# Difference between revisions of "Equivalence Relations"

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* affine (linear) equivalent if there exist <math>A_1,A_2</math> affine (linear) permutations of <math>\mathbb{F}_2^m</math> and <math>\mathbb{F}_2^n</math> respectively, such that <math>F'=A_1\circ F\circ A_2</math>; | * affine (linear) equivalent if there exist <math>A_1,A_2</math> affine (linear) permutations of <math>\mathbb{F}_2^m</math> and <math>\mathbb{F}_2^n</math> respectively, such that <math>F'=A_1\circ F\circ A_2</math>; | ||

* extended affine equivalent (shortly EA-equivalent) if there exists <math>A:\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m</math> affine such that <math>F'=F''+A</math>, with <math>F''</math> affine equivalent to <math>F</math>; | * extended affine equivalent (shortly EA-equivalent) if there exists <math>A:\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m</math> affine such that <math>F'=F''+A</math>, with <math>F''</math> affine equivalent to <math>F</math>; | ||

β | * Carlet-Charpin-Zinoviev equivalent (shortly CCZ-equivalent) if there exists an affine permutation <math>\mathcal{L}</math> of <math>\mathbb{F}_2^n\times\mathbb{F}_2^m</math> such that the image of the graph of <math>F</math> is the graph of <math>F'</math>, i.e. <math>\mathcal{L}(G_F)=G_{F'}</math> with <math>G_F=\{ (x,F(x)) : x\in\mathbb{F}_2^n\}</math> and <math>G_{F'}=\{ (x,F'(x)) : x\in\mathbb{F}_2^n\}</math>. | + | * Carlet-Charpin-Zinoviev equivalent (shortly CCZ-equivalent) if there exists an affine permutation <math>\mathcal{L}</math> of <math>\mathbb{F}_2^n\times\mathbb{F}_2^m</math> such that the image of the graph of <math>F</math> is the graph of <math>F'</math>, i.e. <math>\mathcal{L}(G_F)=G_{F'}</math> with <math>G_F=\{ (x,F(x)) : x\in\mathbb{F}_2^n\}</math> and <math>G_{F'}=\{ (x,F'(x)) : x\in\mathbb{F}_2^n\}</math> ([[:File:CCZeq2.txt|magma code]]). |

Clearly, it is possible to estend such definitions also for maps <math>F,F':\mathbb{F}_p^n\rightarrow\mathbb{F}_p^m</math>, for π a general prime number. | Clearly, it is possible to estend such definitions also for maps <math>F,F':\mathbb{F}_p^n\rightarrow\mathbb{F}_p^m</math>, for π a general prime number. |

## Latest revision as of 16:58, 11 October 2019

## Contents

# Some known Equivalence Relations

Two vectorial Boolean functions are called

- affine (linear) equivalent if there exist affine (linear) permutations of and respectively, such that ;
- extended affine equivalent (shortly EA-equivalent) if there exists affine such that , with affine equivalent to ;
- Carlet-Charpin-Zinoviev equivalent (shortly CCZ-equivalent) if there exists an affine permutation of such that the image of the graph of is the graph of , i.e. with and (magma code).

Clearly, it is possible to estend such definitions also for maps , for π a general prime number.

## Connections between different relations

Such equivalence relations are connected to each other. Obviously affine equivalence is a general case of linear equivalence and they are both a particular case of the EA-equivalence. Moreover, EA-equivalence is a particular case of CCZ-equivalence and each permutation is CCZ-equivalent to its inverse.

In particular we have that CCZ-equivalence coincides with

- EA-equivalence for planar functions,
- linear equivalence for DO planar functions,
- EA-equivalence for all Boolean functions,
- EA-equivalence for all bent vectorial Boolean functions,
- EA-equivalence for two quadratic APN functions.

# Invariants

- The algebraic degree (if the function is not affine) is invariant under EA-equivalence but in general is not preserved under CCZ-equivalence.
- The differential uniformity is invariant under CCZ-equivalence. (CCZ-equivalence relation is the most general known equivalence relation preserving APN and PN properties)

## Invariants in even characteristic

We consider now functions over π½_{2}^{π}.

- The nonlinearity and the extended Walsh spectrum are invariant under CCZ-equivalence.
- The Walsh spectrum is invariant under affine equivalence but in general not under EA- and CCZ-equivalence.
- For APN maps we have also that Ξ-rank and Ξ-rank are invariant under CCZ-equivalence.

To define such ranks let consider πΉ a (π,π)-function and associate a group algebra element πΊ_{πΉ}in π½[π½_{2}^{π}Γπ½_{2}^{π}], We have that for πΉ APN there exists some subset π·_{πΉ}βπ½_{2}^{π}Γπ½_{2}^{π}β{(0,0)} such that The Ξ-rank is the dimension of the ideal generated by πΊ_{πΉ}in π½_{2}[π½_{2}^{π}Γπ½_{2}^{π}] and the Ξ-rank is the dimension of the ideal generated by π·_{πΉ}in π½_{2}[π½_{2}^{π}Γπ½_{2}^{π}]. Equivalently we have that * the Ξ-rank is the rank of the incidence matrix of dev(πΊ_{πΉ}) over π½_{2}, * the Ξ-rank is the π½_{2}-rank of an incidence matrix of dev(π·_{πΉ}). For the incidence structure dev(πΊ_{πΉ}) with blocks for π,πβπ½_{2}^{π}, the related incidence matrix is constructed, indixed by points and blocks, as follow: the (π,π΅)-entry is 1 if point π is incident with block π΅, is 0 otherwise. The same can be done for π·_{πΉ}.

- For APN maps, the multiplier group β³(πΊ
_{πΉ}) is CCZ-invariant.

The multiplier group β³(πΊ_{πΉ}) of dev(πΊ_{πΉ}) is the set of automorphism Ο of π½_{2}^{2π}such that Ο(πΊ_{πΉ})=πΊ_{πΉ}β (π’,π£) for some π’,π£βπ½_{2}^{π}. Such automorphisms form a group contained in the automorphism group of dev(πΊ_{πΉ}).