APN Permutations

Characterization of Permutations

Component Functions

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

Autocorrelation Functions of the Directional Derivatives

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

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

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

Equivalently , $F$ is a permutation if and only if

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

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

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. This implies that, in even dimension, no component can be of degree 2.

Autocorrelation Functions of the Directional Derivatives

An $(n,n)$-function $F$ is an APN permutation if and only if

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

and

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

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