APN Permutations: Difference between revisions
(Created page with "= Characterization of Permutations = == Component Functions == An <math>(n,n)</math>-function <math>F</math> is a permutation if and only if all of its components <math>F_\l...") |
mNo edit summary |
||
| Line 20: | Line 20: | ||
= Characterization of APN Permutations = | = 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 <i>n</i> 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. | |||
== Autocorrelation Functions of the Directional Derivatives == | == Autocorrelation Functions of the Directional Derivatives == | ||
Revision as of 13:29, 23 September 2019
Characterization of Permutations
Component Functions
An 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 (n,n)} -function 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 F} is a permutation if and only if all of its components 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 F_\lambda} for 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 \lambda \in \mathbb{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
for any 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 \lambda \in \mathbb{F}_{2^n}^*} .
Equivalently [1], 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 F} is a permutation if and only if
for any 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 \lambda \in \mathbb{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[2]. This implies that, in even dimension, no component can be of degree 2.
Autocorrelation Functions of the Directional Derivatives
An 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 (n,n)} -function 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 F} is an APN permutation if and only if [1]
and
for any 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 a \in \mathbb{F}_{2^n}^*} .
- ↑ 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
