Dual bent functions on finite groups and C-algebras

Journal of Pure and Applied Algebra 220 (2016) 1055–1073

Contents lists available at ScienceDirect

Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa

Dual bent functions on finite groups and C-algebras Bangteng Xu Department of Mathematics & Statistics, Eastern Kentucky University, Richmond, KY 40475, USA

a r t i c l e

i n f o

Article history: Received 31 March 2014 Received in revised form 25 June 2015 Available online 21 August 2015 Communicated by S. Donkin MSC: 43A30; 20C99; 11T71

a b s t r a c t The dual of a (bent) function on a finite abelian group is a natural concept. In this paper we study the dual bent functions on finite nonabelian groups. A more general algebraic structure of a C-algebra provides a better and natural context for this purpose. We will first study Fourier transforms, bent functions, and dual bent functions on C-algebras. Then as an application, we obtain the properties of dual bent functions on finite nonabelian groups. Examples of bent functions on C-algebras are also presented. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The notion of a Boolean bent function was introduced by Rothaus (cf. [12]). This concept was first generalized to the bent function on an arbitrary finite abelian group by Logachev, Salnikov, and Yashchenko (cf. [6]), and then to the bent function on a finite nonabelian group by Poinsot (cf. [9]). Researches on bent functions and perfect nonlinear functions on arbitrary finite abelian groups and finite nonabelian groups can also be found in [3,7,8,10,11,14,15]. These functions and their generalizations are actively studied by many researchers; see [13] for a systematic survey of recent development in this field. Let G be a finite abelian group, and f : G → C a function, where C is the complex field. The (irreducible)  is defined as a function  the dual group of G. The Fourier transform of f , f, characters of G form a group G,  It is well known that G  is isomorphic to G (cf. [5]). Via the isomorphism between G and G,  f can on G.   also be regarded as a function on G. By the use of f, the dual of f , f , is defined as a function on G in a natural way. Furthermore, f is a bent function if and only if f is also a bent function (cf. [6]). In this paper we study the duals of (bent) functions on finite nonabelian groups. The Fourier transform of a function f on a finite nonabelian group G is defined at all irreducible representations of G. However, the set of all irreducible representations of G does not have an algebraic structure that can be regarded as a dual of G. The Fourier transform of f can also be defined at all irreducible characters of G. The set of irreducible characters of G, Irr(G), is a basis of the ring Ch(G) of complex valued class functions on G. On the other hand, the set Cla(G) of conjugacy class sums of G is a basis of Z(CG), the center of the group E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jpaa.2015.08.007 0022-4049/© 2015 Elsevier B.V. All rights reserved.

1056

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

algebra CG. The duality between Irr(G) and Cla(G) is well known. Since f can be linearly extended to a function on Cla(G), by regarding f as a function on Cla(G), and its Fourier transform as a function on Irr(G), we will define the dual of f as a function on Irr(G) via the duality between Cla(G) and Irr(G). C-algebras abstract common features of centers of group algebras and rings of complex valued class functions on finite groups, and of the Bose–Mesner algebras of commutative association schemes, etc. C-algebras and their applications to the theory of finite groups as well as commutative association schemes have been studied in many papers. Every C-algebra has a dual C-algebra, and the abstract structure of a C-algebra provides a better and natural context for the discussions of dual (bent) functions on finite nonabelian groups. However, the Fourier transforms and bent functions on C-algebras defined below are not formal generalizations of those on finite groups. These definitions in the context of a C-algebra provide a different point of view of the concepts. In the following we will first define the Fourier transform of a function on a C-algebra in Section 2, and prove that with modifications, the Fourier transforms on C-algebras have similar properties to those on finite groups. Then in Section 3, we introduce the notion of bent functions on C-algebras, and discuss their characterizations. The dual of a function on a C-algebra will also be defined in this section, and properties of dual bent functions are studied. Some of our results can be regarded as generalizations of those for finite groups, some of them cannot. Finally, we discuss applications to dual bent functions on arbitrary finite groups in Section 4, and present examples of bent functions on C-algebras in Section 5. 2. Fourier transforms on C-algebras In this section we introduce the Fourier transform of a function on a C-algebra, and discuss its basic properties. There are different definitions and notation for C-algebras and their duals in the literature. In this paper we will always use the following definitions and notation. Let A be a finite dimensional, associative, and commutative algebra over the complex numbers C, with a distinguished basis B = {b0 , b1 , b2 , . . . , bd } such that b0 = 1A , the identity element of A. If the following properties (i)–(iv) hold, then (A, B) is called a standard C-algebra (SCA), and B a standard C-basis of A. d (i) The structure constants for B are real numbers; that is, bi bj = h=0 λijh bh with λijh ∈ R, for all bi , bj ∈ B. (ii) There is an algebra automorphism (denoted by  ) of A such that (a ) = a for all a ∈ A and bi ∈ B for all bi ∈ B. (Hence i is defined by bi := bi .) (iii) For all bi , bj ∈ B, λij0 = 0 if j = i ; and λii 0 > 0. (iv) There is an algebra homomorphism A → C such that bi → λii 0 , 0 ≤ i ≤ d. Let (A, B) be a SCA, with B := {b0 = 1A , b1 , . . . , bd }. Then ki := λii 0 is called the order of bi , and  o(B) := bi ∈B ki is called the order of B. The C-algebras defined in [2] are SCAs. The next example includes some examples of SCAs. Example 2.1. (i) Let G be a finite abelian group, and CG the group algebra of G over C. Then (CG, G) is a SCA, with the automorphism  extended linearly from inversion in G. (ii) Let G be a finite group, and CG the group algebra of G over C. Let C0 , C1 , . . . , Cd be the conjugacy  classes of G, and Ci+ := x∈Ci x ∈ CG, 0 ≤ i ≤ d. Let Z(CG) be the center of CG, and Cla(G) := {Ci+ | 0 ≤ i ≤ d}. Then (Z(CG), Cla(G)) is a SCA, with the automorphism  extended linearly from inversion in G. Furthermore, the order of Ci+ (as an element in the standard C-basis Cla(G)) is |Ci |, 0 ≤ i ≤ d, and o(Cla(G)) = |G|. (iii) Let G be a finite group, Ch(G) the ring of complex valued class functions on G, Irr(G) the set of irreducible characters of G, and Irr(G)S := {χ(1)χ | χ ∈ Irr(G)}, where 1 is the identity element of G.

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1057

Then (Ch(G), Irr(G)S ) is a SCA, with the automorphism  extended linearly from complex conjugation of characters of G. Note that the order of χ(1)χ (as an element in the standard C-basis Irr(G)S ) is χ(1)2 , for any χ ∈ Irr(G), and o(Irr(G)S ) = |G|. Let (A, B) be a SCA, with B := {b0 = 1A , b1 , . . . , bd }. For any z ∈ C, its complex conjugate is denoted d d by z. For any a := i=0 xi bi ∈ A, let a := i=0 xi bi . If a = 0, then aa = 0, and hence (aa )(aa ) = 0. So a2 = 0, and A is semisimple. Let e0 , e1 , . . . , ed be the primitive idempotents of A such that e0 = d o(B)−1 i=0 bi . Then {e0 , e1 , . . . , ed } is another basis of A. Assume that bi =

d 

pi (j)ej

and ei = o(B)−1

j=0

d 

qi (j)bj ,

0 ≤ i ≤ d.

(2.1)

j=0

Let P and Q be the (d +1) ×(d +1) matrices whose (j, i)-entries are pi (j) and qi (j), respectively, 0 ≤ i, j ≤ d. Then P is called the first eigenmatrix of (A, B), and Q the second eigenmatrix of (A, B). Also ki = pi (0), 0 ≤ i ≤ d. Let mi := qi (0), 0 ≤ i ≤ d. Then m0 , m1 , . . . , md are called the multiplicities of (A, B). Furthermore, from [1] we will always assume that the primitive idempotents e1 , · · · , ed are indexed in such a way that ei = ei . Then by [1, (2.9)], pi (j) = pi (j  ) = pi (j),

for any 0 ≤ i, j ≤ d.

(2.2)

The next lemma collects some basic properties of C-algebras that will be needed later. Lemma 2.2(i), (ii) are immediate from (2.1), and Lemma 2.2(iii), (iv) are adopted from Proposition 5.8, p. 96 and Theorem 5.5, p. 94 of [2], respectively. Lemma 2.2. Let (A, B) be a SCA. Then the following hold. (i) (ii) (iii) (iv)

P Q = QP = o(B)I, where I is the identity matrix. d For all 0 ≤ i, r, h ≤ d, ph (i)pr (i) = j=0 λhrj pj (i). For all 0 ≤ i, r, h ≤ d, λhrj kj = λh jr kr . For all 0 ≤ i, j ≤ d, pj (i)/kj = qi (j)/mi .

Two SCAs (A, B) and (U, V) are called exactly isomorphic, and denoted by (A, B) ∼ =x (U, V) or simply ∼ B =x V, if there is an algebra isomorphism ϕ : A → U such that ϕ(B) = V, where ϕ(B) := {ϕ(bi ) | bi ∈ B}. A SCA (A∗ , B∗ ) is called a standard dual C-algebra of (A, B) if the first (second) eigenmatrix of (A, B) is equal to the second (first) eigenmatrix of (A∗ , B∗ ). Let (A, B) be a SCA, and A∗ the set of all linear functionals on A. For any 0 ≤ i ≤ d, let b∗i be the linear functional on A defined by b∗i : A → C,

bj → qi (j), 0 ≤ j ≤ d.

Let B∗ := {b∗0 = 1A∗ , b∗1 , . . . , b∗d }. Then (A∗ , B∗ ) is a SCA, with primitive idempotents e∗0 , e∗1 , . . . , e∗d defined by e∗i : A → C,

bj → δij , 0 ≤ j ≤ d,

where δij is the Kronecker delta. By [2, p. 98, Theorem 5.9], (A∗ , B∗ ) is a standard dual C-algebra of (A, B). d Note that e∗0 = o(B∗ )−1 i=0 b∗i , and b∗i =

d  j=0

qi (j)e∗j

and e∗i = o(B∗ )−1

d  j=0

pi (j)b∗j ,

0 ≤ i ≤ d.

(2.3)

1058

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

Hence, for the orders k0∗ , k1∗ , . . . , kd∗ and multiplicities m∗0 , m∗1 , . . . , m∗d of (A∗ , B∗ ), it follows that ki∗ = mi , m∗i = ki , 0 ≤ i ≤ d, and o(B) = o(B∗ ).

(2.4)

Note that the standard dual C-algebra of (A, B) is unique up to exact isomorphism. Furthermore, if (A∗∗ , B∗∗ ) is the standard dual C-algebra of (A∗ , B∗ ), then (A, B) ∼ =x (A∗∗ , B∗∗ ). That is, (A, B) is the standard dual of (A∗ , B∗ ). The notation and assumptions in the above paragraphs about a SCA and its standard dual C-algebra will be used throughout the paper. For more properties of C-algebras and dual C-algebras, the reader is referred to [1,2,16,17]. Definition 2.3. Let (A, B) be a SCA, and f : B → C a function. Let (A∗ , B∗ ) be the standard dual of (A, B). Then for any b∗i ∈ B∗ , the Fourier transform of f at b∗i is defined as √ f(b∗i ) := mi

d  f (bj )pj (i)

kj

j=0

.

(2.5)

Remark 2.4. (i) The order of an element b in a C-basis is usually denoted by o(b) in the literature (cf. [1] etc.). Thus, mi = o(b∗i ), kj = o(bj ), and (2.5) becomes d   f (bj )pj (i) ∗ . o(bi ) o(bj ) j=0

f(b∗i ) =

(ii) Since for any 0 ≤ i, j ≤ d, pj (i)/kj = qi (j)/mi by Lemma 2.2(iv), the Fourier transform of f at b∗i is also given by d 1  f (bj )qi (j). f(b∗i ) = √ mi j=0

(iii) (A, B) has d + 1 irreducible characters χ0 , χ1 , . . . , χd such that χi (bj ) = pj (i), 0 ≤ i, j ≤ d. Hence, d f(b∗i )  f (bj ) = χi (bj ). √ kj mi j=0

The next proposition gives the inversion formula for Fourier transforms. Proposition 2.5. Let (A, B) be a SCA, with standard dual (A∗ , B∗ ), and let f : B → C be a function. Then ki  f(b∗j )qj (i) , √ o(B) j=0 mj d

f (bi ) = Proof. For any 0 ≤ i, h ≤ d, implies that

d

j=0 qj (i)ph (j)

for any bi ∈ B.

= o(B)δih by Lemma 2.2(i). Hence for any bi ∈ B, (2.5)

d d d ki   f (bh )ph (j) ki  f(b∗j )qj (i) = qj (i) √ mj o(B) j=0 o(B) j=0 kh h=0

=

d ki  f (bh ) o(B)δih = f (bi ). o(B) kh h=0

So the proposition holds. 2

(2.6)

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1059

As in Remark 2.4(i), (2.6) can also be written as f (bi ) =

d o(bi )  f(b∗j )qj (i)  , o(B) j=0 o(b∗ )

for any bi ∈ B.

j

Let (A, B) be a SCA. Then a function f : B → C can be linearly extended to a linear functional (still denoted by f ) on A. So for any functions f, g : B → C, the convolution f ∗ g can be defined as f ∗ g : B → C,

bi →

d  f (bj )g(bj bi )

kj

j=0

.

The next proposition says that the convolution of functions on C-algebras has the similar property of the convolution of functions on finite groups. Also note that for the standard dual (A∗ , B∗ ) of (A, B), any function B∗ → C is also regarded as a linear functional on A∗ . Proposition 2.6. Let (A, B) be a SCA, with standard dual (A∗ , B∗ ), and let f, g : B → C be functions. Then  ∗   ∗   ∗  b b bi   = f √i g √ i , for any b∗i ∈ B∗ . (f ∗ g) √ mi mi mi Proof. For any 0 ≤ j, h ≤ d, it follows from bh bj = any b∗i ∈ B∗ , √  (f ∗ g)(b∗i ) = mi

d r=0

λh jr br that g(bh bj ) =

d r=0

λh jr g(br ). Hence for

d d d  √   f (bh )g(bh bj )pj (i) (f ∗ g)(bj )pj (i) = mi kj kh kj j=0 j=0 h=0

=

√

mi

d  d d   f (bh )g(br )λh jr pj (i)

kh kj

j=0 h=0 r=0

Note that λh jr /kj = λhrj /kr by Lemma 2.2(iii), and ph (i)pr (i) = √  (f ∗ g)(b∗i ) = mi

d d  d   f (bh )g(br )λhrj pj (i) j=0 h=0 r=0

kh kr

=

√

.

d j=0

mi

λhrj pj (i) by Lemma 2.2(ii). Hence,

d d   f (bh )g(br )ph (i)pr (i) h=0 r=0

kh kr

1 = √ f(b∗i ) g (b∗i ). mi  Since (f ∗ g), f, and g are linear functionals on A∗ , the proposition holds. 2 Let (A, B) be a SCA, and f : B → C a function. Then define a function f  : B → C,

bi → f (bi ).

(2.7)

For any z ∈ C, its absolute value is denoted by |z|. The next two propositions will be needed in the next section. Proposition 2.7. Let (A, B) be a SCA, with standard dual (A∗ , B∗ ). Let f : B → C be a function, and f  the same as in (2.7). Then  ∗  2  ∗  b  = (f f √bi , ∗f ) √i mi mi

for any b∗i ∈ B∗ .

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1060

Proof. Since for any 0 ≤ j ≤ d, kj = kj  , (2.2) implies that √ f(b∗i ) = mi

d  f (bj )pj (i) j=0

=

f  (b∗i ),

kj

=

√

mi

d  f (bj )pj  (i) j=0

kj

=

√

mi

d  f (bj )pj (i) j=0

kj

for any b∗i ∈ B∗ .

Hence by Proposition 2.6, √  ∗ 2

∗ f  )(b∗i ), f (bi ) = f(b∗i )f(b∗i ) = f(b∗i ) f  (b∗i ) = mi (f

for any b∗i ∈ B∗ .

So the proposition holds. 2 Proposition 2.8. Let (A, B) be a SCA, with standard dual (A∗ , B∗ ). Let f : B → C be a function, and α ∈ C. Then the following are equivalent. f (bi ) = α. ki f(b∗ ) (ii) For any 1 ≤ i ≤ d, √ i = f (b0 ) − α. mi (i) For any 1 ≤ i ≤ d,

Proof. (i) ⇒ (ii) Note that for any 1 ≤ i ≤ d, by (2.5),

d j=0

pj (i) = 0 and p0 (i) = 1 by (2.1). Also k0 = 1. Hence

d d  f(b∗i )  f (bj )pj (i) = =α pj (i) + f (b0 ) − α = f (b0 ) − α, √ mi kj j=0 j=0

1 ≤ i ≤ d.

So (ii) holds. d (ii) ⇒ (i) For any 1 ≤ i ≤ d, j=0 qj (i) = 0 and q0 (i) = 1 by (2.3). Also m0 = 1. Hence by Proposition 2.5, o(B)

d d  f (bi )  f(b∗j )qj (i) = = (f (b0 ) − α) qj (i) + f(b∗0 ) − f (b0 ) + α √ m ki j j=0 j=0

= f(b∗0 ) − f (b0 ) + α,

1 ≤ i ≤ d.

Let β = (f(b∗0 ) − f (b0 ) + α)/o(B). Then for all 1 ≤ i ≤ d, f (bi )/ki = β. Thus, it follows from o(B) that √ f(b∗0 ) = m0

d  f (bj )pj (0) j=0

kj

=β

d 

d j=0

pj (0) =

pj (0) + f (b0 ) − β = (o(B) − 1)β + f (b0 ).

j=0

Hence, we have that o(B)β = f(b∗0 ) − f (b0 ) + β. So β = α, and (i) holds. 2 3. Bent and dual bent functions on C-algebras In this section we first introduce bent functions on C-algebras, and study their characterizations. We will also discuss the connections between the Fourier transform and the derivatives of a function. Then for a standard C-algebra (A, B) and a function f : B → C, we define the dual f ∗ : B∗ → C, and prove that f ∗∗ = f , and f is a bent function if and only if f ∗ is also a bent function.

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1061

Definition 3.1. Let (A, B) be a SCA, with standard dual (A∗ , B∗ ). Let f : B → C be a function. If for any bi ∈ B and b∗i ∈ B∗ , f(b∗ ) 2 i √ = o(B), mi

f (bi ) ki = 1 and then f is called a bent function.

d Let (A, B) be a SCA, with B = {b0 = 1A , b1 , . . . , bd }, and let f : B → C be a function. If i=0 f (bi ) = 0, then f is said to be balanced. Since pi (0) = ki for all i, and m0 = 1, it follows from (2.5) that √ f(b∗0 ) = m0

d  f (bi )pi (0)

ki

i=0

=

d 

f (bi ).

i=0

Thus, f is balanced if and only if f(b∗0 ) = 0.

(3.1)

Since f is also regarded as a linear functional on A, for any bi ∈ B, the derivative of f in direction bi is defined as df : B → C, dbi

bj →

f (bj )f (bi bj ) . kj

Furthermore, for any 0 ≤ i ≤ d, let Bi be the (d + 1) × (d + 1) matrix whose (j, h)-entry is λijh . Also let K := diag{k0 , k1 , . . . , kd } be the diagonal matrix with diagonal entries k0 , k1 , . . . , kd . Theorem 3.2. Let (A, B) be a SCA, and let f : B → C be a function such that for any bi ∈ B, f (bi )/ki = 1. Then the following are equivalent. (i) f is a bent function. (ii) For any 1 ≤ i ≤ d, df /dbi is balanced. (iii) For any 1 ≤ i ≤ d, X(K −1 Bi )X  = 0, where X = (f (b0 ), f (b1 ), . . . , f (bd )), and X  is the conjugate transpose of X. Proof. Let f  be the same as in (2.7). Then by linear extension, f  is a linear functional on A. That is, for any d d d a = i=0 αi bi ∈ A, where αi ∈ C for all i, f  (a) = i=0 αi f  (bi ). For any 0 ≤ i, j ≤ d, bi bj = h=0 λijh bh  d implies that bj bi = h=0 λijh bh . Thus, since the structure constants of B are real numbers, it follows that f  (bj bi ) =

d 

λijh f  (bh ) =

h=0

d 

λijh f (bh ) = f (bi bj ).

h=0

Hence, (f ∗ f  )(bi ) =

d  f (bj )f  (bj bi ) j=0

kj

=

d  f (bj )f (bi bj ) j=0

kj

=

d  df (bj ), dbi j=0

for any bi ∈ B.

In particular, f (bj )/kj = 1 for all j imply that (f ∗ f  )(b0 ) = o(B).

(3.2)

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1062

Furthermore, since i → i is a permutation on {1, 2, . . . , d}, it follows that df /dbi are balanced for all 1 ≤ i ≤ d

⇔

(f ∗ f  )(bi ) = 0 for all 1 ≤ i ≤ d.

(3.3)

(i) ⇒ (ii) Proposition 2.7 and (3.2) imply that (f ∗ f  )(b∗i ) = o(B) = (f ∗ f  )(b0 ), √ mi

for all 0 ≤ i ≤ d.

Hence by Proposition 2.8, (f ∗ f  )(bi ) = 0 for all 1 ≤ i ≤ d. Thus, df /dbi are balanced for all 1 ≤ i ≤ d by (3.3), and (ii) holds. (ii) ⇒ (i) It follows from (3.3) that (f ∗ f  )(bi ) = 0 for all 1 ≤ i ≤ d. Hence, (3.2) implies that d (f ∗ f  )(b∗i )  (f ∗ f  )(bj )pj (i) = = (f ∗ f  )(b0 ) = o(B), √ mi k j j=0

0 ≤ i ≤ d.

So f is a bent function by Proposition 2.7, and (i) holds. d (ii) ⇔ (iii) For any 0 ≤ i, j ≤ d, bi bj = h=0 λijh bh implies that d d d  d    λijh df f (bj )f (bi bj ) (bj ) = = f (bj ) f (bh ) dbi kj kj j=0 j=0 j=0 h=0

= X(K

−1



Bi )X .

So the equivalence of (ii) and (iii) holds. 2 As a direct consequence of Theorem 3.2 and (3.1), we have the following Corollary 3.3. Let (A, B) be a SCA, and let f : B → C be a function such that for any bi ∈ B, f (bi )/ki = 1. Then f is a bent function if and only if  df (b∗0 ) = 0, dbi

for all 1 ≤ i ≤ d.

The next theorem discusses connections between the derivatives and the Fourier transform of a function. Theorem 3.4. Let (A, B) be a SCA, and f : B → C a function. Then d d  1   ∗ 2 df (bj ) = f (br ) pi (r), dbi o(B) r=0 j=0

0 ≤ i ≤ d,

(3.4)

and d d  ∗ 2   df (bj )qr (i), f (br ) = dbi i=0 j=0

Proof. Since bi bj =

d s=0

0 ≤ r ≤ d.

λijs bs , Proposition 2.5 implies that

f (bi bj ) =

d  s=0

λijs f (bs ) =

d d 1  f(b∗t )qt (s) λijs ks √ . o(B) s=0 t=0 mt

(3.5)

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

Note that ks qt (s) = mt ps (t) by Lemma 2.2(iv), and pi (t)pj (t) = So d 

λijs ks qt (s) =

s=0

d 

d s=0

1063

λijs ps (t) for all t by Lemma 2.2(ii).

λijs ps (t)mt = pi (t)pj (t)mt .

s=0

Thus, f (bi bj ) =

d √ 1  ∗ f (bt )pi (t)pj (t) mt . o(B) t=0

Therefore, the above equality and Proposition 2.5 yield that d d   df f (bj )f (bi bj ) (bj ) = dbi kj j=0 j=0

=

But

d j=0

d d d √ 1    f(b∗r )qr (j)  ∗ f (bt )pi (t)pj (t) mt . √ o(B)2 j=0 r=0 t=0 mr

pj (t)qr (j) = o(B)δrt by Lemma 2.2(i). Thus, d d d  √ 1   f(b∗r )  ∗ df (bj ) = √ f (bt )pi (t) mt δrt dbi o(B) r=0 t=0 mr j=0

=

d 1   ∗ 2 f (br ) pi (r). o(B) r=0

So (3.4) holds. Now (3.5) follows from (3.4) and Lemma 2.2(i), and the theorem holds. 2 d Note that for any 1 ≤ i ≤ d, r=0 mr pi (r) = 0 by [2, Theorem 5.5, p. 94]. So the above theorem implies the equivalence of (i) and (ii) in Theorem 3.2, and gives a direct connection between them. Let (A, B) be a SCA. The set of all functions from B to C is denoted by F(B, C). For any f, g ∈ F(B, C) and α ∈ C, define f + g, f g, αf ∈ F(B, C) by (f + g)(bi ) := f (bi ) + g(bi ), (f g)(bi ) := f (bi )g(bi ), (αf )(bi ) = αf (bi ),

for all bi ∈ B.

Then F(B, C) is an algebra over C of dimension |B|. There is an inner product on F(B, C) defined by f, g :=

d  1 f (bi )g(bi ), k i=0 i

for all f, g ∈ F(B, C).

In the following we give a characterization of bent functions via the inner product defined above. We need the next lemma first. Lemma 3.5. Let (A, B) be a SCA, and let f : B → C be a function such that for any bi ∈ B, f (bi )/ki = 1. Then for any 0 ≤ i, j ≤ d, 

df df , dbi dbj

=

d  s=0

λ

ij  s

d  df (br ). db s r=0

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1064

Proof. For any 0 ≤ i, j ≤ d, it follows from f (bh )/kh = 1 for all h that 

df df , dbi dbj

=

d d   1 df df 1 f (bh )f (bi bh ) f (bh )f (bj bh ) (bh ) (bh ) = kh dbi dbj kh kh kh

h=0

=

h=0

d  1 f (bi bh )f (bj bh ). kh

h=0

For any 0 ≤ r ≤ d, since bi br =

d h=0

λi rh bh and bj bi =

bj bi br

=

d 

λ

i rh

d s=0

bj bh =

d 

λji s bs , we see that λji s bs br .

s=0

h=0

So f a linear functional on A implies that d 

λi rh f (bj bh ) = f (bj bi br ) =

d 

λji s f (bs br ).

s=0

h=0

Note that λihr /kh = λi rh /kr by Lemma 2.2(iii), and bi bh =

d r=0

λihr br . Thus,

d d d  d d    1 λihr f (br )  f (bi bh )f (bj bh ) = f (br )f (bj bh ) = λi rh f (bj bh ) kh kh kr r=0 r=0

h=0

h=0

=

h=0

d d  f (br )  r=0

kr

λji s f (bs br ) =

s=0

d 

λji s

s=0

d  df (br ). db s r=0

That is, we have proved that 

df df , dbi dbj

=

d 

λji s

s=0

d  df (br ). db s r=0

Therefore, by applying the above equality to df /dbj , df /dbi , we see that 

df df , dbi dbj



 =

df df , dbj dbi

=

d  s=0

λ

ij  s

d  df (br ), db s r=0

and the lemma holds. 2 Theorem 3.6. Let (A, B) be a SCA, and let f : B → C be a function such that for any bi ∈ B, f (bi )/ki = 1. Then f is a bent function if and only if

 df df df , ,..., db0 db1 dbd is an orthogonal basis of F(B, C). Proof. First assume that f is a bent function. Then by Theorem 3.2, df /dbs are balanced for all 1 ≤ s ≤ d. Thus, Lemma 3.5 implies that  d  df df df = λij  0 , (br ) = o(B)λij  0 . dbi dbj db0 r=0 Since λij  0 = 0 if and only if j = i, {df /db0 , df /db1 , . . . , df /dbd } is an orthogonal basis of F(B, C).

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1065

Now assume that {df /db0 , df /db1 , . . . , df /dbd } is an orthogonal basis of F(B, C). Then for any 1 ≤ i ≤ d, since λ0i s = 0 if and only if s = i , and λ0i i = 1, Lemma 3.5 implies that  0=

df df , db0 dbi

=

d  df (br ), db i r=0

for any 1 ≤ i ≤ d.

That is, df /dbi is balanced for any 1 ≤ i ≤ d. So df /dbi are balanced for all 1 ≤ i ≤ d, and f is a bent function by Theorem 3.2. 2 It is clear that Theorem 3.6 can be regarded as a generalization of [15, Theorem 4.7] by Proposition 4.3 in the next section. Let (A, B) be a SCA, and let f : B → C be a function such that for any bi ∈ B, f (bi )/ki = 1. Then from Theorem 3.6 and its proof, f is a bent function if and only if for any 1 ≤ i ≤ d, df /db0 and df /dbi are orthogonal. More generally, Lemma 3.5 and Theorem 3.6 imply the following Corollary 3.7. Let (A, B) be a SCA, and let f : B → C be a function such that for any bi ∈ B, f (bi )/ki = 1. Assume that Bh is invertible for some 1 ≤ h ≤ d, where Bh is the same as in Theorem 3.2. Then f is a bent function if and only if for any j = h, df /dbh and df /dbj are orthogonal. Now let us turn to the dual functions on C-algebras. Definition 3.8. Let (A, B) be a SCA, and f : B → C a function. Then the dual of f , f ∗ , is defined by f ∗ : B∗ → C,

√ mi  ∗ b∗i →  f (bi ). o(B)

That is,  ∗ o(b ) f ∗ (b∗i ) =  i f(b∗i ), o(B)

for any b∗i ∈ B∗ .

Proposition 3.9. Let (A, B) be a SCA, and f : B → C a function. Let f ∗∗ : B → C be the dual of f ∗ : B∗ → C. Then f ∗∗ = f . Proof. For any bi ∈ B, √ √ d d ki  f(b∗j )qj (i) ki ∗ ki   f ∗ (b∗j )qj (i) f (bi ) =  ki = f (bi ) =  √ mj o(B) j=0 mj o(B) o(B) j=0 ∗∗

= f (bi ).

(by Proposition 2.5)

So the proposition holds. 2 Let (A, B) be a SCA, f : B → C a function. Then we have two functions f ∗ : B → C and f∗ : B → C. Since √ d ki  f(b∗j )qj (i) f ∗ (bi ) =  √ mj o(B) j=0

ki and f∗ (bi ) =  o(B)

d  ∗  f (bj )qj (i) j=0

mj

,

for any bi ∈ B,

it follows that f ∗ =  f∗ in general. But if all ki = 1, then all mi = 1, and hence f ∗ = f∗ .

1066

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

Theorem 3.10. Let (A, B) be a SCA, f : B → C a function, and let f ∗ : B∗ → C be the dual of f . Then f is a bent function if and only if f ∗ is also a bent function. √ 2 Proof. First assume that f is a bent function. Then it follows from the definition of f ∗ and f(b∗i )/ mi = o(B) for all i that ∗ ∗ (b∗ ) f (bi ) f i ∗ ∗ =  √ mi o(B) m = 1, for any bi ∈ B . i  √

∗ (bi )/ ki = Furthermore, from the proof of Proposition 3.9, o(B)f (bi )/ki for all bi ∈ B. Hence, f f (bi )/ki = 1 implies that 2 f ∗ (b ) 2  f (bi ) i = o(B), √ = o(B) ki ki

for any bi ∈ B.

Thus, f ∗ is a bent function. On the other hand, if f ∗ is a bent function, then f ∗∗ is also a bent function by what we have just proved. That is, f is a bent function by Proposition 3.9. 2 4. Dual bent functions on finite groups As an application of the dual functions on C-algebras, in this section we present the dual of a function on an arbitrary finite group. Let G be a finite group, and let GL(m, C) be the group of all invertible m × m matrices over C. Then a homomorphism ρ : G → GL(m, C) is called a representation of G, and m is called the degree of ρ. In the following the degree of a representation ρ will be denoted by dρ . Let f : G → C be a function. Then the Fourier transform of f at an irreducible representation ρ is defined as f(ρ) :=



f (x)ρ(x).

x∈G

Let χ be an irreducible character of G. Then the Fourier transform of f at χ is defined as f(χ) :=

1  f (x)χ(x). χ(1) x∈G

It is proved in [15, Corollary 3.5] that f is a class function if and only if for any irreducible representation ρ of G, f(ρ) = f(χ)Idρ , where χ is the irreducible character afforded by ρ, and Idρ is the dρ × dρ identity matrix. Let G be a finite group, and T := {z ∈ C |z| = 1}. A function f : G → T is called a bent function (cf. [9, Definition 9]) if for any irreducible unitary representation ρ of G,  ∗ f(ρ) f(ρ) = |G|Idρ ,

(4.1)

 ∗  If G is abelian, then f is a bent function if and only if where f(ρ) is the conjugate transpose of f(ρ). 2 for any (irreducible) character χ of G, f(χ) = |G| (cf. [6]). However, for an arbitrary finite group G, 2 if for any irreducible character χ of G, f(χ) = |G|, then f is called a character-bent function (cf. [15, Definition 5.1]). It is proved in [15, Theorem 5.3] that if f is a class function, then f is a bent function if and only if f is a character-bent function. Let G be a finite group, and CG the group algebra of G over C. Then (Z(CG), Cla(G)) is a SCA (cf. Example 2.1(ii)). Let P = (pi (j))j,i be the first eigenmatrix of (Z(CG), Cla(G)), and Ci , 0 ≤ i ≤ d, the

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1067

conjugacy classes of G. Then the irreducible characters χ0 , χ1 , . . . , χd of G can be indexed in such a way that χi (xj ) =

χi (1)pj (i) , |Cj |

xj ∈ Cj , 0 ≤ i, j ≤ d.

(4.2)

Let Irr(G)S := {χi (1)χi | 0 ≤ i ≤ d}, and let Ch(G) be the ring of complex valued class functions on G. Then (Ch(G), Irr(G)S ) is a SCA (cf. Example 2.1(iii)). (Ch(G), Irr(G)S ) is the standard dual C-algebra of (Z(CG), Cla(G)) (see the construction of a dual C-algebra in Section 2 and [2, Section 2.7]). Moreover, the multiplicities of (Z(CG), Cla(G)) are χi (1)2 , 0 ≤ i ≤ d. The reader is referred to [2] for more details. To simplify notation, the following notation will be used in this section. Notation 4.1. Let G be a finite group. Let (A, B) := (Z(CG), Cla(G)), (A∗ , B∗ ) := (Ch(G), Irr(G)S ), bi := Ci+ , and b∗i = χi (1)χi , 0 ≤ i ≤ d. Hence ki = |Ci |, and mi := χi (1)2 , 0 ≤ i ≤ d. Let G be a finite group, and f : G → C a function. Then f can be linearly extended to a function (still  denoted by f ) f : Cla(G) → C. That is, for any Ci+ ∈ Cla(G), f (Ci+ ) = x∈Ci f (x). Hence f(b∗i ) is defined for all i by Definition 2.3. Lemma 4.2. With the notation in Notation 4.1, let G be a finite group, and f : G → C a function. Then by regarding f as a function on Cla(G) by linear extension, √ f(b∗i )/ mi = f(χi ),

0 ≤ i ≤ d.

(4.3)

Proof. For each 0 ≤ j ≤ d, let xj be a representative of the conjugacy class Cj . Then (2.5) and (4.2) imply that for any 0 ≤ i ≤ d, f(b∗i ) = χi (1)

d  f (bj ) j=0

=

d 

kj

pj (i) = χi (1)

f (bj )χi (xj ) =

j=0

d  f (bj ) kj χi (xj ) j=0



kj

χi (1)

f (x)χi (x)

x∈G

= χi (1)f(χi ). So (4.3) holds. 2 Let G be a finite group, and f : G → T a function. Then as above, f is also a function on Cla(G) by linear extension. As a function on G, the bentness of f is defined by (4.1). But as a function on Cla(G), the bentness of f is defined by Definition 3.1. The next proposition gives the connection between these two concepts. Proposition 4.3. Let G be a finite group, and f : G → T a function. Then the following are equivalent. (i) As a function on G, f is a class bent function. (ii) As a function on Cla(G), f is a bent function. Proof. (i) ⇒ (ii) Since f is a class function, it is clear that f (bi )/ki = 1 for all 0 ≤ i ≤ d. Also f a class bent function on G yields that |f(χi )|2 = |G| for all irreducible characters χi by [15, Theorem 5.3]. So

1068

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

√ 2 by (4.3), f(b∗i )/ mi = |f(χi )|2 = |G| = o(Cla(G)) for all i. Hence, f is a bent function on Cla(G), and (ii) holds.   (ii) ⇒ (i) Since f is T -valued, for any 0 ≤ i ≤ d, f (bi )/ki = 1 yields that x∈Ci f (x) = x∈Ci |f (x)|. Thus, f is constant on each conjugacy class. That is, f is a class function on G. Furthermore, since f is a bent function on Cla(G), (4.3) implies that f is a character-bent function on G. So f is a bent function on G by [15, Theorem 5.3], and (i) holds. 2 Let G be a finite group, and f : G → C a function. Since f is a function on Cla(G) by linear extension, the dual of f , f ∗ , is defined as a function on Irr(G)S by Definition 3.8. Let Irr(G) be the set of all irreducible characters of G. Then by Definition 3.8 and Lemma 4.2, f ∗ is a function defined on Irr(G) as follows. Definition 4.4. Let G be a finite group, and f : G → C a function. Then the dual of f , f ∗ , is defined by χi (1) χi →  f(χi ). |G|

f ∗ : Irr(G) → C,

 be the dual group of G. Remark 4.5. Let G be a finite abelian group, and f : G → C a function. Let G ∼  a → χa is an   That is, G consists of (irreducible) characters of G. Note that G = G. Assume that G → G,  isomorphism. Then it is natural to define the dual of f , f , as a function on G by (cf. [6]) f : G → C,

1 a →  f(χa ). |G|

The dual of a function on an arbitrary finite group defined by Definition 4.4 generalizes the notion of the dual function on a finite abelian group. Let G be a finite group, and g : Irr(G) → C a function. Then g is also a function on Irr(G)S by linear extension. Thus, the Fourier transform and the dual of g are defined at any conjugacy class of G by Definitions 2.3 and 3.8. So we have the following Lemma 4.6. Let G be a finite group, and g : Irr(G) → C a function. Then the Fourier transform of g at the conjugacy class Ci+ is given by 1  g(Ci+ ) =  g(χj )χj (Ci+ ), |Ci | j=0 d

(4.4)

and the dual of g is given by 1  Ci+ →  g(χj )χj (Ci+ ). |G| j=0 d

g ∗ : Cla(G) → C,

(4.5)

Proof. Let Q = (qi (j))j,i be the second eigenmatrix of (Z(CG), Cla(G)). With the notation in Section 2 and Notation 4.1, it follows from (2.5) that g(bi ) =



ki

d  g(b∗j )qj (i) j=0

mj

,

for all 0 ≤ i ≤ d.

But by Lemma 2.2(iv) and (4.2), qj (i) pi (j) χj (Ci+ ) , = = mj ki |Ci |χj (1)

for all 0 ≤ i, j ≤ d.

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1069

Thus, g(Ci+ )

=



d 

1  χj (Ci+ ) = |Ci | χj (1)g(χj ) g(χj )χj (Ci+ ), |C |χ (1) |Ci | j=0 i j j=0 d

So (4.4) holds. Now (4.5) is clear by Definition 3.8.

0 ≤ i ≤ d.

2

Remark 4.7. Let G be a finite group, and g : Irr(G) → C a function. Then we can define a function 1  x →  g(χj )χj (x). |G| j=0 d

g : G → C,

Note that g is a class function on G. Let f : G → C be a function. For any x ∈ G, let Cx be the conjugacy class of x in G. Then we can show that f∗ (x) =

1 f (Cx+ ), |Cx |

for any x ∈ G,

where f ∗ : Irr(G) → C is the dual of f . Hence, for any conjugacy class Ci of G, f (Ci+ ) = f∗ (Ci+ ). If f is a class function, then f = f∗ . The next theorem is a direct consequence of Proposition 3.9, Theorem 3.10, and Proposition 4.3. It can also be proved directly by applying Lemma 4.6 and the Orthogonality Relations of Characters of finite groups. Theorem 4.8. Let G be a finite group, and f : G → C a function. Then the following hold. (i) As functions on Cla(G), f = f ∗∗ . (ii) If f is T -valued, then f is a class bent function on G if and only if f ∗ is a bent function on Irr(G)S . A related concept is the so-called difference set of a finite group. Let G be a group of order n and D a subset   of G with k elements. Let D+ := g∈D g and D(−) := g∈D g −1 . Then D is called an (n, k, λ)-difference set in G if D+ D(−) = λG+ + (k − λ)1G . The relations between bent functions and difference sets have been studied in many papers. A remarkable known result is that for a finite abelian group G, a function f : G → {1, −1} is a bent function if and only if f −1 (1) is a (4u2 , 2u2 ± u, u2 ± u)-difference set in G, where u is a positive integer, and |G| = 4u2 . Difference sets can be defined for a SCA in a similar way. Let (A, B) be a SCA, and S a nonempty subset    of B. Let S+ := bi ∈S bi , (S )+ := bi ∈S bi , and o(S) := bi ∈S ki . If S+ (S )+ = λB+ + (o(S) − λ)b0

for some λ ∈ R,

then S is called a (o(B), o(S), λ)-difference set in B. The above result on the difference sets and bent functions for finite abelian groups can be generalized to SCAs as follows. Theorem 4.9. Let (A, B) be a SCA, and f : B → C a function such that for any bi ∈ B, f (bi ) = ±ki . Let S := {bi ∈ B | f (bi ) > 0}. Then f is a bent function if and only if S is a (4u2 , 2u2 ± u, u2 ± u)-difference  set in B, where u := o(B)/2.

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1070

Proof. Since bj =

d i=0

pj (i)ei and bj =

d i=0

pj  (i)ei =

d i=0

pj (i)ei by (2.2), we see that

2 d   ei . S+ (S )+ = pj (i)ei · pj (i)ei = p (i) j i=0 bj ∈S i=0 bj ∈S i=0 bj ∈S d  

d  

So S is a (4u2 , 2u2 ± u, u2 ± u)-difference set if and only if 

pj (0) = 2u2 ± u

and

bj ∈S

But it follows from f (bi ) = ±ki and

d j=0

2  pj (i) = u2 for i = 1, 2, . . . , d. bj ∈S

(4.6)

pj (i) = δi0 o(B) that

d  f(b∗i )  f (bj ) = pj (i) = 2 pj (i) − δi0 o(B), √ mi kj j=0

0 ≤ i ≤ d.

bj ∈S

√ 2 Thus, f(b∗i )/ mi = o(B) for all i if and only if (4.6) holds. That is, f is a bent function if and only if S is a (4u2 , 2u2 ± u, u2 ± u)-difference set. 2 As an immediate application of Proposition 4.3 and Theorem 4.9, we have the following Corollary 4.10. Let G be an arbitrary finite group, and let f : G → {1, −1} be a class function. Then f is a bent function if and only if f −1 (1) is a (4u2 , 2u2 ± u, u2 ± u)-difference set in G, where u is a positive integer and |G| = 4u2 . 5. Examples In this section we present some examples of bent functions on SCA’s. The SCA’s in the examples below are exactly isomorphic to the Bose–Mesner algebras of the association schemes that can be found in [4]. Example 5.1. Let (A, B) be a SCA, where B := {b0 = 1A , b1 , b2 }, and the multiplication is given by b21 = b0 , b1 b2 = b2 , b22 = 14b0 + 14b1 + 12b2 . The first eigenmatrix of (A, B) is ⎛

1 ⎝ P = 1 1

⎞ 14 1 1 −2 ⎠ . −1 0

The orders and multiplicities are k0 = k1 = 1, k2 = 14 and m0 = 1, m1 = 7, m2 = 8, respectively, and o(B) = 16. For any function f : B → C such that |f (bi )/ki | = 1, 2 f(b∗ )   f (b0 ) f (b1 ) f (b )p (2) j j 2 ≤ 2 < o(B). = − √ = k0 m2 kj k1 j=0 So there is no bent function on (A, B). However, let S := {b0 , b2 }. Then S = S, o(S) = 15, and S+ S+ = 15b0 + 14b1 + 14b2 = 14B+ + b0 = λB+ + (o(S) − λ)b0 , where λ = 14.

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1071

That is, S is a (16, 15, 14)-difference set of B. Theorem 4.9 says that there is a correspondence between some bent functions and special difference sets. This example does not contradict Theorem 4.9, because the  parameters of the difference set S is not (4u2 , 2u2 ± u, u2 ± u), where u := o(B)/2 = 2. Example 5.2. Let (A, B) be a SCA, where B := {b0 = 1A , b1 , . . . , b6 }, and the multiplication is given by b21 = b0 , b1 b2 = b2 , b1 b3 = b3 , b1 b4 = b4 , b1 b5 = b6 , b1 b6 = b5 , b22 = b23 = b24 = 2b0 + 2b1 , b25 = b26 = 4b0 + 2b2 + 2b3 + 2b4 , b2 b3 = 2b4 , b2 b4 = 2b3 , b3 b4 = 2b2 , b2 b5 = b2 b6 = b3 b5 = b3 b6 = b5 + b6 , b4 b5 = b4 b6 = b5 + b6 , b5 b6 = 4b1 + 2b2 + 2b3 + 2b4 . The first eigenmatrix of (A, B) is ⎛

1 1 ⎜1 1 ⎜ ⎜1 1 ⎜ ⎜ P = ⎜1 1 ⎜ ⎜1 1 ⎜ ⎝ 1 −1 1 −1

2 2 2 2 2 2 2 −2 −2 −2 2 −2 −2 −2 2 0 0 0 0 0 0

⎞ 4 4 −4 −4 ⎟ ⎟ 0 0 ⎟ ⎟ ⎟ 0 0 ⎟. ⎟ 0 0 ⎟ ⎟ 2 −2 ⎠ −2 2

The orders and multiplicities are k0 = k1 = 1, k2 = k3 = k4 = 2, k5 = k6 = 4 and m0 = m1 = 1, m2 = m3 = m4 = 2, m5 = m6 = 4, respectively, and o(B) = 16. Let f : B → C be a function defined by f (b0 ) = f (b1 ) =

√ √ 1 1 1 f (b2 ) = f (b3 ) = − f (b4 ) = e −1θ1 , f (b5 ) = −f (b6 ) = 4e −1θ2 , 2 2 2

where θ1 , θ2 are any real numbers. Then it is straightforward to check that f is a bent function. Example 5.3. Let (C, D) be a SCA, where D := {d0 = 1C , d1 , d2 , d3 }, and the multiplication is given by d21 = 5d0 + 2d2 + 2d3 , d1 d2 = 2d1 + 2d2 + d3 , d1 d3 = 2d1 + d2 + 2d3 , d22 = 5d0 + 2d1 + 2d3 , d2 d3 = d1 + 2d2 + 2d3 , d23 = 5d0 + 2d1 + 2d2 . The first eigenmatrix of (C, D) is ⎛

1 5 ⎜1 1 P =⎜ ⎝1 1 1 −3

⎞ 5 5 1 −3 ⎟ ⎟. −3 1 ⎠ 1 1

The orders and multiplicities are k0 = 1, k1 = k2 = k3 = 5 and m0 = 1, m1 = m2 = m3 = 5, respectively, and o(D) = 16. Let g : D → C be a bent function. Then 3 g(d∗i )  g(dj )pj (i) √ = = 4, mi kj j=0

0 ≤ i ≤ 3.

Let g(di ) = xi , 0 ≤ i ≤ 3. Then |x0 | = 1, |x1 | = |x2 | = |x3 | = 5, and the above equations yield that

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1072

x 0 + x1 + x2 + x 3 1 1 3 x0 + x1 + x2 − x3 5 5 5 1 3 1 x0 + x1 − x2 + x3 5 5 5 3 1 1 x0 − x1 + x2 + x3 5 5 5

= 4ε0 , = 4ε1 , = 4ε2 , = 4ε3 ,

where εi ∈ C and |εi | = 1, 0 ≤ i ≤ 3. Therefore,  51 5 ε0 + ε1 + ε2 + ε3 , x1 = (ε0 + ε1 + ε2 − 3ε3 ), 4 5 4 5 5 x2 = (ε0 + ε1 − 3ε2 + ε3 ), x3 = (ε0 − 3ε1 + ε2 + ε3 ), 4 4

x0 =

and

1 ε0 + ε1 + ε2 + ε3 = 4 , |ε0 + ε1 + ε2 − 3ε3 | = |ε0 + ε1 − 3ε2 + ε3 | = |ε0 − 3ε1 + ε2 + ε3 | = 4. 5 5 √

So there are infinitely many bent functions on D. For example, let ε3 = −ε0 = e −1γ1 , and ε1 = −ε2 = √ e −1γ2 , where γ1 , γ2 are any real numbers. Then we have a bent function g : D → C defined by √ √ 1 g(d0 ) = − g(d1 ) = e −1γ1 , g(d2 ) = −g(d3 ) = 5e −1γ2 . 5

Here is another bent function h : D → C be defined by h(d0 ) = 1, h(d1 ) = 5θ, h(d2 ) = 5θ2 , h(d3 ) = 5θ3 , √ √ where θ = −1 or − −1. Let S := {d0 , d2 }. Then  S is a (16, 6, 2)-difference set of D. That is, S is a 2 2 2 (4u , 2u − u, u − u)-difference set of D, where u = o(D)/2 = 2. Note that there are bent functions other than those given in Theorem 4.9 corresponding to the difference set S. Let (A, B) and (C, D) be SCA’s, where B := {b0 = 1A , b1 , . . . , br }, and D := {d0 = 1C , d1 , . . ., ds }. Then the tensor product (A⊗C C, B⊗D) is also a SCA, where B⊗D := {bi ⊗dj | 0 ≤ i ≤ r, 0 ≤ j ≤ s}. Let (A∗ , B∗ ) and (C ∗ , D∗ ) be standard dual C-algebras of (A, B) and (C, D), respectively. Then (A∗ ⊗C C ∗ , B∗ ⊗ D∗ ) is a standard dual C-algebra of (A ⊗C C, B ⊗ D). Let P and U be the first eigenmatrices of (A, B) and (C, D), respectively. We may assume that the elements in B ⊗ D and B∗ ⊗ D∗ are ordered in such a way that the first eigenmatrix of (A ⊗C C, B ⊗ D) is P ⊗ U , the Kronecker product of P and U . The next theorem gives a construction of new bent functions from the old ones. However, the question whether some of the classical constructions of bent functions (e.g. Maiorana–McFarland) can carry over to SCA’s or not remains open. Theorem 5.4. With the notation and assumption in the above paragraph, let f : B → C and g : D → C be bent functions. Then f ⊗ g : B ⊗ D → C, bi ⊗ dα → f (bi )g(dα ), 0 ≤ i ≤ r, 0 ≤ α ≤ s is also a bent function. Proof. First of all, it is clear that (f ⊗ g)(bi ⊗ dα ) f (bi )g(dα ) = o(bi ⊗ dα ) o(bi )o(dα ) = 1, 0 ≤ i ≤ r, 0 ≤ α ≤ s.

B. Xu / Journal of Pure and Applied Algebra 220 (2016) 1055–1073

1073

Let pj (i) and uβ (α) be the (i, j)-entry and (α, β)-entry of P and U , respectively. Then the ((i, α), (j, β))-entry of P ⊗ U is ρj,β (i, α) := pj (i)uβ (α). Thus, for any b∗i ⊗ d∗α ∈ B∗ ⊗ D∗ ,  f (bj )pj (i)  g(dβ )uβ (α) f ⊗ g(b∗ ⊗ d∗α )  (f ⊗ g)(bj ⊗ dβ )ρj,β (i, α)  ∗i = = o(bj ⊗ dβ ) o(bj ) o(dβ ) o(bi ⊗ d∗α ) j j,β β f(b∗ ) g(d∗ ) =  i∗  α . o(bi ) o(d∗α ) Hence, f ⊗ g is a bent function. 2 Example 5.5. Let (A, B) and f : B → C be the same as in Example 5.2, and let (C, D) and g : D → C be the same as in Example 5.3. Then by Theorem 5.4, f ⊗ g : B ⊗ D → C, bi ⊗ dα → f (bi )g(dα ), 0 ≤ i ≤ 6, 0 ≤ α ≤ 3 is a bent function. Acknowledgements The author would like to thank Xiang-dong Hou for useful discussions, especially for his suggestions to study the dual of a function on an arbitrary finite group. The author would also like to thank the referee for the useful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

H. Blau, Quotient structures in C-algebras, J. Algebra 177 (1995) 297–337. E. Bannai, T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, CA, 1984. C. Carlet, C. Ding, Highly nonlinear mappings, J. Complex. 20 (2004) 205–244. A. Hanaki, http://math.shinshu-u.ac.jp/~hanaki/as/. B. Huppert, Character Theory of Finite Groups, Walter de Gruyter & Co., Berlin, 1998. O.A. Logachev, A.A. Salnikov, V.V. Yashchenko, Bent functions on a finite abelian group, Discrete Math. Appl. 7 (1997) 547–564. A. Pott, Nonlinear functions in abelian groups and relative difference sets, in: Optimal Discrete Structures and Algorithms, ODSA 2000, Discrete Appl. Math. 138 (2004) 177–193. L. Poinsot, Non Abelian bent functions, Cryptogr. Commun. 4 (2012) 1–23. L. Poinsot, Bent functions on a finite nonabelian group, J. Discrete Math. Sci. Cryptogr. 9 (2006) 349–364. L. Poinsot, Multidimensional bent functions, GESTS Internat. Trans. Comput. Sci. Engrg. 18 (2005) 185–195. L. Poinsot, A. Pott, Non-Boolean almost perfect nonlinear functions on non-Abelian groups, Int. J. Found. Comput. Sci. 22 (2011) 1351–1367. O.S. Rothaus, On bent functions, J. Comb. Theory, Ser. A 20 (1976) 300–305. N. Tokareva, Generalizations of bent functions: a survey of publications, J. Appl. Ind. Math. 5 (2011) 110–129. B. Xu, Multidimensional Fourier transforms and nonlinear functions on finite groups, Linear Algebra Appl. 452 (2014) 89–105. B. Xu, Bentness and nonlinearity of functions on finite groups, Des. Codes Cryptogr. 76 (2015) 409–430, http://dx.doi. org/10.1007/s10623-014-9968y. B. Xu, Morphisms and functor kerf in C-algebras, Algebra Colloq. 15 (2008) 145–166. B. Xu, The dual morphisms in C-algebras, Commun. Algebra 32 (2004) 345–361.