Equivalence Relations

Some known Equivalence Relations

Two vectorial Boolean functions [math]\displaystyle{ F,F':\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m }[/math] are called

• affine (linear) equivalent if there exist [math]\displaystyle{ A_1,A_2 }[/math] affine (linear) permutations of [math]\displaystyle{ \mathbb{F}_2^m }[/math] and [math]\displaystyle{ \mathbb{F}_2^n }[/math] respectively, such that [math]\displaystyle{ F'=A_1\circ F\circ A_2 }[/math];
• extended affine equivalent (shortly EA-equivalent) if there exists [math]\displaystyle{ A:\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m }[/math] affine such that [math]\displaystyle{ F'=F''+A }[/math], with [math]\displaystyle{ F'' }[/math] affine equivalent to [math]\displaystyle{ F }[/math];
• Carlet-Charpin-Zinoviev equivalent (shortly CCZ-equivalent) if there exists an affine permutation [math]\displaystyle{ \mathcal{L} }[/math] of [math]\displaystyle{ \mathbb{F}_2^n\times\mathbb{F}_2^m }[/math] such that the image of the graph of [math]\displaystyle{ F }[/math] is the graph of [math]\displaystyle{ F' }[/math], i.e. [math]\displaystyle{ \mathcal{L}(G_F)=G_{F'} }[/math] with [math]\displaystyle{ G_F=\{ (x,F(x)) : x\in\mathbb{F}_2^n\} }[/math] and [math]\displaystyle{ G_{F'}=\{ (x,F'(x)) : x\in\mathbb{F}_2^n\} }[/math] (magma code).

Clearly, it is possible to estend such definitions also for maps [math]\displaystyle{ 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 𝔽₂^𝑛.

• 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], [math]\displaystyle{ G_F=\sum_{v\in\mathbb{F}_2^n}(v,F(v)). }[/math]
We have that for 𝐹 APN there exists some subset 𝐷𝐹∈𝔽2𝑛×𝔽2𝑛∖{(0,0)} such that [math]\displaystyle{ G_F\cdot G_F =2^n\cdot(0,0)+2\cdot D_F. }[/math]
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 [math]\displaystyle{ G_F\cdot(a,b)=\{(x+a,F(x)+b) : x\in\mathbb{F}_2^n \}, }[/math] 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.

• For APN maps, the multiplier group ℳ(𝐺_𝐹) is CCZ-invariant.

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).