# Background and definition

Almost perfect nonlinear (APN) functions are the class of $(n,n)$ Vectorial Boolean Functions that provide optimum resistance to against differential attack. Intuitively, the differential attack against a given cipher incorporating a vectorial Boolean function $F$ is efficient when fixing some difference $\delta$ and computing the output of $F$ for all pairs of inputs $(x_{1},x_{2})$ whose difference is $\delta$ produces output pairs with a difference distribution that is far away from uniform.

The formal definition of an APN function $F:\mathbb {F} _{2^{n}}\rightarrow \mathbb {F} _{2^{n}}$ is usually given through the values

$\Delta _{F}(a,b)=|\{x\in \mathbb {F} _{2^{n}}:F(x)+F(a+x)=b\}|$ which, for $a\neq 0$ , express the number of input pairs with difference $a$ that map to a given $b$ . The existence of a pair $(a,b)\in \mathbb {F} _{2^{n}}^{*}\times \mathbb {F} _{2^{n}}$ with a high value of $\Delta _{F}(a,b)$ makes the function $F$ vulnerable to differential cryptanalysis. This motivates the definition of differential uniformity as

$\Delta _{F}=\max\{\Delta _{F}(a,b):a\in \mathbb {F} _{2^{n}}^{*},b\in \mathbb {F} _{2^{n}}\}$ which clearly satisfies $\Delta _{F}\geq 2$ for any function $F$ . The functions meeting this lower bound are called almost perfect nonlinear (APN).

# Characterizations

## Walsh transform

Any $(n,m)$ -function $F$ satisfies

$\sum _{a\in \mathbb {F} _{2^{n}},b\in \mathbb {F} _{2^{m}}^{*}}W_{F}^{4}(a,b)\geq 2^{2n}(3\cdot 2^{n+m}-2^{m+1}-2^{2n})$ with equality characterizing APN functions.

In particular, for $(n,n)$ -functions we have

$\sum _{a\in \mathbb {F} _{2^{n}},b\in \mathbb {F} _{2^{n}}^{*}}W_{F}^{4}(a,b)\geq 2^{3n+1}(2^{n}-1)$ with equality characterizing APN functions.

Sometimes, it is more convenient to sum through all $b\in \mathbb {F} _{2^{m}}$ instead of just the nonzero ones. In this case, the inequality for $(n,m)$ -functions becomes

$\sum _{a\in \mathbb {F} _{2^{n}},b\in \mathbb {F} _{2^{m}}}W_{F}^{4}(a,b)\geq 2^{2n+m}(3\cdot 2^{n}-2)$ and the particular case for $(n,n)$ -functions becomes

$\sum _{a,b\in \mathbb {F} _{2^{n}}}W_{F}^{4}(a,b)\geq 2^{3n+1}(3\cdot 2^{n-1}-1)$ with equality characterizing APN functions in both cases.

## Autocorrelation functions of the directional derivatives 

Given a Boolean function $f:\mathbb {F} _{2^{n}}\rightarrow \mathbb {F} _{2}$ , the autocorrelation function of $f$ is defined as

${\mathcal {F}}(f)=\sum _{x\in \mathbb {F} _{2^{n}}}(-1)^{f(x)}=2^{n}-2wt(f).$ Any $(n,n)$ -function $F$ satisfies

$\sum _{\lambda \in \mathbb {F} _{2^{n}}}{\mathcal {F}}(D_{a}f_{\lambda })=2^{2n+1}$ for any $a\in \mathbb {F} _{2^{n}}^{*}$ . Equality occurs if and only if $F$ is APN.

This allows APN functions to be characterized in terms of the sum-of-square-indicator $\nu (f)$ defined as

$\nu (f)=\sum _{a\in \mathbb {F} _{2^{n}}}{\mathcal {F}}^{2}(D_{a}F)=2^{-n}\sum _{a\in \mathbb {F} _{2^{n}}}{\mathcal {F}}^{4}(f+\varphi _{a})$ for $\varphi _{a}(x)={\rm {Tr}}(ax)$ .

Then any $(n,n)$ function $F$ satisfies

$\sum _{\lambda \in \mathbb {F} _{2^{n}}^{*}}\nu (f_{\lambda })\geq (2^{n}-1)2^{2n+1}$ and equality occurs if and only if $F$ is APN.

Similar techniques can be used to characterize permutations and APN functions with plateaued components.

# Characterization of Plateaued Functions

## Characterization by the Derivatives 

### First characterization

Using the fact that two integer-valued functions over $\mathbb {F} _{2}^{m}$ are equal precisely when their Fourier transforms are equal, one can obtain the following characterization.

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

- $F$ is plateaued 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 above set depends neither on $x$ nor on $v$ when $v\neq 0$ .

Moreover:

- for any $(n,m)$ -function $F$ , the value distribution of $D_{a}D_{b}F(x)$ equals the value distribution of $D_{a}F(b)+D_{a}F(x)$ as $(a,b)$ ranges over $(\mathbb {F} _{2}^{n})^{2}$ ;

- if two plateuaed functions have the same distribution, then for every $u$ , their component functions at $u$ have the same amplitude.

### Characterization in the Case of Unbalanced Components

Let $F$ be an $(n,m)$ -function. Then $F$ is plateuaed with all components unbalanced if and only if, for every $v,x\in \mathbb {F} _{2}^{n}$ , we have

$|\{(a,b)\in (\mathbb {F} _{2}^{n})^{2}:D_{a}D_{b}F(x)=v\}|=|\{(a,b)\in (\mathbb {F} _{2}^{n})^{2}:F(a)+F(b)=v\}|.$ Moreover, $F$ is plateuaed with single amplitude if and only if, in addition, this value does not depend on $v$ for $v\neq 0$ .

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