On design-theoretic aspects of Boolean and vectorial bent functions
Alexandr Polujan Alexander Pott
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On design-theoretic aspects
of Boolean and vectorial bent functions
    Alexandr A. Polujan, Alexander Pott
The authors are with the Institute of Algebra and Geometry, Faculty of Mathematics, Otto von Guericke University Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany (email: alexandr.polujan@ovgu.de; alexander.pott@ovgu.de).

    December 30, 2023. .../latex/6he-sdm-2/6he-sdm-2-3.tex
=============================================================================================================================================================================================================================================================================




There are two construction methods of designs from (n,m)-bent functions, known as translation and addition designs. In this paper we analyze, which equivalence relation for Boolean bent functions, i.e. (n,1)-bent functions, and vectorial bent functions, i.e. (n,m)-bent functions with 2≤ m≤ n/2, is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs. First, we observe that similar to the Boolean bent functions, extended-affine equivalence of vectorial (n,m)-bent functions and isomorphism of addition designs are the same concepts for all even n and m≤ n/2. Further, we show that extended-affine inequivalent Boolean bent functions in n variables, whose translation designs are isomorphic, exist for all n≥6. This implies, that isomorphism of translation designs for Boolean bent functions is a coarser equivalence relation than extended-affine equivalence. However, we do not observe the same phenomenon for vectorial bent functions in a small number of variables. We classify and enumerate all vectorial bent functions in six variables and show, that in contrast to the Boolean case, one cannot exhibit isomorphic translation designs from extended-affine inequivalent vectorial (6,m)-bent functions with m∈{ 2,3 }.



Bent Function, Extended-Affine Equivalence, Combinatorial Design, Linear Code, Difference Set, Relative Difference Set.





§ INTRODUCTION


Boolean and vectorial bent functions, also known as perfect nonlinear functions <cit.>, are special mappings of finite fields, which have the maximum Hamming distance from the set of all affine functions. Being optimal discrete structures, they have numerous applications in combinatorics, cryptography, coding and design theory. Particularly, the interaction between design theory and the theory of perfect nonlinear functions is of special interest. For instance, any new construction of bent functions may lead to a new construction of certain designs. On the other hand, combinatorial invariants of incidence structures constructed from functions over finite fields serve as good distinguishers between inequivalent functions and even classes of functions <cit.>. Before we briefly mention the main constructions of designs from bent functions and their most notable applications, we would like to point the reader's attention, that the notation we use below for translation and addition designs of bent functions will be introduced in details in the following sections.

A translation design of a function F_2^n→_2^m (not necessarily perfect nonlinear) is defined as the development (A) of a certain set A, which is constructed from the function F and has a nice combinatorial structure <cit.>. The classical choice of a set A for a Boolean bent function f_2^n→_2 is either the support f, a (2^n,2^n-1± 2^n/2-1,2^n-2± 2^n/2-1) difference set, or the graph f, a (2^n, 2, 2^n, 2^n-1) relative difference set, while for the vectorial function F_2^n→_2^m one considers only the graph F, which is a (2^n, 2^m, 2^n, 2^n-m) relative difference set. The addition design (F) of a perfect nonlinear function F_2^n→_2^m is defined as the design, supported by codewords of the minimum weight of the first-order Reed-Muller code, appended by the function F, see <cit.>. 
In this way, one can construct three designs from a Boolean bent function f on _2^n: two translation designs (f),(f) and one addition design (f). However, for a vectorial bent function F_2^n→_2^m there are only two options: one translation design (F) and one addition design (F).

So far, translation and addition designs are used in the context of extended-affine equivalence of Boolean bent functions, which is known as the most general equivalence relation for Boolean functions. For instance, Weng et al. in <cit.> used the 2-rank of the translation design (f) to prove, that almost every Desarguesian partial spread bent function is not extended-affine equivalent to a Maiorana-McFarland bent function. Recently, the authors of this paper in <cit.> used algebraic invariants of  (f) and (f) to show inequivalence of certain homogeneous cubic bent functions. Bending in <cit.> proved that extended-affine equivalence of bent functions coincides with isomorphism of addition designs. As a useful application of this result, one can use computer algebra systems, e.g.  Magma <cit.> and GAP <cit.>, to check effectively the equivalence of bent functions in a small number of variables via the isomorphism of addition designs. Bending in <cit.> used invariants of the addition design (f) to derive a necessary condition for a bent function f on _2^n to be extended-affine equivalent to a Maiorana-McFarland bent function. Despite the fact, that the translation and the addition  designs of vectorial bent functions are defined in the same way as for Boolean bent functions, there are no similar applications for vectorial bent functions so far.

The main goal of this paper is to compare Boolean and vectorial bent functions from the point of view of differences between extended-affine equivalence and isomorphism of the addition and the translation designs. For instance, in the Boolean bent case isomorphism of addition designs carries all the information about the extended-affine equivalence of Boolean bent functions, and vice versa. Our first objective is to prove, that the same phenomenon occurs for addition designs of vectorial bent functions. In general, isomorphic incidence structures do not necessarily come from equivalent difference sets: Edel and Pott in <cit.> observed an example of extended-affine inequivalent Boolean bent functions f and f' on _2^6, whose translation designs (f) and (f') are isomorphic. However, these incidence structures do not have a proper generalization for the vectorial case. Our second objective is to extend the observation of Edel and Pott for translation designs (f) and (f') of Boolean functions, i.e. find a pair of Boolean bent functions f,f' on _2^n for any n≥ 6, which are extended-affine inequivalent but their translation designs (F) and (f') are isomorphic. Since the translation design (f) is invariant for the extended-affine equivalence, it will imply that isomorphism of translation designs (F) and (f') of Boolean functions f and f' on _2^n is a coarser equivalence relation than extended-affine equivalence. The third objective of this paper is to show, that in contrast to the Boolean case, isomorphism of designs (F) and (F') of vectorial bent functions in six variables coincides with the extended-affine equivalence. 

After introducing the necessary background on bent functions and designs in Subsection <ref>, we consider addition designs of vectorial bent functions. In Section <ref> we prove that similarly to Boolean bent functions, extended-affine equivalence of vectorial bent functions coincides with the isomorphism of addition designs. In this way, we solve a recent open problem, addressed by Ding, Munemasa and Tonchev in <cit.>. In Section <ref> we first provide examples of extended-affine inequivalent Boolean bent functions f,f' on _2^n, whose translation designs (f) and (f') are isomorphic. Consequently, we prove that for any n≥6 the isomorphism of translation designs (f) and (f') of Boolean bent functions f and f' on _2^n is coarser than extended-affine equivalence. In Section <ref> we show that the similar phenomenon does not occur for vectorial bent functions in a small number of variables. We classify and enumerate all vectorial bent functions in six variables and  observe, that in contrast to the Boolean case, vectorial bent functions F and F' on _2^6 are extended-affine equivalent if and only if their translation designs (F) and (F') are isomorphic. In Section <ref> we give concluding remarks and raise some open problems on bent functions and their designs. In Appendix <ref> we list algebraic normal forms of the obtained representatives of equivalence classes of vectorial bent functions in six variables together with their invariants.



 §.§ Preliminaries



Let _2={0,1} be the finite field with two elements and let _2^n be the vector space of dimension n over _2. Mappings F_2^n→_2^m are called (n,m)-functions. The single-output case m=1 corresponds to Boolean functions, while in the multi-output case m≥2 one deals with vectorial functions. The graph F of an (n,m)-function F is the set F:={(, F()) ∈_2^n}. The support f of a Boolean function f on _2^n is the set f:={𝐱∈_2^n f(𝐱)=1}. Any Boolean function f_2^n→_2 can be uniquely represented as a multivariate polynomial in the ring _2[x_1,…,x_n]/(x_1⊕ x_1^2,…,x_n⊕ x_n^2). This representation is called the algebraic normal form (ANF for short) and given by


    f(𝐱)=⊕_𝐯∈_2^nc_𝐯( ∏_i=1^n x_i^v_i),


where = (x_1,…, x_n)∈_2^n,  c_∈_2 and = (v_1,…, v_n)∈_2^n.  

There are several criteria, which an (n,m)-function has to satisfy in order to be considered as a good cryptographic primitive, among them are high algebraic degree and high nonlinearity. The algebraic degree of a Boolean function f_2^n→_2, denoted by (f), is the algebraic degree of its ANF as a multivariate polynomial. This definition can essentially be extended to the vectorial case. Any vectorial function F_2^n→_2^m can be uniquely (up to the choice of basis of _2^m) associated with m coordinate Boolean functions f_i_2^n→_2 for 1≤ i≤ m as a column-vector F(𝐱):=(f_1(𝐱),…,f_m(𝐱))^T. In this way, the algebraic degree of an (n,m)-function F is defined by (F):=max_1≤ i≤ m(f_i). Clearly, the algebraic degree of an (n,m)-function F can not exceed n.

The nonlinearity of a Boolean function f_2^n→_2 is a measure of distance between the function f and the set of all affine functions 𝒜_n:={ l_2^n→_2 | (l) ≤ 1 }. Formally, it is defined as (f):=min_l∈𝒜_n d_H(f,l), where d_H(f,g):=|{∈_2^n f()≠ g() }| is the Hamming distance between functions f and g on _2^n. This definition can be extended for the vectorial case using the notion of component functions. Recall that for an (n,m)-function F, the component function F_ is the Boolean function F__2^n→_2, given by F_():=⟨, F() ⟩_m, where ⟨·,·⟩_m is a non-degenerate bilinear form on _2^m. In this way, the nonlinearity of a vectorial (n,m)-function F is the minimum nonlinearity of all its component functions and is given by (F):=min_l∈𝒜_n,∈_2^m ∖{0} d_H(F_,l). The main tool to compute the nonlinearity of an (n,m)-function F is the Walsh transform W_F_2^n×_2^m→, defined by
W_F(,):= W_F_() and

    W_F_():= ∑_∈_2^n (-1)^F_()⊕⟨, ⟩_n

for ∈_2^n and ∈_2^m. Using the Walsh transform, the nonlinearity of an (n,m)-function F can be computed as 
(F):=2^n-1- 1/2·max_∈_2^n,∈_2^m ∖{0}| W_F(,) |.
The upper bound on nonlinearity of an (n,m)-function F is given by ( F ) ≤ 2 ^  n - 1  - 2 ^  n / 2 -1 and functions, achieving this bound are called perfect nonlinear.	

	An (n,m)-function F is called bent or perfect nonlinear if ( F ) = 2 ^  n - 1  - 2 ^  n / 2  - 1.


	Throughout the paper we will call single-output bent functions, i.e. m=1 Boolean bent functions, while multi-output bent functions, i.e. m≥2, vectorial bent functions. One can show that an (n,m)-function F is bent if for all ∈_2^n and all ∈_2^m with ≠0 the Walsh transform satisfies W_F(,)= ± 2^n/2. Boolean bent functions exist on _2^n if and only if n is even. Vectorial (n,m)-bent functions exist if and only if m≤ n/2, as it was shown by Nyberg in <cit.>. The algebraic degree of an (n,m)-bent function is at most n/2, see <cit.>.

On the set of all (n,m)-functions we introduce an equivalence relation in the following way. We say that two (n,m)-functions F,F' are extended-affine equivalent (EA-equivalent for short), if there exist a linear permutation A_1 of _2^m, an affine permutation A_2 of _2^n and an affine function A_3_2^n→_2^m such that F=A_1∘ F' ∘ A_2 ⊕ A_3. Further we will study equivalence of (n,m)-functions in connection with the equivalence of the associated linear codes and designs. We refer to <cit.> and <cit.> for extensive references on the subject.		

A linear code  over _2 is a vector subspace ⊆_2^n. Elements of a linear code 𝒞 are called codewords. The number of nonzero coordinates of a codeword ∈𝒞 is called the weight of  and is denoted by (). The minimum distance of a linear code is the minimum weight of its nonzero codewords. We say, that 𝒞⊆_2^n is an [n,k,d]-linear code, if 𝒞 has dimension k and the minimum distance d. The support of a codeword =(c_1,…,c_n)∈𝒞 is defined by ()={1 ≤ i ≤ n: c_i≠ 0}⊆{1,2,3, …, n}. Two linear codes  and ' are permutation equivalent provided there is a permutation of coordinates which sends the code  to '.

An incidence structure ℐ=(𝒫, ℬ) is called a 2-(v, k, λ) design, if the cardinality of the point set 𝒫 is v, the set of blocks ℬ is a collection of k-subsets of 𝒫 and every 2-subset of points { p,q }⊂𝒫 is contained in exactly λ blocks of ℬ.  There are several generalizations of 2-designs, one of them is a divisible design. For instance, the incidence structure ℐ=(𝒫, ℬ) is called a (μ, ν, k, λ) divisible design, if the point set 𝒫 with |𝒫|=v=μ·ν elements is divided into μ point classes of size ν each, the block set ℬ is a collection of k-subsets of 𝒫 and the number of blocks, containing any 2-subset { p,q }⊂𝒫 depends on the relation between points p and q in the following way: if p and q are in the same point class, the 2-subset { p, q } is not contained in a block; otherwise it is contained in exactly λ blocks. All the information about an incidence structure ℐ is contained in its incidence matrix M(ℐ)=(m_i,j), which is a binary b× v matrix with m_i,j=1 if p_j∈ B_i and m_i,j=0 otherwise. In this way, two incidence structures ℐ and ℐ' are isomorphic, if there exist permutation matrices P and Q such that M(ℐ)=P· M(ℐ') · Q.



§ ADDITION DESIGNS OF BOOLEAN AND VECTORIAL BENT FUNCTIONS


Dillon and Schatz in <cit.> and Bending in <cit.> proved independently that Boolean bent functions f and f' on _2^n are extended-affine equivalent if and only if their addition designs (f) and (f') are isomorphic. In this section we show that similar to the Boolean case, vectorial (n,m)-bent functions F and F' are extended-affine equivalent if and only if their addition designs (F) and (F') are isomorphic. We also use the result of Bending <cit.> to show, how one can construct an incidence matrix of the addition design of a vectorial bent function with the help of component functions and their duals. First, we give the definition of the addition design of a bent function.

	Let F be an (n,m)-bent function and C_F be an (n+m+1) × 2^n-matrix  over 𝔽_2, given by
	
    C_F=([   ⋯   1   ⋯;   ⋯       ⋯;   ⋯ F()   ⋯ ])_∈𝔽_2^n.

	We define the linear code 𝒞(F) over 𝔽_2, as the row space of the matrix C_F. It is not difficult to check, that the linear code 𝒞(F) is a [2^n,n+m+1,2^n-1-2^n/2-1]-code.
	Further, we define two sets 𝒫={∈_2^n } and ℬ={(f) f ∈𝒞(F),  (f)=2^n-1-2^n/2-1}. The addition design of an (n,m)-bent function F is the incidence structure (F)=(𝒫,ℬ), which is supported by codewords of the minimum weight of the linear code 𝒞(F). Ding, Munemasa and Tonchev in <cit.> proved that the addition design (F) of an (n,m)-bent function F is a 2-(2^n,2^n-1- 2^n/2-1,(2^m-1)·(2^n-2 - 2^n/2-1)) design. 



	The designs (F) for vectorial (n,m)-bent functions F were introduced recently by Ding, Munemasa and Tonchev in <cit.>. Throughout the paper we will call these objects “addition design”, motivated by terminology introduced by Bending in his thesis <cit.> for the designs (f) of Boolean bent functions f on _2^n. The term “addition” means, that blocks of the design (f) are formed by supports of bent functions, obtained via addition of the original bent function f_2^n→_2 to those affine functions l_2^n→_2, which satisfy (f⊕ l)=2^n-1- 2^n/2-1.



	An incidence matrix of the addition design (F) of an (n,m)-vectorial bent function, similarly to the Boolean case, can be constructed without the use of the linear code 𝒞(F). Recall that the dual of a Boolean bent function f_2^n→_2 is a bent function f̃_2^n→_2, defined by W_f()=2^n/2(-1)^f̃(). Bending in <cit.> proved, that an incidence matrix of the design (f) can be constructed with the help of the dual function f̃ in the following way (without loss of generality we assume, that f(0)=0):
	
    M((f))=    (m_,)_,∈_2^n,
     m_,=    f̃()⊕ f()⊕⟨,⟩_n⊕f̃(0).

	In this way, an incidence matrix of the addition design (F) of an (n,m)-bent function F (w.l.o.g. we assume F(0)=0) can be constructed as the concatenation of incidence matrices of addition designs (F_𝐛) of the nonzero component functions F_𝐛 of F, namely:
	
    M((F))=[     M((F_𝐛_1));     M((F_𝐛_2));              ⋮; M((F_𝐛_2^n-1)) ].



Recently, Ding, Munemasa and Tonchev conjectured <cit.>, that extended-affine equivalence of vectorial bent functions, similarly to the Boolean case <cit.>, coincides with the isomorphism of their addition designs. In the following theorem we show that this conjecture is true.


	Let F and F' be two (n,m)-bent functions. Bent functions F and F' are extended-affine equivalent if and only if addition designs 𝔻(F) and 𝔻(F') are isomorphic. 
	
		First, let us recall the definition of the CCZ-equivalence (abbreviation from Carlet-Charpin-Zinoviev). Two (n,m)-functions F and F' are called CCZ-equivalent, if their graphs F and F' are affine equivalent, i.e. there exists an affine permutation A of _2^n×_2^m s.t. A(F)=F'. The CCZ-equivalence is known as the most general equivalence relation for (n,m)-functions, however as it was shown in <cit.> two (n,m)-bent functions F,F' are extended-affine equivalent if and only they are CCZ-equivalent. By <cit.> functions F and F' are CCZ-equivalent if and only if the linear codes 𝒞(F) and 𝒞(F') are equivalent. The proof of the statement now follows from <cit.>, since linear codes 𝒞(F) and 𝒞(F') of (n,m)-bent functions F and F' are equivalent if and only if the addition designs 𝔻(F) and 𝔻(F') are isomorphic, since an incidence matrix of the addition design (F) is a generator matrix of the code 𝒞(F).
	




§ TRANSLATION DESIGNS OF BOOLEAN BENT FUNCTIONS

In this section we prove that isomorphism of translation designs (G_f) and (G_f') of Boolean bent functions f,f'_2^n→_2 is a coarser equivalence relation for Boolean bent functions than extended-affine equivalence. First, we give a general definition of the translation design.

	For a subset A of an additive group (G,+) the development (A) of A is an incidence structure, whose points are the elements in G, and whose blocks are the translates A + g := {a+g:a∈ A }. For a Boolean function f on _2^n there are two ways to construct a translation design, see <cit.>:
	
		
  * (f), which is a 2-(2^n,2^n-1± 2^n/2-1,2^n-2± 2^n/2-1) design for a bent function f  on _2^n, with the  “+” sign if f(0)=1, and “-” otherwise;
		
  * (f), which is a (2^n, 2, 2^n, 2^n-1) divisible design for a bent function f on _2^n.
	
	It seems, there is no proper generalization of the translation designs (f) for vectorial bent functions, while the second design (f) is defined in the same way. Thus, the translation design of an (n,m)-function F is defined as:
	
		
  * (F), which is a (2^n, 2^m, 2^n, 2^n-m) divisible design for an (n,m)-bent function F.
	



	Despite translation and addition designs (f) and (f) of a Boolean bent function f on _2^n have the same parameters (up to a complement), in general they are non-isomorphic. However, for a quadratic bent function f_2^n→_2 the designs (f) and (f) are isomorphic, see <cit.>.
	
Further we denote by 𝐉_2^n the all-one-matrix of order 2^n and by A ⊗ B the Kronecker product of matrices A and B. In the following proposition we observe, that from isomorphism of designs (f) and (f') of Boolean (not necessarily bent) functions f,f' on _2^n follows the isomorphism of designs (f) and (f'). 

	Let f,f'_2^n→_2 be two Boolean functions. If (f) and (f') are isomorphic, then (f) and (f') are isomorphic too.
	
		First, we denote the complement of a Boolean function f by f̅:= f⊕ 1 and by M_f an incidence matrix of the translation design (f), which can be computed as follows M_f:=(f(⊕))_,∈_2^n, see <cit.>. With the use of incidence matrices M_f and M_f of translation designs (f) and (f̅), respectively, one can decompose the incidence matrix M((f)) of a Boolean function f_2^n→_2 in the following way  <cit.>:
		
		
    M((f))=
    		(
    		[ M_f M_f; M_f M_f;     ]).

		
		Since (f) and (f') are isomorphic, there exist permutation matrices P and Q, such that M_f=P· M_f'· Q. Clearly, (f) and (f') are isomorphic with the same permutation matrices P and Q, as one can see from the following calculations
		
    M_f=     M_f⊕_2^n=P· M_f'· Q ⊕_2^n
    =     P·(M_f'⊕_2^n)· Q=P· M_f'· Q.

		
		Finally, since M((f))=(_2⊗ P)· M((f')) · (_2⊗ Q), we conclude that (f) and (f') are isomorphic.
	



	The converse of the previous statement is not true in general. A simple argument to see it, is that the design (f) of a Boolean function f on _2^n is invariant for affine equivalence <cit.>, that is f()=f'( A⊕) for a non-degenerate n× n matrix A, but not extended-affine equivalence <cit.>. In general, there are many examples of non-isomorphic translation designs (f) and (f⊕ l), obtained by addition of an affine (and even linear) function l to a bent function f on _2^n, as it was mentioned by Dempwolff to the second author of this paper in a private communication. At the same time, the design (F) of an (n,m)-function F is invariant for CCZ-equivalence and, hence, extended-affine equivalence, see <cit.>. In view of this remark we define isomorphic (n,m)-functions in the following way.



	Two (n,m)-functions F,F' are isomorphic, if translation designs (F) and (F') are isomorphic.



	Let f be a quadratic and f' be a cubic Maiorana-McFarland bent functions on _2^6, given by their ANFs
	
    f()=    x_1 x_2 ⊕ x_3 x_4 ⊕ x_5 x_6,
    
    	f'()=    x_1 x_2 ⊕ x_3 x_4 ⊕ x_5 x_6 ⊕ x_1 x_3 x_5.

	
	Edel and Pott in <cit.> observed that the designs (f) and (f') are isomorphic. By Proposition <ref> the divisible designs (f) and (f') are isomorphic too, and hence the functions f and f' are isomorphic in the sense of Definition <ref>.

In the following proposition we show that using the direct sum construction one can always extend a pair of isomorphic incidence structures derived from Boolean and vectorial functions to an infinite family.

	Let f,f'_2^n→_2 be two Boolean functions and let F,F' be two (n,m)-functions.
	
		
  * Let h be a Boolean function on _2^k. If translation designs (f) and (f') are isomorphic, then the translation designs (f⊕ h) and (f'⊕ h) of functions f⊕ h and f'⊕ h on _2^n×_2^k are isomorphic too.
		
  * Let H be a (k,m)-function. If translation designs (F) and (F') are isomorphic, then the translation designs (F⊕ H) and (F'⊕ H) of (n+k,m)-functions F⊕ H and F'⊕ H are isomorphic too.
	
	
		1. Let ,∈_2^n and ,∈_2^k. For any fixed ,∈_2^k the entry of the incidence matrix M_f⊕ h of the translation design (f⊕ h) labeled by ((,), (,)) is f(⊕) ⊕ h(⊕). In this way, the incidence matrix M_f⊕ h has the following form
		
		
    M_f⊕ h=(_2^k⊗ M_f) ⊕ (M_h⊗_2^n).

		Since (f) and (f') are isomorphic, there exist permutation matrices P and Q, such that M_f=P· M_f'· Q. Finally, from the following equality
		
		
    M_f⊕ h=(_2^k⊗ P) · M_f'⊕ h·(_2^k⊗ Q)
 
		one can see that that designs (f⊕ h) and (f'⊕ h) are isomorphic.
		
		2. Let ,∈_2^n, ,∈_2^k and ,∈_2^m. The point (,,) is incident to the block (,,)⊕F⊕ H of the translation design (F⊕ H) if and only if the point (,) is incident to the block (,⊕ H(⊕))⊕F of the translation design (F). The statement now follows from the fact that for a block B the mapping ρ:B ↦ B ⊕ (0,H(⊕)) is an automorphism of designs (F) and (F'), which are isomorphic.
	

Further we show, that isomorphism of divisible designs for Boolean bent functions is a coarser equivalence relation than extended-affine equivalence.

	Boolean bent functions, which are extended-affine inequivalent but isomorphic exist on _2^n for all n≥ 6.
	
		Let g be a quadratic bent function on _2^k and let f and f' be bent functions from the Example <ref>. By Proposition <ref> Boolean functions f⊕ g and f'⊕ g on _2^n with n=k+6 are isomorphic. Clearly, direct sums f⊕ g and f'⊕ g are bent, since all the functions f,f' and g are bent. Finally, since (f⊕ g)=2 and (f'⊕ g)=3, we get that functions f⊕ g and f'⊕ g are extended-affine inequivalent on _2^n.
	


	Extended-affine inequivalent Boolean bent functions f and f' on _2^6 from Example <ref> define isomorphic designs (f) and (f') with 2-transitive automorphism group. According to Kantor <cit.> any 2-(2^n,2^n-1- 2^n/2-1,2^n-2 - 2^n/2-1) design with a 2-transitive automorphism group is unique up to isomorphism. In general, if a design has a large automorphism group, it is more likely that it can be represented by several inequivalent difference sets (bent functions) due to the large symmetry. In this way, one may think that the reason why functions from Example <ref> have isomorphic translation designs is the 2-transitivity of the automorphism group. In the following example we show that isomorphic translation designs (f) and (f') of EA-inequivalent bent functions f and f' do not necessarily need to have a 2-transitive automorphism group.



	Let f,f' be two Maiorana-McFarland bent functions on _2^10 given by	
	
    f() =     x_1 x_6 ⊕ x_2 x_7 ⊕ x_3 x_8⊕ x_4 x_9 ⊕ x_5 x_10
    ⊕     x_1 x_2 x_3 x_4 x_5, 
    
    	f'() =     f() ⊕ x_4 ⊕ x_6 ⊕ x_8 ⊕ x_10⊕ x_1 x_2 ⊕ x_2 x_3 
    ⊕    x_1 x_2 x_3 ⊕ x_2 x_4 x_5 ⊕ x_1 x_2 x_4 x_5 ⊕ x_2 x_3 x_4 x_5.

	With Magma <cit.> one can check that |(𝒞(f))|=2^30· 3^2·5· 7· 31 and |(𝒞(f'))|=2^30· 3^2· 7, what implies that functions f and f' are extended-affine inequivalent. However, the designs (f) and (f') are isomorphic. First, we observe that in general designs (f) and (f') are isomorphic if and only if there exist a pair of permutations π,σ_2^n→_2^n, such that f(π()⊕σ())=f'(⊕) holds for all ,∈_2^n, since an incidence matrix M_f of the translation design (f) can be computed as M_f:=(f(⊕))_,∈_2^n. It is easy to check, that the following nonlinear functions π,σ_2^10→_2^10, given by  algebraic normal forms
	
    π()=    (x_1, x_2, x_3, x_4, x_1  ⊕  x_5, x_1  ⊕  x_10⊕  x_2 x_3  ⊕  x_5  ⊕  x_6, 
        (x_1 x_3  ⊕  x_7, x_1 x_2  ⊕  x_8, x_9, x_1  ⊕  x_10),
    σ() =     ⊕ (1, 0, 1, 0, y_1, y_1 ⊕ y_10⊕ y_2 ⊕ y_2 y_3 ⊕ y_5, 
        ( y_1 ⊕ y_3 ⊕ y_1 y_3, y_2 ⊕ y_1 y_2, 1 , 1 ⊕ y_1),

	are permutations and satisfy f(π()⊕σ())=f'(⊕) for all ,∈_2^10. In this way, designs (f) and (f') are isomorphic. Further we observe that the 2-rank of any 2-(2^n,2^n-1- 2^n/2-1,2^n-2 - 2^n/2-1) design D (i.e. __2M(D)) with a 2-transitive automorphism group equals n+2. Any such a design is isomorphic to (g) of a quadratic bent function g on _2^n, and ((g))=n+2 as it was shown in <cit.>. Since f is a Maiorana-McFarland bent function of the form ⟨',”⟩_n/2⊕ h(”), where ',”∈_2^n/2 and h is a monomial function on _2^n/2 with (h)>3, we have ((f))=n-2(h) + 2^(h) by <cit.>. In this way, (D)=12, which is different from ((f))=((f'))=32, from what follows that automorphism groups of designs (f) and (f') are not 2-transitive.



§ TRANSLATION DESIGNS OF VECTORIAL BENT FUNCTIONS


In the previous section we showed that extended-affine inequivalent Boolean bent functions can give isomorphic translation designs. Further we show that the same phenomenon does not occur for the vectorial bent functions in 6 variables. We classify and enumerate all (6,m)-vectorial bent functions and show, that two vectorial bent functions F and F' in six variables are EA-equivalent if and only if their translation designs (F) and (F') are isomorphic. 	


 §.§ Extension invariants of bent functions


We denote by  the set of all (n,m)-bent functions, by  the set of all (n,m)-affine functions and by  the set of affine-free (n,m)-bent functions, i.e. any f∈ contains no affine terms in its ANF. Since bentness is invariant with respect to the addition of affine terms, the cardinalities of these three sets are related as follows ||=||· ||.

For the sake of convenience we denote by C^m_i an i-th EA-equivalence class of (n,m)-bent functions.	On the set ⋃_m=1^n/2 we introduce the order relation “≺” in the following way. Let m<l and C^m_i and C^l_j be two equivalence classes of (n,m)- and (n,l)-bent functions, respectively.  We say that a function F∈ C^m_i is contained in G∈ C^l_j and write F≺ G, if the first m coordinate functions of G()=(g_1(),…,g_l())^T form a function F, that is F()=(g_1(),…,g_m())^T. Similarly, we say that F∈ C^m_i is contained in the equivalence class C^l_j and write F ≺ C^l_j, if there exist a representative G∈ C^l_j, such that F≺ G. Finally, we say that the equivalence class C^m_i is contained in C^l_j and denote it by C^m_i≺ C^l_j if there exist F∈ C^m_i, such that F ≺ C^l_j.

	An (n,m)-bent function F is called extendable, if the there exists a Boolean bent function f_2^n→_2, such that the function G∈_2^n↦(F(),f())^T is (n,m+1)-bent. If no such a bent function f exists, the function F is called non-extendable.


	The problem of the existence of non-extendable bent functions F_p^n→_p^m has mostly been studied for the case p odd, see <cit.>. The particular case of this problem, namely p=2 and m=1, is closely related to the Tokareva's conjecture <cit.>, that any Boolean function on _2^n of degree at most n/2 can be represented as the sum of two Boolean bent functions on _2^n. For instance, a single example of a non-extendable Boolean bent function would disprove the Tokareva's conjecture.
	

	Let F be an (n,m)-bent function. Further we define the following two sets
	
    ℱ(F):=    { f∈n1  (F,f)^T  (n,m+1)}, 
    (F):=     { (F,f)^T  f ∈ℱ(F) },

	namely, ℱ(F) is the set of affine-free Boolean bent functions, which can extend an (n,m)-bent function F to an (n,m+1)-bent function and (F) is the set of extensions of a function F. Clearly, different extensions may lead to different equivalence classes. In this way, we define
	
    ℱ(F,C^m+1_j):=     { f∈n1  (F,f)^T ∈ C^m+1_j . 
       .   (n,m+1)}

	as the set of affine-free Boolean bent functions, which can extend an (n,m)-bent function F to the equivalence class C^m+1_j. Similarly, we define the set of extensions of the function F, which belong to the equivalence class C^m+1_j, that is
	
    (F,C^m+1_j):={ (F,f)^T  f ∈ℱ(F,C^m+1_j) }.

	Clearly, the collection of sets (F,C^m+1_j) forms a partition of (F), namely
	
    (F)=_j:F ≺ C^m+1_j(F,C^m+1_j).



	Non-extendable (n,m)-bent functions F are also called lonely, see <cit.>. In this way, it is essential to call the following sets:
	
		
  * ℱ(F) — the set of bent friends of a bent function F;
		
  * ℱ(F,C^m+1_j) — the set of bent friends of F, leading to the equivalence class C^m+1_j.
	
	Indeed, according to Definition <ref>, a bent function F is lonely, if it has no bent friends, that is |ℱ(F)|=0. We also call (n,n/2)-bent functions absolutely non-extendable (lonely), since (n,m)-bent functions do not exist for m>n/2 due to the Nyberg bound <cit.>.
	

	For an (n,m+1)-function G we denote by 𝒮(G) a set of equivalence classes of (n,m)-bent functions F,F' of the form F∈_2^n↦ A_F ∘ G() and F'∈_2^n↦ A_F'∘ G(), where A_F,A_F'_2^m+1→_2^m are surjective linear mappings and the equivalence relation “∼” is defined as follows: F ∼ F' if {F_∈_2^m }={F'_∈_2^m }, i.e. functions F and F' have the same component functions. We will call the set 𝒮(G) the set of different (n,m)-bent spaces of a function G. Clearly, its cardinality is given by |𝒮(G)|=m+1  [ ] m_2=2^m+1-1. Finally, for an (n,m)-bent function F≺ G we denote by 𝒮(F,G):={ H∈𝒮(G) H  F } the set of different (n,m)-bent spaces of G, which are EA-equivalent to F.


Further, we show that cardinalities of the sets ℱ(F,C^m+1_j) and 𝒮(F,G) do not depend on representatives of equivalence classes and thus are invariants for extended-affine equivalence.

 Let F,F'∈ C^m_i≺ C^m+1_j be two (n,m)-bent functions and G,G'∈ C^m+1_j be two (n,m+1)-bent functions. Then the following hold.
	
		
  * |ℱ(F,C^m+1_j)|=|ℱ(F',C^m+1_j)|;
		
  * If F≺ G and F'≺ G', then  |𝒮(F, G)|=|𝒮(F', G')|.
	
	
		1. Let F and F' be EA-equivalent, i.e. F=A_1∘ F' ∘ A_2 ⊕ A_3. Clearly, if f is a bent friend of F, then f':=f∘ A_2 is a bent friend of F'. Moreover, the non-degenerate affine transformation A_2 maps different bent friends to different ones. 
		
		2. Assume that H∈𝒮(F,G), i.e. there exist non-degenerate linear mapping A_H_2^m+1→_2^m such that H=A_H ∘ G=B_1∘ F∘ B_2 ⊕ B_3,
		since H,F ∈ C^m_i. Further, we may assume that G'=A_1∘ G∘ A_2 ⊕ A_3, since G,G'∈ C^m+1_j. Multiplying the latter equality by A_H∘ A_1^-1 from left and substituting it the second last, one gets A_H∘ A_1^-1∘ G' = H∘ A_2⊕ A_H ∘ A_1^-1∘ A_3. Finally,
		denoting by A_H':=A_H∘ A^-1_1, we get that the function H':=A_H'∘ G' is EA-equivalent to F', and hence H'∈𝒮(F',G').
	

In this way, for two equivalence classes C^m_i ≺ C^m+1_j we denote by |ℱ(C^m_i, C^m+1_j)|, the number of Boolean bent functions, which can extend any representative of C^m_i to the class C^m+1_j and by |𝒮(C^m_i, C^m+1_j)|, the number of different bent spaces contained in C^m+1_j, which represent the equivalence class C^m_i, that is

    |ℱ(C^m_i,C^m+1_j)|:=    |ℱ(F,C^m+1_j)|
     |𝒮(C^m_i,C^m+1_j)|:=    |𝒮(F,C^m+1_j)|  F∈ C^m_i.

In the next subsection we will use the number of bent friends |ℱ(C^m_i,C^m+1_j)| in order to enumerate all vectorial bent functions in six variables and the number of bent spaces |𝒮(C^m_i,C^m+1_j)| in order to verify these computations.


 §.§ Classification and enumeration of vectorial bent functions in six variables


Now we describe how to determine the cardinality of the equivalence class C^m+1_j, provided its structure is known.

	Let C^m_1,…, C^m_k ≺ C^m+1_j be all equivalence classes of (n,m)-bent functions, contained in C^m+1_j. Then the cardinality of the class C^m+1_j is equal to
	
    |C^m+1_j|=2^n+1·∑_i=1^k |C^m_i|· |ℱ(C^m_i,C^m+1_j)|.

	
		Any function G∈ C^m+1_j can be considered as an extension of a function F ∈ C^m_i ≺ C^m+1_j, that is  G=(F,f)^T∈ C^m+1_j for f∈ℱ(F,C^m+1_j). There are k ways to select an equivalence class C^m_i ≺ C^m+1_j, such that F∈ C^m_i, and there are |C^m_i| ways to choose a representative F. Finally, for any representative F∈ C^m_i there exist exactly 2^n+1·|ℱ(C^m_i,C^m+1_j)| ways to extend it to a function G ∈ C^m+1_j, since bentness is invariant with respect to addition of affine terms.
	

Further we summarize the above ideas in the form of a recursive algorithm.
	
Applying the Algorithm <ref> for Boolean bent functions in six variables, we obtain the main result of this section.

	For vectorial bent functions in 6 variables the following hold.
	
		
  * There are 23,392,233,361,244,160≈2^54.37 vectorial (6,2)-bent functions, which are divided into 9 extended-affine equivalence classes.
		
  * There are 121,282,113,886,947,901,440≈2^66.71 vectorial (6,3)-bent functions, which are divided into 13 extended-affine equivalence classes.
	
	Moreover, if a (6,m)-bent function F is non-extendable, then F is absolutely non-extendable, i.e. it has m=3.
	
		Further we discuss the main steps of the Algorithm <ref> and explain how one can verify our computational results.
		
		Input. For the input of the Algorithm <ref> one has to provide the pairs (F^1_i ∈ C^1_i,|C^1_i|) for all equivalence classes C^1_i, which form the partition of the set of Boolean bent functions 61. The representatives of 4 equivalence classes are well-known and could be found in <cit.>. For the cardinalities of the equivalence classes we refer to <cit.>.
		
		Output. For the computation of the collections ℱ(F^m_i) one first has to construct all affine-free Boolean bent functions 61, which can be efficiently listed as described in <cit.>. Further, for a given representative F^m_i∈ C^m_i we construct the set ℱ(F^m_i), by checking directly the characteristic property in (<ref>). The classification of functions G∈(F^m_i) is carried out with Magma <cit.>, by checking equivalence of linear codes 𝒞(G) introduced in Definition <ref>.
		
		In this way, Algorithm <ref> constructs n/2-1 layers of the weighted Hasse diagram, given in Figure <ref> as follows. For all 2≤ m≤ n/2-1 we draw an edge between equivalence classes C^m_i and C^m+1_j if C^m_i ≺ C^m+1_j and assign two weights with it. The first number closer to the equivalence class C^m_i is the number of bent spaces |𝒮(C^m_i,C^m+1_j)| and the second number, closer to C^m+1_j, is the number of bent friends |ℱ(C^m_i,C^m+1_j)|. Note that, if C^m_1,…, C^m_k ≺ C^m+1_j are all equivalence classes, contained in C^m+1_j, then the following relation holds 
		
    ∑_i=1^k|𝒮(C^m_i,C^m+1_j)|= m+1  [ ] m_2=2^m+1-1.

		In Figure <ref> we list exact cardinalities |C^1_i| for all equivalence classes C^1_i,  while for equivalence classes C^m≥ 2_i, due to the lack of a space, we give only approximate values. Note that, the exact values |C^m≥ 2_i| can be recovered with the Proposition <ref>.
		
		
		Finally, we give the total number of bent functions in six variables in Table <ref> and provide algebraic normal forms of representatives of the equivalence classes together with their invariants in Appendix <ref>.
			
		
		Verification. First, one may observe that cardinalities of all equivalence classes and hence of the sets 6m are divisible by the order of the general linear group (m,2), which is given by |(m,2)|=∏ _k=0^m-1(2^m-2^k). We also observe that the number of affine-free quadratic (6,2)-bent functions established with Algorithm <ref> coincides with the theoretically computed value |n2|, given in <cit.>. Further we note that for equivalence classes C^m_i,C^m_i'≺ C^m+1_j, contained in C^m+1_j the following relation holds
		
    |C^m_i| ·|ℱ(C^m_i,C^m+1_j)|/|𝒮(C^m_i,C^m+1_j)|=|C^m_i'| ·|ℱ(C^m_i',C^m+1_j)|/|𝒮(C^m_i',C^m+1_j)|,

		since the portion of (n,m)-bent functions from the class C^m_i, contained in the equivalence class C^m+1_j equals to
		
    |𝒮(C^m_i,C^m+1_j)|/(2^m+1-1)· 2^n+1· |ℱ(C^m_i,C^m+1_j)|.

		Finally, from Figure <ref> one can see that the only non-extendable bent functions in 6 variables are those, which achieve the Nyberg bound, i.e. (6,3)-bent functions.
	


We also checked that the only equivalence classes of Boolean bent functions, which lead to isomorphic translation designs are C^1_1 and C^1_2, as one can see from Example <ref> and Table <ref>. Surprisingly, in contrast to the Boolean case, one cannot construct isomorphic translation designs from extended-affine inequivalent vectorial bent functions.

	Let F and F' be two (6,m)-bent functions with m≥2. The following statements are equivalent.
	
		
  * Bent functions F and F' are extended-affine equivalent.
		
  * Divisible designs (F) and (F') are isomorphic.
	
	
		All computations about equivalence and isomorphism are carried out with Magma <cit.>. Invariants of equivalence classes and their translation designs are listed in Table <ref> and Table <ref>.
	




	
	It is well-known, that all Boolean bent functions in six variables up to EA-equivalence can be described by two classical constructions: Maiorana-McFarland ℳ, and Desarguesian partial spread 𝒫𝒮_ap, which have straightforward generalizations to the vectorial case, see <cit.>. We endow _2^n/2 with the structure of the finite field (_2^n/2,+,·) and identify _2^n with _2^n/2×_2^n/2. The strict Maiorana-McFarland class ℳ of vectorial bent functions is the set of (n,m)-functions F of the form F(x,y)=L(x·π (y))+G(y), where L_2^n/2→_2^m is a linear or an affine function, π_2^n/2→_2^n/2 is a permutation, and G_2^n/2→_2^m is an arbitrary (n/2,m)-function. The 𝒫𝒮_ap class of vectorial bent functions is the set of (n,m)-bent functions F of the form F(x, y):=H(x· y^2^n/2-2)=H(x/y) with x/y=0 if y=0 for x, y ∈_2^n/2 and H is
	a balanced (n/2, m)-function (or, equivalently, permutation if m=n/2).
	
	In the following table we list equivalence classes C^m_i of (6,3)-bent functions, which can be described by ℳ and 𝒫𝒮_ap classes. Note that (6,2)-bent functions from ℳ and 𝒫𝒮_ap can be constructed as proper bent subspaces of (6,3)-bent from ℳ and 𝒫𝒮_ap classes.
	

	
	
	In this way, from Figure <ref> and Table <ref> one can see that the only “missing” equivalence classes of (6,3)-bent functions are C^3_3,C^3_6,C^3_7,C^3_9 and of (6,2)-bent functions are C^2_4,C^2_6. In view of this observation we conclude, that in contrast to the Boolean case, vectorial versions of the classical Maiorana-McFarland and Desarguesian partial spread constructions do not cover the whole set of vectorial bent functions in six variables. 




§ SUMMARY AND CONCLUDING REMARKS


In this paper we compared different concepts of equivalence relations for Boolean and vectorial bent functions: extended-affine equivalence of functions, isomorphism of translation designs and isomorphism of addition designs. We summarize our results in the following table.

Finally, we would like to mention some open problems on bent functions and their translation designs, which the reader is invited to attack.

	As one can see from Examples <ref> and <ref>, it is possible to construct EA-inequivalent but isomorphic Boolean bent functions, by taking proper Maiorana-McFarland bent functions and extending them to infinite families using the Proposition <ref>. So far, this approach does not seem to work for vectorial bent functions:
	
		
  * there is only one up to EA-equivalence vectorial bent function in 4 variables, from what follows that all derived translation designs are isomorphic;
		
  * by Theorem <ref> all isomorphic vectorial bent functions in 6 variables are also EA-equivalent. 
	
	As one can see from Proposition <ref>, a single example of EA-inequivalent but isomorphic vectorial bent functions will lead to an infinite family and, consequently, will prove that for vectorial bent functions the isomorphism of translation designs is a coarser equivalence relation than EA-equivalence. However, since one still does not have an example of such functions, it is essential to ask, whether EA-inequivalent but isomorphic vectorial bent functions may in general exist.


	There are very few symmetric designs with a 2-transitive automorphism group, as it was shown by Kantor in <cit.>. One of them is 2-(2^n,2^n-1- 2^n/2-1,2^n-2 - 2^n/2-1) design, which can be constructed as the addition (f) or as the translation (D_f) design of a bent function f on _2^n. While in the case of addition designs (f) a bent function f on _2^n has to be quadratic, from Example <ref> one can see that for the translation design (f) one still has some freedom to choose a function f. We conjecture, that a translation design (f) of a bent function f on _2^n has 2-transitive automorphism group if and only if function f is EA-equivalent to a Maiorana-McFarland bent function of the form ⟨, ⟩_n/2⊕ g() with (g)≤ 3.




§ ACKNOWLEDGMENT

The authors would like to thank anonymous referees for their comments that helped to improve the presentation of the results.

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§ APPENDIX: ALGEBRAIC NORMAL FORMS OF BENT FUNCTIONS IN SIX VARIABLES



For any equivalence class C^m_i of (6,m)-bent functions we compute the following invariants:

	
  * |(C^m_i)|:=|(𝒞(F^m_i))|, that is the order of the automorphism group of the linear code 𝒞(F^m_i), for a representative F^m_i∈ C^m_i listed in Appendix <ref>;
	
  * |((C^m_i))|:=|((F^m_i))|, that is the order of the automorphism group of the translation design (F^m_i), for a representative F^m_i∈ C^m_i listed in Appendix <ref>;
	
  * (C^m_i):=(F^m_i), that is the Smith normal form of the incidence matrix M((F^m_i)), given by the multiset (F^m_i)={*d_1^e_1,…, d_k^e_k*}, where consecutive elementary divisors d_k and d_k+1 satisfy d_k|d_k+1, and e_k is the multiplicity of the elementary divisor d_k.


Note that the multiplicity of one in the Smith normal form (F^m_i) is the (F^m_i) and similarly to <cit.> one can show, that all elementary divisors d_k in the (F) of an (n,m)-bent function F are powers of two. We also observe that any two different equivalence classes C^m_i and C^m_j of bent functions in six variables have different pairs of invariants 

    (|(C^m_i)|,(C^m_i)) ≠ (|(C^m_j)|,(C^m_j)).

In this way, the reader can be sure that all the representatives of equivalence classes listed in Appendix <ref> are extended-affine inequivalent.


For quadratic vectorial bent functions from equivalence classes C^m_1 with m=2,3 we have |((C^m_1))| =2^n+m· |(C^m_1)|· 7^1, where n=6. For the rest of vectorial (n,m)-bent functions in six variables we have |((C^m_i))| =2^n+m· |(C^m_i)|. In this way, translation designs of vectorial bent functions from equivalence classes C^3_10 and C^3_11 can be distinguished by the orders of their automorphism groups, despite the Smith normal forms coincide.

Finally, we list algebraic normal forms of representatives of EA-equivalence classes of bent functions in 6 variables. The representatives F^m_i∈ C^m_i and F^m+1_j∈ C^m+1_j are selected in such a way, that F^m_i ≺ F^m+1_j as on Figure <ref>. We abbreviate 1≤ i≤6 for the variable x_i. 



Alexandr A. Polujan
	received Diploma in mathematics and system analysis from the Faculty of Mechanics and Mathematics of the Belarusian State University, Minsk, Belarus, in 2015. Since 2016, he has been pursuing the Ph.D. degree under the supervision of Alexander Pott at the Otto von Guericke University, Magdeburg, Germany. His research interests include mainly perfect nonlinear functions.

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Alexander Pott
	received his Ph.D. degree in Mathematics in 1988 from the Justus-Liebig-University Giessen. He held visiting positions in Dayton (Ohio) and Duisburg (Germany). He was professor at the University of Augsburg (Germany), and presently he is full professor at the Otto von Guericke University Magdeburg, Germany. His research interests include Finite Geometry (difference sets), Algebraic Coding Theory (sequences, Boolean functions) and Finite Fields.



