# APN Permutations

## Contents

# Characterization of Permutations

## Component Functions

An -function is a permutation if and only if all of its components for are balanced.

## Autocorrelation Functions of the Directional Derivatives

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

for any .

Equivalently ^{[1]}, is a permutation if and only if

for any .

# 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 -function is an APN permutation if and only if ^{[1]}

and

for any .

- ↑
^{1.0}^{1.1}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 - ↑ Marco Calderini, Massimiliano Sala, Irene Villa,
*A note on APN permutations in even dimension*, Finite Fields and Their Applications, vol. 46, 1-16, 2017