Difference between revisions of "Equivalence Relations"

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m (Invariants in even characteristic)
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* The nonlinearity and the extended Walsh spectrum are invariant under CCZ-equivalence.
 
* 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.
 
* The Walsh spectrum is invariant under affine equivalence but in general not under EA- and CCZ-equivalence.
* For APN maps we have also that Δ- and Γ-ranks are invariant under CCZ-equivalence.
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* For APN maps we have also that [[:File:DeltaRank.txt|Δ-rank]] and [[:File:GammaRank.txt|Γ-rank]] are invariant under CCZ-equivalence.
 
  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 exists some subset 𝐷<sub>𝐹</sub>∈𝔽<sub>2</sub><sup>𝑛</sup>×𝔽<sub>2</sub><sup>𝑛</sup>∖{(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 exists some subset 𝐷<sub>𝐹</sub>∈𝔽<sub>2</sub><sup>𝑛</sup>×𝔽<sub>2</sub><sup>𝑛</sup>∖{(0,0)} such that <math>G_F\cdot G_F =2^n\cdot(0,0)+2\cdot D_F.</math>
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  the (𝑝,𝐵)-entry is 1 if point 𝑝 is incident with block 𝐵, is 0 otherwise.
 
  the (𝑝,𝐵)-entry is 1 if point 𝑝 is incident with block 𝐵, is 0 otherwise.
 
  The same can be done for 𝐷<sub>𝐹</sub>.
 
  The same can be done for 𝐷<sub>𝐹</sub>.
* For APN maps, the multiplier group ℳ(𝐺<sub>𝐹</sub>) is CCZ-invariant.
+
* For APN maps, the [[:File:MGF.txt|multiplier group ℳ(𝐺<sub>𝐹</sub>)]] is CCZ-invariant.
 
  The multiplier group ℳ(𝐺<sub>𝐹</sub>) of dev(𝐺<sub>𝐹</sub>) is the set of automorphism φ of 𝔽<sub>2</sub><sup>2𝑛</sup> such that φ(𝐺<sub>𝐹</sub>)=𝐺<sub>𝐹</sub>⋅(𝑢,𝑣) for some 𝑢,𝑣∈𝔽<sub>2</sub><sup>𝑛</sup>. Such automorphisms form a group contained in the automorphism group of dev(𝐺<sub>𝐹</sub>).
 
  The multiplier group ℳ(𝐺<sub>𝐹</sub>) of dev(𝐺<sub>𝐹</sub>) is the set of automorphism φ of 𝔽<sub>2</sub><sup>2𝑛</sup> such that φ(𝐺<sub>𝐹</sub>)=𝐺<sub>𝐹</sub>⋅(𝑢,𝑣) for some 𝑢,𝑣∈𝔽<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 14:24, 11 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 Δ-rank and Γ-rank 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 exists some subset 𝐷𝐹∈𝔽2𝑛×𝔽2𝑛∖{(0,0)} such that 
The Γ-rank is the dimension of the ideal generated by 𝐺𝐹 in 𝔽2[𝔽2𝑛×𝔽2𝑛] and the Δ-rank is the dimension of the ideal generated by 𝐷𝐹 in 𝔽2[𝔽2𝑛×𝔽2𝑛].
Equivalently we have that
* the Γ-rank is the rank of the incidence matrix of dev(𝐺𝐹) over 𝔽2,
* the Δ-rank is the 𝔽2-rank of an incidence matrix of dev(𝐷𝐹).
For the incidence structure dev(𝐺𝐹) with blocks  for 𝑎,𝑏∈𝔽2𝑛, the related incidence matrix is constructed, indixed by points and blocks, as follow:
the (𝑝,𝐵)-entry is 1 if point 𝑝 is incident with block 𝐵, is 0 otherwise.
The same can be done for 𝐷𝐹.
The multiplier group ℳ(𝐺𝐹) of dev(𝐺𝐹) is the set of automorphism φ of 𝔽22𝑛 such that φ(𝐺𝐹)=𝐺𝐹⋅(𝑢,𝑣) for some 𝑢,𝑣∈𝔽2𝑛. Such automorphisms form a group contained in the automorphism group of dev(𝐺𝐹).