# 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. .

- ↑ Camion P, Carlet C, Charpin P, Sendrier N. On Correlation-immune functions. InAdvances in Cryptology—CRYPTO’91 1992 (pp. 86-100). Springer Berlin/Heidelberg.