is plateaued if and only if, for every , we have

Furthermore, is plateaued with single amplitude if and only if, for every , we have

Alternatively, is plateuaed if and only if, for every , we have

## Characterization by the Means of the Power Moments of the Walsh Transform

### First Characterization

A Boolean function is plateuaed if and only if, for every , we have

An -function is plateuaed if and only if for every and , we have

Furthermore, is plateaued with single amplitude if and only if, in addition, the sum does not depend on for .

### Second Characterization

A Boolean function is plateuaed if and only if, for every , we have

An -function is plateuaed if and only if, for every and every , we have

Moreover, is plateaued with single amplitude if and only if the two sums above do not depend on for .

### Third Characterization

Any Boolean function in variables satisfies

with equality if and only if is plateuaed.

Any -function satisfies

with equality if and only if is plateuaed.

In addition, every -function satisfies

with equality if and only if is plateuaed.

# Characterization of APN among Plateaued Functions

## Deciding APN-ness of Plateaued Functions

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 in order to decided the APN-ness of a given function . More precisely, a plateuaed function is APN if and only if the equation

has at most two solutions for any .

## Characterization by the Walsh Transform

Suppose is a plateaued function with . Then is APN if and only if

or, equivalently,