APN Permutations: Difference between revisions
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== Component Functions == | == Component Functions == | ||
An | An (𝑛,𝑛)-function 𝐹 is a permutation if and only if all of its components 𝐹<sub>λ</sub> for λ ∈ 𝔽*<sub>2<sup>𝑛</sup></sub> are balanced. | ||
== Autocorrelation Functions of the Directional Derivatives == | == Autocorrelation Functions of the Directional Derivatives == | ||
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<div><math>\sum_{a \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n</math></div> | <div><math>\sum_{a \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n</math></div> | ||
for any < | for any λ ∈ 𝔽*<sub>2<sup>𝑛</sup></sub>. | ||
Equivalently <ref name="bercanchalai2006"> 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</ref>, | Equivalently <ref name="bercanchalai2006"> 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</ref>, 𝐹 is a permutation if and only if | ||
<div><math>\sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n</math></div> | <div><math>\sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n</math></div> | ||
for any < | for any λ ∈ 𝔽*<sub>2<sup>𝑛</sup></sub>. | ||
= Characterization of APN Permutations = | = Characterization of APN Permutations = | ||
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Clearly we have that no component function can be of degree 1. (This result is true for general APN maps) | Clearly we have that no component function can be of degree 1. (This result is true for general APN maps) | ||
For | For 𝑛 even we have also that no component can be partially-bent<ref name="CalSalVil"> Marco Calderini, Massimiliano Sala, Irene Villa, ''A note on APN permutations in even dimension'', Finite Fields and Their Applications, vol. 46, 1-16, 2017</ref>. | ||
This implies that, in even dimension, no component can be of degree 2. | This implies that, in even dimension, no component can be of degree 2. | ||
== Autocorrelation Functions of the Directional Derivatives == | == Autocorrelation Functions of the Directional Derivatives == | ||
An | An (𝑛,𝑛)-function 𝐹 is an APN permutation if and only if <ref name="bercanchalai2006"></ref> | ||
<div><math>\sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n</math></div> | <div><math>\sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n</math></div> | ||
and | and | ||
<div><math>\sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}^2(D_af_\lambda) = 2^{2n}</math></div> | <div><math>\sum_{\lambda \in \mathbb{F}_{2^n}^*} \mathcal{F}^2(D_af_\lambda) = 2^{2n}</math></div> | ||
for any < | for any 𝑎 ∈ 𝔽*<sub>2<sup>𝑛</sup></sub>. | ||
Revision as of 13:05, 11 October 2019
Characterization of Permutations
Component Functions
An (𝑛,𝑛)-function 𝐹 is a permutation if and only if all of its components 𝐹λ for λ ∈ 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
[math]\displaystyle{ \sum_{a \in \mathbb{F}_{2^n}^*} \mathcal{F}(D_af_\lambda) = -2^n }[/math]
for any λ ∈ 𝔽*2𝑛.
Equivalently [1], 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*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]
[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*2𝑛.
- ↑ 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
- ↑ Marco Calderini, Massimiliano Sala, Irene Villa, A note on APN permutations in even dimension, Finite Fields and Their Applications, vol. 46, 1-16, 2017