# Characterization of Permutations

## Component Functions

An ${\displaystyle (n,n)}$-function ${\displaystyle F}$ is a permutation if and only if all of its components ${\displaystyle F_{\lambda }}$ for ${\displaystyle \lambda \in \mathbb {F} _{2^{n}}^{*}}$ are balanced.

## Autocorrelation Functions of the Directional Derivatives

The characterization in terms of the component functions given above can be equivalently expressed as

${\displaystyle \sum _{a\in \mathbb {F} _{2^{n}}^{*}}{\mathcal {F}}(D_{a}f_{\lambda })=-2^{n}}$

for any ${\displaystyle \lambda \in \mathbb {F} _{2^{n}}^{*}}$.

Equivalently [1], ${\displaystyle F}$ is a permutation if and only if

${\displaystyle \sum _{\lambda \in \mathbb {F} _{2^{n}}^{*}}{\mathcal {F}}(D_{a}f_{\lambda })=-2^{n}}$

for any ${\displaystyle \lambda \in \mathbb {F} _{2^{n}}^{*}}$.

# Characterization of APN Permutations

## On the component functions

Clearly we have that no component function can be of degree 1. (This result is true for general APN maps)

For n even we have also that no component can be partially-bent[2]. This implies that, in even dimension, no component can be of degree 2.

## Autocorrelation Functions of the Directional Derivatives

An ${\displaystyle (n,n)}$-function ${\displaystyle F}$ is an APN permutation if and only if [1]

${\displaystyle \sum _{\lambda \in \mathbb {F} _{2^{n}}^{*}}{\mathcal {F}}(D_{a}f_{\lambda })=-2^{n}}$

and

${\displaystyle \sum _{\lambda \in \mathbb {F} _{2^{n}}^{*}}{\mathcal {F}}^{2}(D_{a}f_{\lambda })=2^{2n}}$

for any ${\displaystyle a\in \mathbb {F} _{2^{n}}^{*}}$.

1. Thierry Berger, Anne Canteaut, Pascale Charpin, Yann Laigle-Chapuy, On Almost Perfect Nonlinear Functions Over GF(2^n), IEEE Transactions on Information Theory, 2006 Sep,52(9),4160-70
2. Marco Calderini, Massimiliano Sala, Irene Villa, A note on APN permutations in even dimension, Finite Fields and Their Applications, vol. 46, 1-16, 2017