# Plateaued Functions

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# Background and Definition

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

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