APN Permutations

Characterization of Permutations

Component Functions

An (𝑛,𝑛)-function 𝐹 is a permutation if and only if all of its components 𝐹_λ for λ ∈ 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

[math]\displaystyle{ \sum_{a \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n }[/math]

for any λ ∈ 𝔽*_2^𝑛.

Equivalently , 𝐹 is a permutation if and only if

[math]\displaystyle{ \sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n }[/math]

for any λ ∈ 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

[math]\displaystyle{ \sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n }[/math]

and

[math]\displaystyle{ \sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}^2(D_af_\lambda) = 2^{2n} }[/math]

for any 𝑎 ∈ 𝔽*_2^𝑛.