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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{a \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n}

for any λ ∈ 𝔽*_2^𝑛.

Equivalently , 𝐹 is a permutation if and only if

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}^2(D_af_\lambda) = 2^{2n}}

for any 𝑎 ∈ 𝔽*_2^𝑛.