Fourier Transforms and Bent Functions on Finite Abelian Group-Acted Sets
aa r X i v : . [ c s . D M ] J un Fourier Transforms and Bent Functions on Finite AbelianGroup-Acted Sets
Yun Fan and Bangteng Xu School of Mathematics and Statistics, Central China Normal University,Wuhan 430079, China Department of Mathematics and Statistics, Eastern Kentucky University,Richmond, KY 40475, USAFebruary 6, 2018
Abstract
Let G be a finite abelian group acting faithfully on a finite set X . As a naturalgeneralization of the perfect nonlinearity of Boolean functions, the G -bentness and G -perfect nonlinearity of functions on X are studied by Poinsot et al. [6, 7] via Fouriertransforms of functions on G . In this paper we introduce the so-called G -dual set b X of X , which plays the role similar to the dual group b G of G , and the Fourier transforms offunctions on X , a generalization of the Fourier transforms of functions on finite abeliangroups. Then we characterize the bent functions on X in terms of their own Fouriertransforms on b X . Bent (perfect nonlinear) functions on finite abelian groups and G -bent( G -perfect nonlinear) functions on X are treated in a uniform way in this paper, andmany known results in [4, 2, 6, 7] are obtained as direct consequences. Furthermore, wewill prove that the bentness of a function on X can be determined by its distance from theset of G -linear functions. In order to explain the main results clearly, examples are alsopresented. Keywords: group actions; G -linear functions; G -dual sets; Fourier transforms on G -sets; bent functions; G -perfect nonlinear functions Bent functions, perfect nonlinear functions, and their generalizations have been studied in manypapers. The notion of a Boolean bent function was introduced by Rothaus [10]. More than adecade ago, Logachev, Salnikov, and Yashchenko [4] generalized this concept to bent functionson finite abelian groups. As a further generalization, Poinsot [5] studied bent functions onfinite nonabelian groups. Recently, a closely related notion, perfect nonlinear functions betweenfinite abelian groups as well as between arbitrary finite groups, has been studied in quite a fewpapers; for example, see [2, 8, 9, 12, 13, 14]. These functions have numerous applications incryptography, coding theory, and other fields. A critical tool in these studies is the Fouriertransforms of functions on finite groups.The perfect nonlinearity of a function f : G → H between finite abelian groups G and H is characterized by its derivatives f ′ α : G → H, x f ( αx ) f ( x ) − , for all non-identity α ∈ G . Email addresses : [email protected] (Yun Fan), [email protected] (Bangteng Xu)
1y observing that, for any α ∈ G , the mapping G → G, x αx is just the regular action of G on its base set G , Poinsot et al. [6, 7] generalized the concept of the perfect nonlinearityto a function g : X → H , where X is a finite set with an action of G on it. The derivativesof g are g ′ α : X → H, x f ( αx ) f ( x ) − , for any α ∈ G . In order to characterize the perfectnonlinearity of functions from the set X to H , the Fourier transforms of functions on the group G are used in [6, 7]. For any x ∈ X , let g x : G → H be the function defined by g x ( α ) = g ( αx ),for any α ∈ G . Let b G, b H be the dual groups of G and H , respectively, and let ζ be the principal(irreducible) character of H . Then g is G -perfect nonlinear (cf. [7, Theorems 2 and 3]) if andonly if for any ζ ∈ b H \{ ζ } ,1 | X | X x ∈ X (cid:12)(cid:12) d ζ ◦ g x ( ξ ) (cid:12)(cid:12) = | G | , for any ξ ∈ b G, where | X | and | G | are the cardinalities of X and G , respectively, and d ζ ◦ g x is the Fouriertransform of ζ ◦ g x on b G . Poinsot [6] also defined the bentness of functions on X in a similarway. Let C be the complex field, T the unit circle in C , and f : X → T a function. For any x ∈ X , let f x be a function on G defined by f x ( α ) := f ( αx ), for any α ∈ G . Then usingthe Fourier transforms b f x of the functions f x on G , the G -bentness of the function f on X isdefined as follows: f is G -bent if 1 | X | X x ∈ X (cid:12)(cid:12) b f x ( ξ ) (cid:12)(cid:12) = | G | , for all ξ ∈ b G. It is proved that f is G -bent if and only if for all non-identity element α ∈ G , the derivatives f ′ α are balanced.As the Fourier transform on finite groups is the key tool in the study of bent and perfectnonlinear functions on finite groups, in this paper for a finite abelian group G acting on a finiteset X ( X is called a G -set ), we will develop the Fourier analysis on X , and use it as our toolto study the bentness and perfect nonlinearity of functions on X . By characterizing the bentfunctions on X in terms of their own Fourier transforms, we are able to treat bent and perfectnonlinear functions on finite abelian groups as well as G -bent and G -perfect nonlinear functionson X in a uniform way.The set of functions from a G -set X to C , denoted by C X , is a C G -module, where C G isthe group algebra of G over C . C X is also a unitary space with a natural G -invariant innerproduct. For each irreducible character ψ ∈ b G , the G -linear component of C X with respect to ψ is the C G -submodule of C X consisting of ψ -linear functions (see Definition 2.2 below). C X can be decomposed into the orthogonal direct sum of its G -linear components (see Proposition2.7 below). A key step is that by using this decomposition we obtain an orthogonal basis b X of C X consisting of G -linear functions such that b X is closed under complex conjugation (seeTheorem 2.10 below). Such a basis b X , called a G -dual set of X , plays a role in C X similarto b G in C G . For any function f ∈ C X , we define the Fourier transform b f of f as a function on b X (see Definition 3.1 below), and define the bentness of f in terms of b f ( λ ) for all λ ∈ b X (seeDefinition 4.1 below). Our definitions of the Fourier transforms and bentness of functions on G -sets are natural generalizations of the Fourier transforms and bentness of functions on finiteabelian groups, respectively.Then using the Fourier analysis on a G -set X , we study the characterizations of bentfunctions on X . We will prove that (Theorem 4.6) a function f : X → T is bent if and onlyif the derivatives of f in all nontrivial directions are balanced. Since the bentness and perfectnonlinearity of functions on finite abelian groups and G -sets are treated in a uniform way,many known results in [4, 2, 6, 7] are obtained as immediate consequences. Furthermore, we2ill prove that (Theorem 4.9) f ∈ T X is a bent function if and only if the distance from f to the set ( C X ) G of G -linear functions reaches the best possible upper bound of the distancebetween ( C X ) G and any function in T X . This result gives another geometric interpretation ofthe importance of bent functions in cryptography. The perfect nonlinearity of functions from X to a finite abelian group H is also characterized in terms of Fourier transforms of functionson X (see Theorem 5.2). To explain the theory established in this paper, several examples arealso included.The rest of the paper is organized as follows. In Section 2 we present the classical decom-position of the C G -module C X , and prove the existence of the G -dual set b X of X . Then inSection 3 we introduce the Fourier transforms of functions in C X , and investigate their basicproperties. Section 4 is devoted to the study of the characterizations of bent functions on X .Finally, G -perfect nonlinear functions are discussed in Section 5, and explanatory examples arepresented in Section 6. G -dual sets Throughout the paper, G is always a finite abelian group of order | G | = m with multiplicativeoperation, X a finite G -set with cardinality | X | = n , C the complex field, and T the unit circlein C . For any sets R and S , by R S we denote the set of functions from S to R . Note that C S is equipped with scalar multiplication, function addition and function multiplication such thatit is a complex algebra. Also for f ∈ C S by f we denote the complex conjugation function,i.e. f ( s ) = f ( s ) for s ∈ S , where f ( s ) is the complex conjugate of f ( s ) ∈ C . Furthermore, C X is a C G -module (see below). In this section we discuss the structure of C X , and prove thatit has a special basis b X , called the G -dual set, which plays a similar role of b G for the Fouriertransforms. Some of the results in this section are known for general C G -modules, but ourtreatment is different. An irreducible character of a finite abelian group G is a homomorphism from G to the multi-plicative group of non-zero complex numbers. By b G we denote the dual group of G , i.e. thegroup consisting of irreducible characters of G . For the fundamentals of representation theoryof finite groups, the reader is referred to [1, 11]. We include some needed known facts here.For any σ ∈ C G we have a function b σ ∈ C b G , called the Fourier transform of σ , defined by b σ ( ψ ) = P α ∈ G σ ( α ) ψ ( α ), for all ψ ∈ b G . On the other hand, for any τ ∈ C b G we have a function b τ ∈ C G , called the Fourier inverse transform of τ , defined as follows: b τ ( α ) = m P ψ ∈ b G τ ( ψ ) ψ ( α )for all α ∈ G . Note that ψ ∈ b G for all ψ ∈ b G , and ψ runs over b G as ψ runs over b G . Also ψ ( α ) = ψ ( α ) − = ψ ( α − ), for all α ∈ G .It is well-known that bb σ = σ for all σ ∈ C G , and bb τ = τ for all τ ∈ C b G . Remark 2.1. By ρ we denote the regular character of G , i.e. ρ ( α ) = ( m, α = 1 , , α = 1; ∀ α ∈ G, where 1 denotes the identity element of G . It is known that ρ = P ψ ∈ b G ψ . Since b G is a basisof the m -dimensional space C G , any σ ∈ C G is a linear combination of b G , and the coefficientsof the linear combination are uniquely determined by σ . As σ = bb σ , by the Fourier inverse3ransform the linear combination of σ is σ = m P ψ ∈ b G b σ ( ψ ) ψ . Note that σ ( α ) = 0 for all α ∈ G \{ } if and only if σ = σ (1) m ρ , where G \{ } denotes the difference set of G removing theidentity element 1. Thus,(i) σ takes zero on G \{ } if and only if b σ is constant on b G . Since G and b G are dual to each other through the Fourier transforms, by interchanging theroles of σ and b σ in (i), we further get(ii) σ is constant on G if and only if b σ takes zero on b G \{ } , where by abuse of notation, 1 denotes the unity (principal) character: 1( α ) = 1 for all α ∈ G .No ambiguity of this notation can arise from the context. G -linear functions and the classical decomposition As mentioned before, X is a G -set with cardinality | X | = n . That is, there is a map G × X → X ,( α, x ) αx , such that for all x ∈ X we have ( αβ ) x = α ( βx ) for all α, β ∈ G , and 1 x = x . Forthe fundamentals of group actions, we refer the reader to [1].If all the values of a function f ∈ C X have length 1, i.e. f ∈ T X , then we say that f is a unitary function .The complex space C X is a C G -module with the following G -action:( αf )( x ) = f ( α − x ) , ∀ f ∈ C X ∀ α ∈ G ∀ x ∈ X. (2.1)In the literature, a C G -module is also called a complex G -space, see [11, Ch 1]. In this paperwe will use both terms. Furthermore, C X is a unitary space with the following inner product: h f, g i = X x ∈ X f ( x ) g ( x ) , ∀ f, g ∈ C X . This inner product is G -invariant in the following sense: h αf, αg i = h f, g i , or equivalently h αf, g i = h f, α − g i , for all f, g ∈ C X and α ∈ G . The length (or norm) | f | of any f ∈ C X isthen defined as | f | = p h f, f i = s X x ∈ X f ( x ) f ( x ) , (2.2)and the distance between f, g ∈ C X is defined as d( f, g ) = | f − g | . Further, for any subsets S , S ⊆ C X we can define the distance between S and S as follows:d( S , S ) = min (cid:8) d( f , f ) (cid:12)(cid:12) f ∈ S , f ∈ S (cid:9) . (2.3) Definition 2.2. (i) A function f ∈ C X is said to be G -linear if there is a ψ ∈ b G such that f ( αx ) = ψ ( α ) f ( x ) , ∀ α ∈ G ∀ x ∈ X, i.e. αf = ψ ( α ) f for all α ∈ G . More precisely, in that case we say that the function f is ψ -linear .(ii) For any ψ ∈ b G , the ψ -component ( C X ) ψ of C X is defined by( C X ) ψ = (cid:8) f (cid:12)(cid:12) f ∈ C X , f is ψ -linear (cid:9) . The ψ -components for all ψ ∈ b G are also simply called G -linear components of C X , withoutmentioning the irreducible characters.(iii) By ( C X ) G we denote the set of G -linear functions on X , i.e.( C X ) G = { f | f ∈ C X , f is G -linear } = [ ψ ∈ b G ( C X ) ψ . emark 2.3. (i) Algebraically, a function f ∈ C X is G -linear if and only if ˜ f : C X → C , P x ∈ X c x x P x ∈ X c x f ( x ) is a C G -homomorphism, where C X is the permutation C G -moduleand C is viewed as a C G -module with respect to some character ψ . There is also a geometricinterpretation of the G -linearity by derivatives; see Lemma 4.4 below.(ii) Let f be ψ -linear. Since ψ ( α ) = 0 for all α ∈ G , the zero-point set Ann( f ) = { x ∈ X | f ( x ) = 0 } of the function f must be G -invariant. Furthermore, if ψ ( α ) = 1 for some α ∈ G ,then f ( αx ) = f ( x ) (and hence αx = x ) for all x ∈ Ann( f ), where Ann( f ) = { x ∈ X | f ( x ) = 0 } denotes the complement of Ann( f ) in X . Lemma 2.4. (i) For any ψ ∈ b G , ( C X ) ψ is a G -invariant subspace ( i.e. C G -submodule ) of C X .(ii) For any ψ, ϕ ∈ b G such that ψ = ϕ , we have the orthogonality: (cid:10) ( C X ) ψ , ( C X ) ϕ (cid:11) = 0 . In particular, ( C X ) ψ ∩ ( C X ) ϕ = { } if ψ = ϕ . Proof. (i) is straightforward. Now we prove (ii). Since ψ = ϕ , there is an α ∈ G such that ψ ( α ) = ϕ ( α ). So for any f ∈ ( C X ) ψ and g ∈ ( C X ) ϕ , ψ ( α ) h f, g i = h ψ ( α ) f, g i = h α − f, g i = h f, αg i = h f, ϕ ( α ) g i = ϕ ( α ) h f, g i . Hence ψ ( α ) = ϕ ( α ) implies that h f, g i = 0. This proves that (cid:10) ( C X ) ψ , ( C X ) ϕ (cid:11) = 0. The rest of(ii) is clear. Remark 2.5.
The zero function 0 is clearly ψ -linear for all ψ ∈ b G . However, for a non-zero G -linear function f , Lemma 2.4 implies that there is a unique ψ ∈ b G such that f is ψ -linear.In other words, the union ( C X ) G = S ψ ∈ b G ( C X ) ψ is a disjoint union, in the sense that only thezero function is in the intersections.For σ ∈ C G and f ∈ C X , we define a function σ ∗ f ∈ C X , called the convolution product of σ and f , as follows: ( σ ∗ f )( x ) = X α ∈ G σ ( α ) f ( α − x ) , ∀ x ∈ X. Specifically, taking X = G to be the regular G -set, then for σ, τ ∈ C G the above formula givesthe usual convolution product τ ∗ σ ∈ C G of functions on the group:( τ ∗ σ )( α ) = X β ∈ G τ ( β ) σ ( β − α ) , ∀ α ∈ G. Lemma 2.6. (i) The following map is bilinear: C G × C X −→ C X , ( σ, f ) σ ∗ f. (ii) ( τ ∗ σ ) ∗ f = τ ∗ ( σ ∗ f ) , ∀ τ, σ ∈ C G , f ∈ C X .(iii) σ ∗ f = σ ∗ f , ∀ σ ∈ C G , f ∈ C X . Proof.
The lemma is true by straightforward computations.Any G -space is decomposed into a direct sum with each summand associated with exactlyone irreducible character; such a decomposition is called the classical decomposition (see [11, § C X . 5 roposition 2.7. (i) For any ψ ∈ b G and f ∈ C X , we have ψ ∗ f ∈ ( C X ) ψ .(ii) For any ψ ∈ b G and f ∈ C X , f ∈ ( C X ) ψ if and only if f = m ψ ∗ f .(iii) For any f ∈ C X , we have f = 1 m X ψ ∈ b G ψ ∗ f .(iv) We have the orthogonal direct sum: C X = L ψ ∈ b G ( C X ) ψ . Proof. (i). For any α ∈ G and x ∈ X , since αβ runs over G as β runs over G , we have( ψ ∗ f )( αx ) = X β ∈ G ψ ( αβ ) f (( αβ ) − αx ) = X β ∈ G ψ ( α ) ψ ( β ) f ( β − x )= ψ ( α ) · X β ∈ G ψ ( β ) f ( β − x ) = ψ ( α ) · ( ψ ∗ f )( x ) . (ii). If f = m ψ ∗ f , then f ∈ ( C X ) ψ by (i). If f ∈ ( C X ) ψ , then for any x ∈ X ,1 m ( ψ ∗ f )( x ) = 1 m X α ∈ G ψ ( α ) f ( α − x ) = 1 m X α ∈ G ψ ( α ) ψ ( α ) f ( x ) = f ( x ) . (iii). Since the regular character ρ satisfies that ρ ( α ) = ( , α = 1; m, α = 1; , we have m ρ ∗ f = f ,for all f ∈ C X . Furthermore, since ρ = P ψ ∈ ˆ G ψ , we get that f = 1 m ρ ∗ f = 1 m (cid:16) X ψ ∈ b G ψ (cid:17) ∗ f = 1 m X ψ ∈ b G ψ ∗ f. (iv) follows directly from (i), (iii), and Lemma 2.4. Definition 2.8. (i) For any f ∈ C X and ψ ∈ b G , the ψ -component of f is f ψ = m ψ ∗ f , andthe classical decomposition of f is f = P ψ ∈ b G f ψ .(ii) The orthogonal direct sum C X = L ψ ∈ b G ( C X ) ψ is called the classical decomposition of the G -space C X ; cf. [11, § G -dual set b X For an unitary space V and a basis u , · · · , u n of V , if there is a non-zero n ∈ C such that h u i , u j i = ( , i = j ; n, i = j ; , then we say that u , · · · , u n is an ¯ n -normal orthogonal basis of theunitary space. Definition 2.9.
A basis b X of the unitary G -space C X is called a G -dual set of X if the followingthree conditions are satisfied:(i) b X is an n -normal orthogonal basis;(ii) any λ ∈ b X is G -linear;(iii) b X is closed under complex conjugation, i.e. λ ∈ b X for all λ ∈ b X .6 heorem 2.10. For any G -set X , there exists a G -dual set b X . Proof.
Since ψ ∈ b G for any ψ ∈ b G , it follows from Lemma 2.6(iii) and Proposition 2.7(ii)that for any f ∈ ( C X ) ψ , f ∈ ( C X ) ψ . That is,( C X ) ψ = ( C X ) ψ , ∀ ψ ∈ b G, where ( C X ) ψ = (cid:8) f (cid:12)(cid:12) f ∈ ( C X ) ψ (cid:9) . For any ψ ∈ b G , it is known that there is an n -normalorthogonal basis ( b X ) ψ for the ψ -component ( C X ) ψ of C X . Hence, ( b X ) ψ = (cid:8) λ (cid:12)(cid:12) λ ∈ ( b X ) ψ (cid:9) isalso an n -normal orthogonal basis of the ψ -component ( C X ) ψ . Thus, if ψ = ψ , then ( b X ) ψ ∪ ( b X ) ψ is an n -normal orthogonal basis of ( C X ) ψ ⊕ ( C X ) ψ .In the following we prove that if ψ = ψ , then there is an n -normal orthogonal basis ( b X ) ψ of ( C X ) ψ such that for any λ ∈ ( b X ) ψ , λ = λ . Let f ∈ ( C X ) ψ such that f = 0. Then atleast one of f + f and √− f − f ) is not zero. Thus, ( C X ) ψ = ( C X ) ψ implies that there is a λ ∈ ( C X ) ψ such that λ = 0, and λ = λ . We may also assume that h λ , λ i = n . Note that( C X ) ψ = C λ ⊕ ( C λ ) ⊥ . Also for any f ∈ ( C λ ) ⊥ , it follows from λ = λ that f ∈ ( C λ ) ⊥ .Hence, if ( C λ ) ⊥ = { } , then as above, there is λ ∈ ( C λ ) ⊥ such that λ = λ , h λ , λ i = n ,and ( C λ ) ⊥ = C λ ⊕ ( C λ ⊕ C λ ) ⊥ . Continuing this process, we see that λ , λ , · · · form an n -normal orthogonal basis of ( C X ) ψ .Therefore, the orthogonal direct sum C X = L ψ ∈ b G ( C X ) ψ implies that the union b X of the n -normal orthogonal bases of the G -linear components of C X chosen in the above two paragraphsis a basis of C X that satisfies the conditions (i), (ii) and (iii) of Definition 2.9. Remark 2.11. (i) If b X is a G -dual set of X , then b Y = { ελ | λ ∈ b X, ε ∈ T } is also a G -dualset of X . We call b Y a rescaling of b X by T .(ii) If X is a transitive G -set, then every G -linear component ( C X ) ψ of C X is 1-dimensional,and hence ( b X ) ψ consists of exactly one function of length √ n . Thus, X has a unique G -dualset b X up to rescaling by T .(iii) In particular, if X = G is the regular G -set, then X has a unique G -dual set up torescaling by T . Usually, the dual group b G is chosen as b X .(iv) However, if the number of the G -orbits of X is greater that 1, then the G -dual set b X isnot unique up to rescaling by T . The proof of Theorem 2.10 provides a way to chose a G -dualset. Later we will show another way to get a G -dual set(see Examples 6.2 and 6.3 below).From now on, for the G -set X we fix a G -dual set b X . Then we have the disjoint union b X = [ ψ ∈ b G ( b X ) ψ , where ( b X ) ψ is an n -normal orthogonal basis of ( C X ) ψ , and ( b X ) ψ = ( b X ) ψ . Thus, the ψ -component of C X is ( C X ) ψ = M λ ∈ ( b X ) ψ C λ, ∀ ψ ∈ b G. (2.4)Note that some subsets ( b X ) ψ may be empty (correspondingly, some component ( C X ) ψ may bezero).Let b X = { λ , · · · , λ n } and X = { x , · · · , x n } . Then we have an n × n matrix Λ = (cid:0) λ i ( x j ) (cid:1) ≤ i,j ≤ n . The n -normal orthogonality of b X implies that Λ · Λ T = nI , where I is the7dentity matrix and Λ T is the conjugate transpose of Λ. Hence we also have Λ T · Λ = nI . Thus,we have the following Lemma 2.12. (Orthogonality Relations)
The following hold: X x ∈ X λ ( x ) µ ( x ) = ( n, λ = µ ;0 , λ = µ ; ∀ λ, µ ∈ b X. (2.5) X λ ∈ b X λ ( x ) λ ( y ) = ( n, x = y ;0 , x = y ; ∀ x, y ∈ X. (2.6) G -sets Given a G -dual set b X of the G -set X , in this section we define the Fourier transform of f ∈ C X on b X , and discuss its basic properties. We will need to consider the space C b X of complexfunctions on b X , which is also a unitary space with the inner product h g, h i = X λ ∈ b X g ( λ )¯ h ( λ ) , ∀ g, h ∈ C b X . For any σ ∈ C G , the Fourier transform b σ of σ at any ψ ∈ b G is b σ ( ψ ) = P α ∈ G σ ( α ) ψ ( α ).The next definition generalizes this notion to the functions on G -sets. Definition 3.1.
For any f ∈ C X , the Fourier transform of f , b f ∈ C b X , is defined as b f ( λ ) = X x ∈ X f ( x ) λ ( x ) , ∀ λ ∈ b X. For any g ∈ C b X , the Fourier inversion of g , b g ∈ C X , is defined as b g ( x ) = 1 n X λ ∈ b X g ( λ ) λ ( x ) , ∀ x ∈ X. It is clear that if X = G is the regular G -set, then the Fourier transform of f ∈ C X in theabove definition is exactly the Fourier transform of functions on G . Remark 3.2.
For x ∈ X we have the characteristic function x (i.e. x ( y ) = 0, if y = x , and x ( x ) = 1), whose Fourier transform is b x ( λ ) = λ ( x ), for any λ ∈ b X . Thus, we can rewrite thedefinitions of b f and b g in Definition 3.1 as follows: b f ( λ ) = h f, λ i , ∀ f ∈ C X , ∀ λ ∈ b X ; b g ( x ) = 1 n h g, b x i , ∀ g ∈ C b X , ∀ x ∈ X. Lemma 3.3. bb f = f, ∀ f ∈ C X , and bb g = g, ∀ g ∈ C b X . Proof.
For any x ∈ X we have bb f ( x ) = 1 n X λ ∈ b X (cid:16) X y ∈ X f ( y ) λ ( y ) (cid:17) λ ( x ) = X y ∈ X f ( y ) · n X λ ∈ b X λ ( y ) λ ( x ) .
8y the second orthogonality relation (2.6), we have bb f ( x ) = f ( x ) for all x ∈ X . Similarly, bythe first orthogonality relation (2.5), we have bb g ( λ ) = g ( λ ) for all λ ∈ b X .For any λ ∈ b X , there is a unique irreducible character ψ λ of G such that λ ∈ ( C X ) ψ λ byRemark 2.5. Also for any g ∈ C b X , the length of g is | g | = p h g, g i . Lemma 3.4.
Let σ ∈ C G and f ∈ C X . Then the following hold.(i) d σ ∗ f ( λ ) = b σ ( ψ λ ) b f ( λ ) for all λ ∈ b X .(ii) If σ ∈ b G , then \ m σ ∗ f ( λ ) = ( b f ( λ ) , λ ∈ ( b X ) σ ;0 , λ / ∈ ( b X ) σ . (iii) | c f ψ | = | \ m ψ ∗ f | = P λ ∈ ( b X ) ψ | b f ( λ ) | , for all ψ ∈ b G . Proof. (i). Since λ ( α − x ) = ψ λ ( α − ) λ ( x ) for α ∈ G and x ∈ X , we have d σ ∗ f ( λ ) = X x ∈ X ( σ ∗ f )( x ) λ ( x ) = X x ∈ X X α ∈ G σ ( α ) f ( α − x ) λ ( x )= X α ∈ G σ ( α ) ψ λ ( α ) X x ∈ X f ( α − x ) λ ( α − x )= b σ ( ψ λ ) b f ( λ ) . (ii). If σ ∈ b G , then by Remark 3.2, b σ ( ψ λ ) = h σ, ψ λ i = ( m, σ = ψ λ ;0 , σ = ψ λ . So (ii) follows from(i). (iii). | c f ψ | = | \ m ψ ∗ f | = P λ ∈ b X | \ m ψ ∗ f ( λ ) | . Since b X = S ψ ∈ b G ( b X ) ψ , by (ii) we get | c f ψ | = P λ ∈ ( b X ) ψ | b f ( λ ) | = P λ ∈ ( b X ) ψ | b f ( λ ) | .The following is an easy but useful fact. Lemma 3.5.
Any function f ∈ C X is a unique linear combination of b X as follows: f = 1 n X λ ∈ b X b f ( λ ) λ. Hence, for the classical decomposition f = P ψ ∈ b G f ψ , the ψ -component f ψ = 1 m ψ ∗ f = 1 n X λ ∈ ( b X ) ψ b f ( λ ) λ. Proof.
For all x ∈ X , Lemma 3.3 implies that f ( x ) = bb f ( x ) = 1 n X λ ∈ b X b f ( λ ) λ ( x ) = 1 n X λ ∈ b X b f ( λ ) λ ( x ) . Since 1 n X λ ∈ ( b X ) ψ b f ( λ ) λ ∈ ( C X ) ψ , Proposition 2.7 and (2.4) imply that the ψ -component of f is f ψ = n P λ ∈ ( b X ) ψ b f ( λ ) λ . 9 emma 3.6. Let f, g ∈ C X and α ∈ G . Then h α − f, g i = 1 n X ψ ∈ b G ψ ( α ) X λ ∈ ( b X ) ψ b f ( λ ) b g ( λ ) . In particular h f, g i = n h b f , b g i . Proof.
Recall that ( α − f )( x ) = f ( αx ) (cf. Eqn (2.1)). So by Lemma 3.5, we have h α − f, g i = X x ∈ X f ( αx ) g ( x ) = X x ∈ X n X λ ∈ b X b f ( λ ) λ ( αx ) 1 n X µ ∈ b X b g ( µ ) µ ( x ) . Note that b X is the disjoint union b X = S ψ ∈ b G ( b X ) ψ , and for λ ∈ ( b X ) ψ we have λ ( αx ) = ψ ( α ) λ ( x ).So h α − f, g i = 1 n X x ∈ X X ψ ∈ b G X λ ∈ ( b X ) ψ b f ( λ ) ψ ( α ) λ ( x ) X µ ∈ b X b g ( µ ) µ ( x )= 1 n X ψ ∈ b G ψ ( α ) X λ ∈ ( b X ) ψ X µ ∈ b X b f ( λ ) b g ( µ ) X x ∈ X λ ( x ) µ ( x ) . By the first orthogonality relation (2.5), we get that h α − f, g i = 1 n X ψ ∈ b G ψ ( α ) X λ ∈ ( b X ) ψ b f ( λ ) b g ( λ ) . Taking α = 1 in the above formula, we have h f, g i = n h b f , b g i .The next corollary is immediate from Lemma 3.6. Corollary 3.7. If f ∈ T X , then h f, f i = n and h b f , b f i = n . For any λ ∈ b X , b λ ∈ C b X , and b λ ( µ ) = ( , µ = λ ; n, µ = λ. So { b λ | λ ∈ b X } is an n -normalorthogonal basis of C b X . G -sets In this section we define the bentness of functions on the G -set X , and study its characteri-zations. For a finite abelian group G , a unitary function f : G → T is called a bent function (cf. [4]) if for any ψ ∈ b G , | b f ( ψ ) | = | G | . The next definition generalizes this notion to unitaryfunctions on G -sets. Definition 4.1.
A unitary function f : X → T on the G -set X is called a bent function if X λ ∈ ( b X ) ψ (cid:12)(cid:12) b f ( λ ) (cid:12)(cid:12) = | X | | G | , for all ψ ∈ b G. If X = G is the regular G -set, then for any ψ ∈ b G , ( b X ) ψ = { ψ } , and the above definitionof a bent function on the G -set X is exactly the same as the definition of a bent function on10 . The bentness of functions on G -sets are also defined in [6, Definition 6], and called G -bentfunctions . But the definition in [6] is different; it uses the Fourier transforms of functions on G .However, we will show that the definition in [6] is equivalent to Definition 4.1 (see Corollary4.12 below).Although the bent function is defined by the use of λ ∈ b X , the next lemma says that thebentness of a function on X is independent of the choice of b X . Recall that the length of afunction is defined by Eqn (2.2). Lemma 4.2.
For a unitary function f : X → T , the following are equivalent.(i) f is a bent function.(ii) For any ψ, ϕ ∈ b G , | b f ψ | = | b f ϕ | .(iii) For any ψ, ϕ ∈ b G , | f ψ | = | f ϕ | . Proof.
By Lemma 3.4, (i) implies (ii). Assume (ii). From Lemma 3.4 and Corollary 3.7we see that X ψ ∈ b G | c f ψ | = X ψ ∈ b G X λ ∈ ( b X ) ψ | b f ( λ ) | = h b f , b f i = n . Hence, for any ψ ∈ b G , P λ ∈ ( b X ) ψ (cid:12)(cid:12) b f ( λ ) (cid:12)(cid:12) = | c f ψ | = n /m , and (i) holds.(ii) and (iii) are equivalent by Lemma 3.6.The zero-point set Ann( f ) of f ∈ C X and its complement Ann( f ) in X are introduced inRemark 2.3. Note that f is non-zero if and only if Ann( f ) = ∅ . Definition 4.3. If f ∈ C X is a non-zero function and Ann( f ) is G -invariant, then f is said tobe differentiable . For any differentiable function f ∈ C X we define a function f ′ α on Ann( f ) asfollows: f ′ α ( x ) = f ( αx ) f ( x ) − , ∀ x ∈ Ann( f ) .f ′ α is called the derivative of f in direction α .Any unitary function f ∈ T X is differentiable and f ′ α ∈ T X . By Remark 2.3(ii), any non-zero G -linear function is differentiable. The following lemma is a geometric explanation of the G -linearity of a function by its derivative. Lemma 4.4.
Let f ∈ C X be differentiable. Then f ′ α is a constant function on Ann( f ) for any α ∈ G if and only if f is G -linear, i.e. there is a unique character ψ ∈ b G such that f ∈ ( C X ) ψ . Proof.
It is clear that if f is ψ -linear for ψ ∈ b G , then for any α ∈ G , f ′ α ( x ) = ψ ( α ) for x ∈ Ann( f ) is a constant function. Now assume that for any α ∈ G , f ′ α ( x ) = ψ f ( α ), for all x ∈ Ann( f ). Then for any α, β ∈ G we have ψ f ( αβ ) = f ′ αβ ( x ) = f ( αβx ) f ( x ) − = f ( α ( βx )) f ( βx ) − · f ( βx ) f ( x ) − = f ′ α ( βx ) · f ′ β ( x ) = ψ f ( α ) ψ f ( β ) . So ψ f is an irreducible character of G , and f ( αx ) = ψ f ( α ) f ( x ) , ∀ x ∈ X. Unitary functions far away from G -linear functions on X are more useful and interestingin cryptography. So by Lemma 4.4 we want to investigate those unitary functions whosederivatives in all nontrivial directions are far away from constant functions. As for unitaryfunctions on finite groups, a unitary function h : X → T is said to be balanced if P x ∈ X h ( x ) = 0.11 efinition 4.5. A unitary function f : X → T is said to have totally balanced derivatives if X x ∈ X f ′ α ( x ) = 0 , ∀ α ∈ G \{ } . Now we are ready to present the characterizations of bent functions on G -sets. Theorem 4.6.
A unitary function f ∈ T X is bent if and only if f has totally balanced deriva-tives. Proof.
Note that P x ∈ X f ′ α ( x ) = P x ∈ X f ( αx ) f ( x ) = h α − f, f i . So by Lemma 3.6 we have X x ∈ X f ′ α ( x ) = 1 n X ψ ∈ b G ψ ( α ) X λ ∈ ( b X ) ψ b f ( λ ) b f ( λ )= 1 n X ψ ∈ b G (cid:16) X λ ∈ ( b X ) ψ | b f ( λ ) | (cid:17) ψ ( α ) . Thus, Lemma 3.4(iii) implies that X x ∈ X f ′ α ( x ) = 1 n X ψ ∈ b G | c f ψ | ψ ( α ) . (4.1)If f has totally balanced derivatives, i.e. P x ∈ X f ′ α ( x ) = 0 for all α ∈ G \{ } , then Eqn (4.1)implies that the function P ψ ∈ b G | c f ψ | ψ on G takes zero on G \{ } , and hence it must be amultiple of the regular character ρ = P ψ ∈ b G ψ of G , cf. Remark 2.1. Thus, for any ψ, ϕ ∈ b G we have | c f ψ | = | c f ϕ | , and f is bent by Lemma 4.2.Conversely, if f is bent, i.e. | c f ψ | = n m for all ψ ∈ b G , then by Eqn (4.1) we have X x ∈ X f ′ α ( x ) = 1 n X ψ ∈ b G n m ψ ( α ) = nm X ψ ∈ b G ψ ( α ) = nm ρ ( α ) . So for all α ∈ G \{ } , P x ∈ X f ′ α ( x ) = 0 by Remark 2.1, and f has totally balanced derivatives. Corollary 4.7.
If there is a ψ ∈ b G such that ( C X ) ψ = 0 (i.e. ( b X ) ψ = ∅ ), then there exists nobent function f ∈ T X . Proof.
In that case | c f ψ | = 0. Remark 4.8.
The above corollary says that the condition “( C X ) ψ = 0 for all ψ ∈ b G ” is anecessary condition for the existence of bent functions.If the G -action on X is not faithful, i.e. the kernel K of the action is nontrivial, thenthere must be an irreducible character ψ of G which takes nontrivial values on K , and hence( C X ) ψ = 0, cf. Remark 2.3(ii). So by the above corollary, there exists no bent functions on X .However, even if the G -action on X is faithful, there may still exist some ψ ∈ b G such that( C X ) ψ = 0, and hence the bent functions on X do not exist. See Example 6.2 below for suchan example. 12ur next characterization of a bent function is given by its distance from the set ( C X ) G of G -linear functions (cf. Definition 2.2). Recall that the distance between two subsets of C X isdefined by Eqn (2.3). The next theorem says that the distance from a bent function to ( C X ) G is greater than the distance from any non-bent unitary function to ( C X ) G . It also says that p ( m − n/m is the best possible upper bound of the distance between any unitary functionand ( C X ) G . Theorem 4.9.
Let f ∈ T X . Then the following hold.(i) d ( f, ( C X ) G ) ≤ q ( m − nm .(ii) f is bent if and only if d ( f, ( C X ) G ) = q ( m − nm . Proof.
We have seen that ( C X ) G = [ ψ ∈ b G ( C X ) ψ . For any G -linear function g there is a ϕ ∈ b G such that g is ϕ -linear, i.e. g = g ϕ ∈ ( C X ) ϕ ,and g ψ = 0 for any ψ ∈ b G such that ψ = ϕ . Since any two different G -linear components areorthogonal to each other, we can compute the distance between f and g as follows:d( f, g ) = | f − g | = (cid:12)(cid:12)(cid:12) X ψ ∈ b G ( f ψ − g ψ ) (cid:12)(cid:12)(cid:12) = | f ϕ − g ϕ | + X ψ = ϕ | f ψ | ≥ X ψ = ϕ | f ψ | ;and the equality holds when g = f ϕ . By Corollary 3.7, X ψ ∈ b G | f ψ | = X ψ ∈ b G h f ψ , f ψ i = * X ψ ∈ b G f ψ , X ψ ∈ b G f ψ + = h f, f i = | f | = n. So according to the definition of the distance in Eqn (2.3), we haved (cid:0) f, ( C X ) ϕ (cid:1) = n − | f ϕ | . Hence the square of the distance between f and ( C X ) G isd (cid:0) f, ( C X ) G (cid:1) = min ϕ ∈ b G (cid:8) n − | f ϕ | (cid:9) = n − max ϕ ∈ b G (cid:8) | f ϕ | (cid:9) . By the equality P ψ ∈ b G | f ψ | = n again, | b G | = m implies thatmax ϕ ∈ b G (cid:8) | f ϕ | (cid:9) ≥ nm , where the equality holds if and only if | f ψ | = | f ϕ | for all ψ, ϕ ∈ b G . In conclusion,d (cid:0) f, ( C X ) G (cid:1) ≤ n − nm = ( m − nm , (4.2)and the equality in (4.2) holds if and only if | f ψ | = | f ϕ | for all ψ, ϕ ∈ b G . By Lemma 4.2, theequality in (4.2) holds if and only if f is bent.By taking X = G as the regular G -set, we have the next corollary from Theorem 4.6,Theorem 4.9 and Lemma 4.2. Note that the equivalence of (i) and (ii) in Corollary 4.10 belowwas proved in [4]. 13 orollary 4.10. Let f : G → T be a unitary function. Then the following are equivalent.(i) f is a bent function.(ii) f has totally balanced derivatives.(iii) Among all functions in T G , f has the greatest distance p | G | − from the set ( C G ) G of G -linear functions.(iv) |h f, ψ i| are equal for all ψ ∈ b G . Proof.
The equivalence of (i), (ii), and (iii) is immediate from Theorems 4.6 and 4.9.Since b G is a basis of C G , we may assume that f = P ψ ∈ b G c ψ ψ , where c ψ ∈ C . Hence, the ψ -component of f is f ψ = c ψ ψ , for any ψ ∈ b G . Thus, |h f, ψ i| = |h c ψ ψ, ψ i| = | c ψ | = q |h f ψ , f ψ i| , for any ψ ∈ b G. So the equivalence of (i) and (iv) holds by Lemma 4.2.
Lemma 4.11.
For any f ∈ C X and x ∈ X , let f x ∈ C G be defined by f x ( α ) = f ( αx ) for all α ∈ G . Then b f x ( ψ ) = mf ψ ( x ) for all ψ ∈ b G . Proof.
It is a straightforward computation to see that b f x ( ψ ) = X α ∈ G f x ( α ) ψ ( α ) = X α ∈ G f ( αx ) ψ ( α − ) = ( ψ ∗ f )( x ) = mf ψ ( x ) . The next corollary is one of the main results of [6, 7], where the G -bentness of f ∈ T X isdefined by the condition (ii) of Corollary 4.12. So Corollary 4.12 implies that the G -bentnessdefined in [6, 7] is equivalent to the bentness defined by Definition 4.1. Corollary 4.12. (Cf. [6, 7])
Let f ∈ T X . Then the following two statements are equivalentto each other:(i) f has totally balanced derivatives. That is, f is a bent function by Definition 4.1.(ii) n P x ∈ X | b f x ( ψ ) | = m for all ψ ∈ b G . That is, f is a G -bent function by [6, Definition6]. Proof.
By Lemmas 4.11 and 3.6, we have1 n X x ∈ X | b f x ( ψ ) | = m n X x ∈ X | f ψ ( x ) | = m n h f ψ , f ψ i = m n h c f ψ , c f ψ i = m n | c f ψ | . Thus, (ii) holds if and only if | c f ψ | = n m for all ψ ∈ b G if and only if f has totally balancedderivatives by Theorem 4.6 and Lemma 4.2. Remark 4.13.
For any f, g ∈ C X , the pseudo-convolution f ⊠ g of f and g is defined as (cf.[7]) f ⊠ g : G → C , α X x ∈ X f ( x ) g ( αx ) . By Lemmas 2.12 and 3.3, it is straightforward to show that \ ( f ⊠ g )( ψ ) = mn X λ ∈ ( b X ) ψ b f ( λ ) b g ( λ ) , for any ψ ∈ b G. The equivalence of (i) and (ii) of Corollary 4.12 can also be proved by the above equality.14
Perfect nonlinear functions on G -sets As an application of the characterizations of bent functions on G -sets, in this section we discussthe characterizations of perfect nonlinear functions from a G -set to an abelian group. Ourapproach here is different from that of [6, 7]. Let X be a G -set as before, and let H be anabelian group with multiplicative operation. Let H X denote the set of all functions from X to H . An f ∈ H X is said to be evenly-balanced (cf. [13, 14]) if | H | divides | X | and (cid:12)(cid:12) { x ∈ X | f ( x ) = h } (cid:12)(cid:12) = | X || H | , for any h ∈ H. An evenly-balanced function is also called a balanced or uniformly distributed function in someliterature. The derivative of f ∈ H X in direction α ∈ G is f ′ α : X → H, x f ( αx ) f ( x ) − . Definition 5.1. (cf. [7, Definition 1]) A function f : X → H is said to be G -perfect nonlinear if for any α ∈ G \{ } , the function f ′ α is evenly-balanced.Any g ∈ H X induces a non-negative integral function g on H as follows: g : H → N ∪ { } , h (cid:12)(cid:12) { x ∈ X | f ( x ) = h } (cid:12)(cid:12) . Hence, g is constant on H if and only if g is evenly-balanced. Thus, a function f : X → H is G -perfect nonlinear if and only if for any α ∈ G , f ′ α is constant on H . Theorem 5.2.
Let f ∈ H X . Then following are equivalent.(i) For any ξ ∈ b H \{ } the composition function ξ ◦ f : X → C has totally balanced deriva-tives.(ii) For any ξ ∈ b H \{ } the composition function ξ ◦ f : X → C is bent.(iii) The function f : X → H is G -perfect nonlinear. Proof.
It is enough to show that (i) ⇔ (iii). Since ( ξ ◦ f )( x ) ∈ T , we have ( ξ ◦ f )( x ) − =( ξ ◦ f )( x ), for any x ∈ X . So X x ∈ X ( ξ ◦ f ) ′ α ( x ) = X x ∈ X ( ξ ◦ f )( αx )( ξ ◦ f )( x )= X x ∈ X ξ (cid:0) f ( αx ) (cid:1) ξ (cid:0) f ( x ) (cid:1) = X x ∈ X ξ (cid:0) f ( αx ) (cid:1) ξ (cid:0) f ( x ) − (cid:1) = X x ∈ X ξ (cid:0) f ( αx ) f ( x ) − (cid:1) = X x ∈ X ξ (cid:0) f ′ α ( x ) (cid:1) . For any h ∈ H , let X ( f ′ α , h ) = { x ∈ X | f ′ α ( x ) = h } . Then X is the disjoint union X = S h ∈ H X ( f ′ α , h ), and the cardinality | X ( f ′ α , h ) | = f ′ α ( h ). So X x ∈ X ( ξ ◦ f ) ′ α ( x ) = X h ∈ H X x ∈ X ( f ′ α ,h ) ξ ( h ) = X h ∈ H f ′ α ( h ) ξ ( h ) = d f ′ α ( ξ ) . (5.1)Thus, ( ξ ◦ f ) ′ α is balanced if and only if d f ′ α ( ξ ) = 0. Hence for any ξ ∈ b H \{ } , the function( ξ ◦ f ) ′ α is balanced if and only if d f ′ α is zero on b H \{ } if and only if f ′ α is constant on H byRemark 2.1(ii). That is, for any ξ ∈ b H \{ } , the function ξ ◦ f has totally balanced derivativesif and only if f is G -perfect nonlinear.Taking X = G to be the regular G -set, we have the next15 orollary 5.3. (Cf. [2]) Let
G, H be abelian groups, and f : G → H a function. Then thefollowing are equivalent.(i) f is perfect nonlinear.(ii) For any ξ ∈ b H \{ } the composition function ξ ◦ f : G → C is bent. Let f ∈ H X . Then for any x ∈ X , there is a function (cf. [6, 7]) f x : G → H, α f ( αx ) . Also for any ξ ∈ b H , there is a function ( ξ ◦ f ) x : G → T, α ( ξ ◦ f )( αx ). Note that ( ξ ◦ f ) x = ξ ◦ f x , for any x ∈ X . The next corollary is immediate from Theorem 5.2 and Corollary 4.12. Corollary 5.4. (cf. [6, Theorems 5 and 7])
Let f ∈ H X . Then the following are equivalent.(i) f is G -perfect nonlinear.(ii) For any ξ ∈ b H \{ } and α ∈ G , | X | X x ∈ X (cid:12)(cid:12)(cid:12) \ ( ξ ◦ f x )( α ) (cid:12)(cid:12)(cid:12) = | G | . In this section we present a few examples that explain the theory developed in the previoussections.
Example 6.1.
Assume that X = G is the regular G -set. As mentioned in Remark 2.11, the G -dual set b X is unique up to rescaling by T , and the typical choice of b X is just the dualgroup b G . So the theory developed in previous sections includes the corresponding theory forfinite abelian groups as a special case. For example, some well-known results in [2, 4, 12] aswell as other properties of bent functions on finite abelian groups are given in Corollary 4.10and Corollary 5.3 as immediate consequences.The next example gives a G -set on which there exists no bent function. Example 6.2.
Let G = { , α, β, γ } be the Klein four group. That is, G is an abelian groupsuch that α = β = γ = 1 , αβ = γ, βγ = α, γα = β. Then b G = { ψ = 1 , ψ , ψ , ψ } is given by Table 6.1.1 α β γψ ψ − − ψ − − ψ − − X = { x , x , x , x } be a faithful G -set with two orbits X and X as follows: • X = { x , x } , 1 and α fix both points x and x , while β and γ interchange the twopoints; 16 X = { x , x } , 1 and β fix both points x and x , while γ and α interchange the twopoints.We can take b X = { λ , λ , λ , λ } as in Table 6.2 (to simplify the table, we list √ λ i instead of λ i ). x x x x √ λ √ λ − √ λ √ λ − G -dual set in Example 6.2Hence, the G -linear components are( b X ) ψ = { λ , λ } , ( b X ) ψ = { λ } , ( b X ) ψ = { λ } , ( b X ) ψ = ∅ . Since one of the G -linear components is empty, there exists no bent function f ∈ T X byCorollary 4.7.The next example gives a G -set X and a bent function on X . Example 6.3.
As above in Example 6.2, let G = { , α, β, γ } be the Klein four group and b G = { ψ , ψ , ψ , ψ } . But this time we consider the G -set X = { x , x , x , x , x , x } withthree orbits: • X = { x , x } , 1 and α fix both points x and x , while β and γ interchange the twopoints; • X = { x , x } , 1 and β fix both points x and x , while γ and α interchange the twopoints; • X = { x , x } , 1 and γ fix both points x and x , while α and β interchange the twopoints.We can take b X = { λ , λ , λ , λ , λ , λ } as in Table 6.3 (to simplify the table, we list √ λ i instead of λ i ). x x x x x x √ λ √ λ − √ λ √ λ − √ λ √ λ − G -dual set in Example 6.3We can check that the G -linear components of C X are( b X ) ψ = { λ , λ , λ } , ( b X ) ψ = { λ } , ( b X ) ψ = { λ } , ( b X ) ψ = { λ } . ω = − √− be a primitive third root of unity. Take f ∈ T X as follows: f ( x j ) = ω (1+( − j ) / = ( , j = 1 , , ω, j = 2 , , . Then X x ∈ X j f ′ α ( x ) = X x ∈ X j f ( αx ) f ( x ) − = ( , j = 1;1 · ω − + ω · − , j = 2 , . So P x ∈ X f ′ α ( x ) = 0. Similarly, P x ∈ X f ′ β ( x ) = P x ∈ X f ′ γ ( x ) = 0. That is, f has totally balancedderivatives.On the other hand, h b f ψ , b f ψ i = X λ ∈ ( b X ) ψ | b f ( λ ) | = X j =1 , , (cid:12)(cid:12) X x ∈ X f ( x ) λ j ( x ) (cid:12)(cid:12) = X j =1 , , | ω | = 3 | ω | = 3 , h b f ψ , b f ψ i = X λ ∈ ( b X ) ψ | b f ( λ ) | = (cid:12)(cid:12) X x ∈ X f ( x ) λ ( x ) (cid:12)(cid:12) = | − ω | = 3 . Similarly, h b f ψ , b f ψ i = h b f ψ , b f ψ i = | − ω | = 3. In conclusion, we have h b f ψ , b f ψ i = 3, ∀ ψ ∈ b G ,and f is a bent function.Finally, we give an example of a G -perfect nonlinear function. Example 6.4.
We continue Example 6.3 and further take H = { , h, h } with h = 1 to be acyclic group of order 3. Let g : X → H be as follows: g ( x j ) = h (1+( − j ) / = ( , j = 1 , , h, j = 2 , , . It is known that b H = { , ξ, ξ } , where ξ ( h i ) = ω i , i = 0 , ,
2. Then the composition function ξ ◦ g : X → C is just the function f in Example 6.3, and hence ξ ◦ g is a bent function on X .Similarly we can check that ξ ◦ g is also a bent function on X . So g : X → H is a G -perfectnonlinear function from the G -set X to the abelian group H . In fact, one can check directlythat g ′ α ( x j ) = g ( αx j ) g ( x j ) − = , j = 1 , h, j = 3 , h , i = 4 , . Hence, g ′ α ( h i ) = 2 for i = 0 , ,
2. Similarly, g ′ β = g ′ γ = 2 are constant functions on H , too. Acknowledgments
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