# APN Permutations

## Contents

# Characterization of Permutations

## Component Functions

An (π,π)-function πΉ is a permutation if and only if all of its components πΉ_{Ξ»} for Ξ» β π½*_{2π} 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 Ξ» β π½*_{2π}.

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

for any Ξ» β π½*_{2π}.

# 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 π 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 (π,π)-function πΉ is an APN permutation if and only if ^{[1]}

and

for any π β π½*_{2π}.

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