Plateaued Functions

Background and Definition

A Boolean function [math]\displaystyle{ f : \mathbb{F}_{2^n} \rightarrow \mathbb{F}_2 }[/math] is said to be plateaued if its Walsh transform takes at most three distinct values, viz. [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ \pm \mu }[/math] for some positive ineger [math]\displaystyle{ \mu }[/math] called the amplitude of [math]\displaystyle{ f }[/math].

This notion can be naturally extended to vectorial Boolean functions by applying it to each component. More precisely, if [math]\displaystyle{ F }[/math] is an [math]\displaystyle{ (n,m) }[/math]-function, we say that [math]\displaystyle{ F }[/math] is plateaued if all its component functions [math]\displaystyle{ u \cdot F }[/math] for [math]\displaystyle{ u \ne 0 }[/math] are plateaued. If all of the component functions are plateaued and have the same amplitude, we say that [math]\displaystyle{ F }[/math] 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.