APN Permutations

From Boolean Functions
Jump to: navigation, search

Characterization of Permutations

Component Functions

An (𝑛,𝑛)-function 𝐹 is a permutation if and only if all of its components 𝐹λ for Ξ» ∈ 𝔽*2𝑛 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» ∈ 𝔽*2𝑛.

Equivalently [1], 𝐹 is a permutation if and only if

for any Ξ» ∈ 𝔽*2𝑛.

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 (𝑛,𝑛)-function 𝐹 is an APN permutation if and only if [1]

and

for any π‘Ž ∈ 𝔽*2𝑛.

  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