# Plateaued Functions

## Contents

# 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*.

## 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 .