APN Permutations

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