Equivalence Relations - Revision history

Ivi062 at 14:58, 11 October 2019

2019-10-11T14:58:40Z

← Older revision		Revision as of 14:58, 11 October 2019
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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>;
	* 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 𝑝 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 𝑝 a general prime number.

Ivi062: /* Invariants in even characteristic */

2019-10-11T13:24:09Z

Invariants in even characteristic

← Older revision		Revision as of 13:24, 11 October 2019
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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.		* 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>).

Ivi062 at 14:34, 2 October 2019

2019-10-02T14:34:05Z

← Older revision		Revision as of 14:34, 2 October 2019
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	* For APN maps we have also that Δ- and Γ-ranks are invariant under CCZ-equivalence.		* For APN maps we have also that Δ- and Γ-ranks 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 ~~exist~~ 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>
	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>		The Γ-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 Δ-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 𝑎,𝑏∈𝔽<sub>2</sub><sup>𝑛</sup>, the related incidence matrix is constructed, indixed by points and blocks, as follow:		Equivalently we have that
			* the Γ-rank is the rank of the incidence matrix of dev(𝐺<sub>𝐹</sub>) over 𝔽<sub>2</sub>,
			* the Δ-rank is the 𝔽<sub>2</sub>-rank of an incidence matrix of dev(𝐷<sub>𝐹</sub>).
			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> for 𝑎,𝑏∈𝔽<sub>2</sub><sup>𝑛</sup>, 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 (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>.
	~~Hence we have that~~
	* the Γ-rank is the rank of the incidence matrix of dev(𝐺<sub>𝐹</sub>) over 𝔽<sub>2</sub>,
	* the Δ-rank is the 𝔽<sub>2</sub>-rank of an incidence matrix of dev(𝐷<sub>𝐹</sub>).
	Equivalently the Γ-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 Δ-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 ℳ(𝐺<sub>𝐹</sub>) is CCZ-invariant.		* For APN maps, the 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>).

Ivi062 at 14:06, 2 October 2019

2019-10-02T14:06:06Z

← Older revision		Revision as of 14:06, 2 October 2019
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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>∈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 exist 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>
	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>
	for 𝑎,𝑏∈𝔽<sub>2</sub><sup>𝑛</sup>, the related incidence matrix is constructed, indixed by points and blocks, as follow:		for 𝑎,𝑏∈𝔽<sub>2</sub><sup>𝑛</sup>, 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 (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>.
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	* the Γ-rank is the rank of the incidence matrix of dev(𝐺<sub>𝐹</sub>) over 𝔽<sub>2</sub>,		* the Γ-rank is the rank of the incidence matrix of dev(𝐺<sub>𝐹</sub>) over 𝔽<sub>2</sub>,
	* the Δ-rank is the 𝔽<sub>2</sub>-rank of an incidence matrix of dev(𝐷<sub>𝐹</sub>).		* the Δ-rank is the 𝔽<sub>2</sub>-rank of an incidence matrix of dev(𝐷<sub>𝐹</sub>).
			Equivalently the Γ-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 Δ-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 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>).

Ivi062 at 13:46, 2 October 2019

2019-10-02T13:46:13Z

← Older revision		Revision as of 13:46, 2 October 2019
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	* 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.		* For APN maps we have also that Δ- and Γ-ranks 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 exist 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 exist 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>
	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 with blocks <math>G_F\cdot(a,b)=\{(x+a,F(x)+b) : x\in\mathbb{F}_2^n \},</math>
	for 𝑎,𝑏∈𝔽<sub>2</sub><sup>𝑛</sup>, the related incidence matrix is constructed, indixed by points and blocks, as follow:		for 𝑎,𝑏∈𝔽<sub>2</sub><sup>𝑛</sup>, 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 (𝑝,𝐵)-entry is 1 if point 𝑝 is incident with block 𝐵, is 0 otherwise.

Ivi062: /* Invariants in even characteristic */

2019-10-02T13:43:49Z

Invariants in even characteristic

← Older revision		Revision as of 13:43, 2 October 2019
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	* 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.		* For APN maps we have also that Δ- and Γ-ranks 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>]		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>
	~~<center>~~<math>G_F=\sum_{v\in\mathbb{F}_2^n}(v,F(v)).</math~~></center~~>		We have that for 𝐹 APN there exist 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 exist some subset 𝐷<sub>𝐹</sub>∈𝔽<sub>2</sub><sup>𝑛</sup>×𝔽<sub>2</sub><sup>𝑛</sup>∖{(0,0)} such that		For the incidence structure with blocks <math>G_F\cdot(a,b)=\{(x+a,F(x)+b) : x\in\mathbb{F}_2^n \},</math>
	~~<center>~~<math>G_F\cdot G_F =2^n\cdot(0,0)+2\cdot D_F.</math~~></center~~>		for 𝑎,𝑏∈𝔽<sub>2</sub><sup>𝑛</sup>, the related incidence matrix is constructed, indixed by points and blocks, as follow:
	For the incidence structure with blocks		the (𝑝,𝐵)-entry is 1 if point 𝑝 is incident with block 𝐵, is 0 otherwise.
	~~<center>~~<math>G_F\cdot(a,b)=\{(x+a,F(x)+b) : x\in\mathbb{F}_2^n \},</math~~></center~~>		The same can be done for 𝐷<sub>𝐹</sub>.
	for 𝑎,𝑏∈𝔽<sub>2</sub><sup>𝑛</sup>, the related incidence matrix is constructed, indixed by points and blocks, as follow:		Hence we have that
	the (𝑝,𝐵)-entry is 1 if point 𝑝 is incident with block 𝐵, is 0 otherwise.		* the Γ-rank is the rank of the incidence matrix of dev(𝐺<sub>𝐹</sub>) over 𝔽<sub>2</sub>,
	The same can be done for 𝐷<sub>𝐹</sub>.		* the Δ-rank is the 𝔽<sub>2</sub>-rank of an incidence matrix of dev(𝐷<sub>𝐹</sub>).
	Hence we have that
	-- the Γ-rank is the rank of the incidence matrix of dev(𝐺<sub>𝐹</sub>) over 𝔽<sub>2</sub>,
	-- the Δ-rank is the 𝔽<sub>2</sub>-rank of an incidence matrix of dev(𝐷<sub>𝐹</sub>).

Ivi062: /* Invariants in even characteristic */

2019-10-02T13:41:33Z

Invariants in even characteristic

← Older revision		Revision as of 13:41, 2 October 2019
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	The same can be done for 𝐷<sub>𝐹</sub>.		The same can be done for 𝐷<sub>𝐹</sub>.
	Hence we have that		Hence we have that
	** the Γ-rank is the rank of the incidence matrix of dev(𝐺<sub>𝐹</sub>) over 𝔽<sub>2</sub>,		-- the Γ-rank is the rank of the incidence matrix of dev(𝐺<sub>𝐹</sub>) over 𝔽<sub>2</sub>,
	** the Δ-rank is the 𝔽<sub>2</sub>-rank of an incidence matrix of dev(𝐷<sub>𝐹</sub>).		-- the Δ-rank is the 𝔽<sub>2</sub>-rank of an incidence matrix of dev(𝐷<sub>𝐹</sub>).

Ivi062: Created page with "=Some known Equivalence Relations = Two vectorial Boolean functions $F,F':\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m$ are called * affine (linear) equivalent if there..."

2019-10-02T13:37:21Z

Created page with "=Some known Equivalence Relations = Two vectorial Boolean functions <math>F,F':\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m</math> are called * affine (linear) equivalent if there..."

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=Some known Equivalence Relations =
Two vectorial Boolean functions <math>F,F':\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m</math> are called
* 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>;
* 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>.

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 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- and Γ-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]
<center><math>G_F=\sum_{v\in\mathbb{F}_2^n}(v,F(v)).</math></center>
We have that for 𝐹 APN there exist some subset 𝐷𝐹∈𝔽2𝑛×𝔽2𝑛∖{(0,0)} such that
<center><math>G_F\cdot G_F =2^n\cdot(0,0)+2\cdot D_F.</math></center>
For the incidence structure with blocks
<center><math>G_F\cdot(a,b)=\{(x+a,F(x)+b) : x\in\mathbb{F}_2^n \},</math></center>
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.
Hence 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).

Equivalence Relations - Revision history

Ivi062 at 14:58, 11 October 2019

Ivi062: /* Invariants in even characteristic */

Ivi062 at 14:34, 2 October 2019

Ivi062 at 14:06, 2 October 2019

Ivi062 at 13:46, 2 October 2019

Ivi062: /* Invariants in even characteristic */

Ivi062: /* Invariants in even characteristic */

Ivi062: Created page with "=Some known Equivalence Relations = Two vectorial Boolean functions F,F':\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m are called * affine (linear) equivalent if there..."

Ivi062: Created page with "=Some known Equivalence Relations = Two vectorial Boolean functions $F,F':\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m$ are called * affine (linear) equivalent if there..."