Plateaued Functions

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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.

The characterization by means of the derivatives below suggests the following definition: a v.B.f. $F$ is said to be strongly-plateuaed if, for every $a$ and every $v$, the size of the set $\{b\in \mathbb {F} _{2}^{n}:D_{a}D_{b}F(x)=v\}$ does not depend on $x$, or, equivalently, the size of the set $\{b\in \mathbb {F} _{2}^{n}:D_{a}F(b)=D_{a}F(x)+v\}$ does not depend on $x$.

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

The Maiorana-MacFarland class of bent functions can be generalized into the class of functions $f_{\phi ,h}$ of the form

$f_{\phi ,h}(x,y)=x\cdot \phi (y)+h(y)$

for $x\in \mathbb {F} _{2}^{r},y\in \mathbb {F} _{2}^{s}$, where $r$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle s} are any positive integers, $n=r+s$, $\phi :\mathbb {F} _{2}^{s}\rightarrow \mathbb {F} _{2}^{r}$ is arbitrary and $h:\mathbb {F} _{2}^{s}\rightarrow \mathbb {F} _{2}$ is any Boolean function.

The Walsh transform of $f_{\phi ,h}$ takes the value

$W_{f_{\phi ,h}}(a,b)=2^{r}\sum _{y\in \phi ^{-1}(a)}(-1)^{b\cdot y+h(y)}$

at $(a,b)$. If $\phi$ is injective, resp. takes each value in its image set two times, then $f_{\phi ,h}$ is plateaued of amplitude $2^{r}$, resp. $2^{r+1}$.

Characterization of Plateaued Functions

Characterization by the Derivatives

Using the fact that a Boolean function $f$ is plateaued if and only if the expression $\sum _{a,b\in \mathbb {F} _{2}^{n}}(-1)^{DaDbf(x)}$ does not depend on $x\in \mathbb {F} _{2}^{n}$, one can derive the following characterization.

Let $F$ be an $(n,m)$-function. Then:

• F is plateuaed if and only if, for every $v\in \mathbb {F} _{2}^{m}$, the size of the set

$\{(a,b)\in (\mathbb {F} _{2}^{n})^{2}:D_{a}D_{b}F(x)=v\}$

does not depend on $x$;

• F is plateaued with single amplitude if and only if the size of the set depends neither on $x$, nor on $v\in \mathbb {F} _{2}^{m}$ for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v\neq 0} .

Moreover:

• for every $F$, the value distribution of $D_{a}D_{b}F(x)$ equals that of $D_{a}F(b)+D_{a}F(x)$ when $(a,b)$ ranges over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{F}_2^n)^2} ;

• if two plateaued functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F,G} have the same distribution, then all of their component functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u \cdot F, u\cdot G} have the same amplitude.

Power Functions

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x) = x^d} . Then, for every $v,x,\lambda \in \mathbb{F}_{2^n}</math> with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda \ne 0} , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | \{ (a,b) \in \mathbb{F}_{2^n}^2 : D_aF(b) + D_aF(x) = v \} | = | \{ (a,b) \in \mathbb{F}_{2^n}^2 : D_aF(b) + D_aF(x/\lambda) = v/\lambda^d \}|.}

Then:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is plateaued if and only if, for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v \in \mathbb{F}_{2^n}} , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | \{ (a,b) \in \mathbb{F}_{2^n}^2 : D_aF(b) + D_aF(1) = v \} | = | \{ (a,b) \in \mathbb{F}_{2^n}^2 : D_aF(b) + D_aF(0) = v \}|;}

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is plateaued with single amplitude if and only if the size above does not, in addition, depend on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v \ne 0} .

Functions with Unbalanced Components

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} be an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n,m)} -function. Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is plateuaed with all components unbalanced if and only if, for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v,x \in \mathbb{F}_{2}^n} , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | \{ (a,b) \in (\mathbb{F}_2^n)^2 : D_aD_bF(x) = v \}| = | \{ (a,b) \in (\mathbb{F}_2^n)^2 : F(a) + F(b) = v \}|.}

Moreover, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is plateaued with single amplitude if and only if this value does not, in addition, depend on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v \ne 0} .

Strongly-Plateaued Functions

A Boolean function is strongly-plateaued if and only if its partially-bent. A v.B.f. is strongly-plateaued if and only if all of its component functions are partially-bent. In particular, bent and quadratic Boolean and vectorial functions are strongly-plateaued.

The image set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm Im}(D_aF)} of any derivative of a strongly-plateaued function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is an affine space.

Characterization by the Auto-Correlation Functions

Recall that the autocorrelation function of a Boolean function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\Delta_f}(a) = \sum_{x \in \mathbb{F}_2^n} (-1)^{f(x) + f(x+a)}} .

An Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -variable Boolean function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is plateaued if and only if, for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbb{F}_2^n} , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^n \sum_{a \in \mathbb{F}_2^n} \Delta_f(a) \Delta_f(a+x) = \left( \sum_{a \in \mathbb{F}_2^n} \Delta_f^2(a) \right) \Delta_f(x).}

An Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n,m)} -function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is plateaued if and only if, for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbb{F}_2^n, u \in \mathbb{F}_2^m} , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^n \sum_{a \in \mathbb{F}_2^n} \Delta_{u \cdot F}(a) \Delta_{u \cdot F}(a+x) = \left( \sum_{a \in \mathbb{F}_2^n} \Delta_{u \cdot F}^2(a) \right) \Delta_{u \cdot F}(x).}

Furthermore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is plateaued with single amplitude if and only if, for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \mathbb{F}_2^n, u \in \mathbb{F}_2^m} , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{a \in \mathbb{F}_2^n} \Delta_{u \cdot F}(a) \Delta_{u \cdot F}(a+x) = \mu^2 \Delta_{u \cdot F}(x).}

Alternatively, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is plateuaed if and only if, for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,v \in \mathbb{F}_2^n} , we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^n | \{ (a,b,c) \in (\mathbb{F}_2^n)^3 : F(a) + F(b) + F(c) + F(a+b+c+x) = v \}| = | \{ (a,b,c,d) \in (\mathbb{F}_2^n)^4 : F(a) + F(b) + F(c) + F(a+b+c) + F(d) + F(d+x) = v \}|.}