# Equivalence Relations

## 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 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 𝔽_{2}^{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- 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_{𝐹} in 𝔽[𝔽_{2}^{𝑛}×𝔽_{2}^{𝑛}]

We have that for 𝐹 APN there exist some subset 𝐷_{𝐹}∈𝔽_{2}^{𝑛}×𝔽_{2}^{𝑛}∖{(0,0)} such that

For the incidence structure with blocks

for 𝑎,𝑏∈𝔽_{2}^{𝑛}, 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 𝐷_{𝐹}.
Hence we have that

-- the Γ-rank is the rank of the incidence matrix of dev(𝐺_{𝐹}) over 𝔽_{2},

-- the Δ-rank is the 𝔽_{2}-rank of an incidence matrix of dev(𝐷_{𝐹}).