Difference between revisions of "Equivalence Relations"

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  To define such ranks let consider 𝐹 a (𝑛,𝑛)-function and associate a group algebra element 𝐺<sub>𝐹</sub> in 𝔽[𝔽<sub>2</sub><sup>𝑛</sup>×𝔽<sub>2</sub><sup>𝑛</sup>], <math>G_F=\sum_{v\in\mathbb{F}_2^n}(v,F(v)).</math>
 
  To define such ranks let consider 𝐹 a (𝑛,𝑛)-function and associate a group algebra element 𝐺<sub>𝐹</sub> in 𝔽[𝔽<sub>2</sub><sup>𝑛</sup>×𝔽<sub>2</sub><sup>𝑛</sup>], <math>G_F=\sum_{v\in\mathbb{F}_2^n}(v,F(v)).</math>
 
  We have that for 𝐹 APN there exist some subset 𝐷<sub>𝐹</sub>βˆˆπ”½<sub>2</sub><sup>𝑛</sup>×𝔽<sub>2</sub><sup>𝑛</sup>βˆ–{(0,0)} such that <math>G_F\cdot G_F =2^n\cdot(0,0)+2\cdot D_F.</math>
 
  We have that for 𝐹 APN there exist some subset 𝐷<sub>𝐹</sub>βˆˆπ”½<sub>2</sub><sup>𝑛</sup>×𝔽<sub>2</sub><sup>𝑛</sup>βˆ–{(0,0)} such that <math>G_F\cdot G_F =2^n\cdot(0,0)+2\cdot D_F.</math>
βˆ’
  For the incidence structure with blocks <math>G_F\cdot(a,b)=\{(x+a,F(x)+b) : x\in\mathbb{F}_2^n \},</math>
+
  For the incidence structure dev(𝐺<sub>𝐹</sub>) with blocks <math>G_F\cdot(a,b)=\{(x+a,F(x)+b) : x\in\mathbb{F}_2^n \},</math>
βˆ’
  for π‘Ž,π‘βˆˆπ”½<sub>2</sub><sup>𝑛</sup>, the related incidence matrix is constructed, indixed by points and blocks, as follow:
+
  for π‘Ž,π‘βˆˆπ”½<sub>2</sub><sup>𝑛</sup>, the related incidence matrix is constructed, indixed by points and blocks, as follow:
 
  the (𝑝,𝐡)-entry is 1 if point 𝑝 is incident with block 𝐡, is 0 otherwise.
 
  the (𝑝,𝐡)-entry is 1 if point 𝑝 is incident with block 𝐡, is 0 otherwise.
 
  The same can be done for 𝐷<sub>𝐹</sub>.
 
  The same can be done for 𝐷<sub>𝐹</sub>.
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  * the Ξ“-rank is the rank of the incidence matrix of dev(𝐺<sub>𝐹</sub>) over 𝔽<sub>2</sub>,
 
  * the Ξ“-rank is the rank of the incidence matrix of dev(𝐺<sub>𝐹</sub>) over 𝔽<sub>2</sub>,
 
  * the Ξ”-rank is the 𝔽<sub>2</sub>-rank of an incidence matrix of dev(𝐷<sub>𝐹</sub>).
 
  * the Ξ”-rank is the 𝔽<sub>2</sub>-rank of an incidence matrix of dev(𝐷<sub>𝐹</sub>).
 +
Equivalently the Ξ“-rank is the dimension of the ideal generated by 𝐺<sub>𝐹</sub> in 𝔽<sub>2</sub>[𝔽<sub>2</sub><sup>𝑛</sup>×𝔽<sub>2</sub><sup>𝑛</sup>] and the Ξ”-rank is the dimension of the ideal generated by 𝐷<sub>𝐹</sub> in 𝔽<sub>2</sub>[𝔽<sub>2</sub><sup>𝑛</sup>×𝔽<sub>2</sub><sup>𝑛</sup>].
 +
* For APN maps, the multiplier group β„³(𝐺<sub>𝐹</sub>) is CCZ-invariant.
 +
The multiplier group β„³(𝐺<sub>𝐹</sub>) of dev(𝐺<sub>𝐹</sub>) is the set of automorphism Ο† of 𝔽<sub>2</sub><sup>2𝑛</sup> such that Ο†(𝐺<sub>𝐹</sub>)=𝐺<sub>𝐹</sub>β‹…(𝑒,𝑣) for some 𝑒,π‘£βˆˆπ”½<sub>2</sub><sup>𝑛</sup>. Such automorphisms form a group contained in the automorphism group of dev(𝐺<sub>𝐹</sub>).

Revision as of 15:06, 2 October 2019

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 𝑝 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 𝔽2𝑛.

  • 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 𝐹 a (𝑛,𝑛)-function and associate a group algebra element 𝐺𝐹 in 𝔽[𝔽2𝑛×𝔽2𝑛], 
We have that for 𝐹 APN there exist some subset π·πΉβˆˆπ”½2𝑛×𝔽2π‘›βˆ–{(0,0)} such that 
For the incidence structure dev(𝐺𝐹) with blocks 
for π‘Ž,π‘βˆˆπ”½2𝑛, the related incidence matrix is constructed, indixed by points and blocks, as follow:
the (𝑝,𝐡)-entry is 1 if point 𝑝 is incident with block 𝐡, is 0 otherwise.
The same can be done for 𝐷𝐹.
Hence we have that
* the Ξ“-rank is the rank of the incidence matrix of dev(𝐺𝐹) over 𝔽2,
* the Ξ”-rank is the 𝔽2-rank of an incidence matrix of dev(𝐷𝐹).
Equivalently the Ξ“-rank is the dimension of the ideal generated by 𝐺𝐹 in 𝔽2[𝔽2𝑛×𝔽2𝑛] and the Ξ”-rank is the dimension of the ideal generated by 𝐷𝐹 in 𝔽2[𝔽2𝑛×𝔽2𝑛].
  • For APN maps, the multiplier group β„³(𝐺𝐹) is CCZ-invariant.
The multiplier group β„³(𝐺𝐹) of dev(𝐺𝐹) is the set of automorphism Ο† of 𝔽22𝑛 such that Ο†(𝐺𝐹)=𝐺𝐹⋅(𝑒,𝑣) for some 𝑒,π‘£βˆˆπ”½2𝑛. Such automorphisms form a group contained in the automorphism group of dev(𝐺𝐹).