# Plateaued Functions

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# Background and Definition

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

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