# Difference between revisions of "Equivalence Relations"

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The same can be done for π·<sub>πΉ</sub>. | The same can be done for π·<sub>πΉ</sub>. | ||

Hence we have that | Hence we have that | ||

β | + | -- the Ξ-rank is the rank of the incidence matrix of dev(πΊ<sub>πΉ</sub>) over π½<sub>2</sub>, | |

β | + | -- the Ξ-rank is the π½<sub>2</sub>-rank of an incidence matrix of dev(π·<sub>πΉ</sub>). |

## Revision as of 14:41, 2 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 .

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- and Ξ-ranks 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 exist some subset π·_{πΉ}βπ½_{2}^{π}Γπ½_{2}^{π}β{(0,0)} such that

For the incidence structure 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}.
Hence 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}).