Some Results on Bent-Negabent Boolean Functions over Finite Fields
aa r X i v : . [ c s . I T ] J un Some Results on Bent-Negabent Boolean Functionsover Finite Fields ∗ Sumanta SarkarCentre of Excellence in CryptologyIndian Statistical Institute, Kolkata, [email protected]
Abstract
We consider negabent Boolean functions that have Trace repre-sentation. We completely characterize quadratic negabent monomialfunctions. We show the relation between negabent functions and bentfunctions via a quadratic function. Using this characterization, wegive infinite classes of bent-negabent Boolean functions over the finitefield F n , with the maximum possible degree, n . These are the firstever constructions of negabent functions with trace representation thathave optimal degree. Keywords:
Negabent function, bent function, quadratic Boolean func-tion, Maiorana-McFarland function, permutation, complete mapping poly-nomial.
Hadamard-Walsh transform is an important tool in characterizing Booleanfunctions. For example, many cryptographic properties can be analyzed bythe Hadamard-Walsh transform. A function on even number of variablesthat has the maximum possible distance from the affine functions is calleda bent function. These functions have equal absolute spectral values un-der the Hadamard-Walsh transform and was first introduced by Rothaus[Rot76]. It is natural to investigate the spectral values of Boolean func-tions under some other Fourier transform. In 2007, Parker and Pott [PP07], ∗ This is a sufficiently revised and extended version of the paper [Sar12]. Section 5 is acompletely new contribution. +
12, Sar09]. As an example, the 6-variable function x ( x x ⊕ x x ⊕ x ⊕ x ) ⊕ x ( x x ⊕ x x + x ) + x ( x ⊕ x ) is a cubic negabentfunction.In [PP07, SPP08] some classes of Boolean functions which are bothbent and negabent (bent-negabent) have been identified. In [SPP08], con-struction of negabent functions has been shown in the class of Maiorana-McFarland bent functions. It is interesting to note that all the affine func-tions (both odd and even variables) are negabent [PP07, Proposition 1]. In[Sar09], symmetric negabent functions have been characterized and shownto be all affine for both odd and even number of variables. The maximumdegree of an n -variable bent-negabent function is n . Very recently construc-tion of bent-negabent functions have been given in [SPT13] with the optimaldegree.In this paper, we characterize the negabent functions which are definedover finite fields, i.e. , functions with Trace representation.Let F n be the vector space formed by the binary n -tuples and F n be thefinite field with 2 n elements. For a set E , the set of non zero elements of E is denoted by E ∗ .In [PP07], quadratic negabent Boolean functions defined over the vectorspace F n were characterized. Any quadratic Boolean function can be writtenas g ( x , . . . , x n ) = X ≤ i 1, is the imaginary unit of the complex number. Note that N f ( λ ) is complex valued.The function f : F n → F is called negabent [PP07] if the magnitudeof N f ( λ ) is 1, i.e. , |N f ( λ ) | = 1 for all λ ∈ F n . In the following theoremwe state an alternate characterization of negabent functions in terms oftheir negaperiodic autocorrelation values which has been shown in [PP07,Theorem 2] and [SGC + 12, Lemma 3]. Theorem 1. A Boolean function f is negabent if and only if X x ∈ F n ( − f ( x ) ⊕ f ( x ⊕ y ) ( − x · y = 0 (4) for all y ∈ F n ∗ . From this theorem, we see that the correlation values between the func-tion f ( x ) and f ( x ⊕ y ) ⊕ y · x are all zero for all y ∈ F n ∗ . Moreover, for evennumber of variables, if a negabent function is also a bent function, then thecorrelation values of the function f ( x ) and f ( x ⊕ y ) are also equal to zero forall y ∈ F n ∗ . Therefore, the functions which are both bent and negabent areinteresting to study. We call these functions bent-negabent . Bent functionscan exist only on even number of variables and it has degree more than1. However, all the affine functions are negabent [PP07] which tells thatnegabent functions exist for both even and odd number of variables.4 Characterization of negabent functions over thefinite field F n Now we consider Boolean functions defined over the field F n and we char-acterize the negabent property of those functions. The vector space F n canbe easily identified with the field F n by choosing a basis of F n over F .The function T r n : F n F is defined as T r n ( x ) = x + x + . . . + x n − . We denote T r n simply by T r and “ + ” is the finite field addition. If wechoose the basis { α , . . . , α n } to be self dual then it can be shown that T r ( xy ) = P ni =1 x i y i .Henceforth, in this paper we choose the basis to be self dual. It is alsonotable that a linear function over F n is given by ℓ ( x ) = T r ( ax ), a = 0.Given a polynomial F ( x ) over F n , we can get a Boolean function f : F n → F defined as f ( x ) = T r ( F ( x )). The highest binary weight of theexponents of F ( x ) is denoted as the algebraic degree of F ( x ), then the degreeof f ( x ) = T r ( F ( x )) is equal to the algebraic degree of F ( x ).With the above discussions it is now clear that we can characterize ne-gabent functions defined over the finite field as follows. Proposition 1. The function f : F n → F is negabent if and only if X x ∈ F n ( − f ( x )+ f ( x + a )+ T r ( ax ) = 0 (5) for all a ∈ F ∗ n . It is known that a bent function f : F n F is bent if and only if X x ∈ F n ( − f ( x )+ f ( x + a ) = 0 (6)for all a ∈ F ∗ n .A function for which both (5) and (6) hold is a bent-negabent function.An n -variable Boolean function φ is called balanced if its weight is 2 n − .Note that P x ∈ F n ( − φ ( x ) = 0 if and only if φ is balanced. Therefore, if f is bent-negabent, then both f ( x ) + f ( x + a ) and f ( x ) + f ( x + a ) + T r ( ax )are balanced for all a ∈ F ∗ n . 5 .1 Linear structure of negabent functions As the sum (5) involves derivatives of f , i.e. , f ( x ) + f ( x + a ), here we brieflydiscuss about the linear structures of negabent functions. Definition 1. An a ∈ F ∗ n is said to be a linear structure of a polynomial F ( x ) over F n if the derivative F ( x ) + F ( x + a ) is constant. From (6), it is clear that a bent function can not have a linear structure.However, a negabent function can have a linear structure. In fact, if f issuch that any a ∈ F ∗ n is a linear structure, then f is negabent. As in thatcase, the term f ( x ) + f ( x + a ) + ax is c + ax , for some constant c , whichis affine, i.e. , balanced. This happens when f ( x ) is an affine polynomial,which proves that if f is affine it is negabent.A polynomial F ( x ) over F n is called a complete mapping polynomialif both F ( x ) and F ( x ) + x are permutation polynomials. We use suchpermutation polynomials in our constructions. We consider quadratic monomials and characterize when they are negabent. Proposition 2. Let f : F n → F be a quadratic function of the form f ( x ) = T r ( λx k +1 ) . Then f is negabent if and only if λ n − k a n − k + λa k + a = 0 (7) for all a ∈ F ∗ n .Proof. From (5) we know that the function f is negabent if and only if forall a ∈ F ∗ n , X x ∈ F n ( − T r ( λx k +1 )+ T r ( λ ( x + a ) k +1 )+ T r ( ax ) = 0 , i.e. , X x ∈ F n ( − T r (cid:16) ( λ n − k a n − k + λa k + a ) x (cid:17) = 0 . (8)Note that (8) is true if and only if λ n − k a n − k + λa k + a = 0 , for all a ∈ F ∗ n . Hence the result. 6 roposition 3. The quadratic function f : F n → F of the form f ( x ) = T r ( λx k +1 ) is negabent if and only if P ( x ) = λ n − k x n − k + λx k + x is a permutation polynomial over F n .Proof. From Proposition 2, we know that f is negabent if and only if λ n − k a n − k + λa k + a = 0for all a ∈ F ∗ n . That means for such λ ∈ F ∗ n , P ( x ) = λ n − k x n − k + λx k + x has no non zero root in F n . Note that P ( x ) is a linearized polynomial andlinearized polynomial is permutation if and only if it has no non zero root.Hence the result.The polynomial P ( x ) is a permutation over F n if and only if the polyno-mial P ( x ) k is a permutation over F n . Therefore, T r ( λx k +1 ) is a negabentif and only if λx + λ k x k + x k is a permutation. There are some resultson the number of solutions of the polynomial λ k x k + x k + λx in [HK10].Let us point out some results related to the equation λ k x k + x k + λx = 0which has been extensively studied in [HK10]. Let gcd( k, n ) = d ≥ n = td for t > 1. A particular sequence of polynomials over F n is introducedas follows. C ( x ) = 1 ,C ( x ) = 1 ,C i +2 ( x ) = C i +1 ( x ) + x ik C i ( x ) for 1 ≤ i ≤ t − . (9)Another polynomial Z n ( x ) over F n is defined as follows. Z ( x ) = 1 ,Z t ( x ) = C n +1 ( x ) + xC k t − ( x ) for t > . (10)Then we have the following result from [HK10, Proposition 2]. Proposition 4. Let gcd( k, n ) = d ≥ and n = td for t > . The equation λ k x k + x k + λx = 0 defined over F n has no non zero solution in F n if and only if Z t ( λ ) = 0 . λ for which Z t ( λ ) = 0 is known, which is as follows. Lemma 1. [HK10, Corollary 1] Let gcd( k, n ) = d ≥ and n = td for t > .Then α is a zero of Z t ( x ) in F n if and only if it is of the form v k +10 ( v + v ) k +1 , (11) where v ∈ F n \ F d and v = v k . The total number of distinct roots are ( n + d − d d − for even t n + d − d d − for odd t. Therefore, we have the following theorem which characterizes the quadraticnegabent monomials. Theorem 2. The function f : x T r ( λx k +1 ) is negabent if and only if λ can not be written as v k +10 ( v + v ) k +1 for v ∈ F n \ F d and v = v k where gcd( k, n ) = d and n = td .Proof. Proposition 3 and Proposition 4 imply that f is negabent if and onlyif λ is not a zero of Z t ( x ) where gcd( k, n ) = d and n = td . From Lemma 1,we know that Z t ( λ ) = 0 if and only if λ is not of the form v k +10 ( v + v ) k +1 where v ∈ F n \ F d and v = v k . We recall the well known result on the quadratic bent monomials. This isdirectly taken from [DL04]. Lemma 2. [DL04] Let λ ∈ F n and n even. The function f : F n → F with f ( x ) = T r ( λx k +1 ) is bent if and only if λ / ∈ { x k +1 | x ∈ F n } . Note that λ n − k x n − k + λx k is a permutation if and only if λ / ∈ { x k +1 | x ∈ F n } . Therefore, if f ( x ) = T r ( λx k +1 ) is negabent, then Proposition3 tells that λ n − k x n − k + λx k + x is also a permutation polynomial, i.e. , λ n − k x n − k + λx k is a complete mapping polynomial. We summarize theseresults as follows. 8 heorem 3. Let λ ∈ F n where n is even. The function f : F n → F with f ( x ) = T r ( λx k +1 ) is bent negabent if and only if one of the following two equivalent statementsholds.1. λ n − k x n − k + λx k is a complete mapping polynomial.2. λ is neither of the form v k +10 ( v + v ) k +1 nor of the form v k +1 for v ∈ F n , v ∈ F n \ F d and v = v k , where gcd( k, n ) = d and n = td . The existence of quadratic bent-negabent functions is known [PP07, The-orem 5]. However, we reprove the same result by simple counting argumentand using the previous characterization of the bent-negabent functions. Theorem 4. For all n ≥ , quadratic bent-negabent functions always exist.Proof. We show that there always exists a λ ∈ F n which satisfies the con-dition 2 of Theorem 3.If gcd(2 k + 1 , n − 1) = 1, then x x k +1 is a bijection. Then for any λ ∈ F n there exists x ∈ F n such that λ = x k +1 . Therefore, if T r ( λx k +1 )is bent then gcd(2 k +1 , n − > 1. Since 2 does not divide both of 2 k +1 and2 n − 1. Therefore, gcd(2 k + 1 , n − ≥ 3. Let S = { x k +1 | x ∈ F n } , then | S | ≤ n − . On the other hand, if gcd( k, n ) = d and n = td , then by Lemma1, we know that the number of possible λ ∈ F n such that λ is of the form v k +10 ( v + v ) k +1 is ( n + d − d d − for even t n + d − d d − for odd t . Let S = { y ∈ F n | y = v k +10 ( v + v ) k +1 } .Note that | S ∪ S | = | S | + | S | − | S ∩ S | . Then | S ∪ S | ≤ n − 13 + 2 n + d − d d − − | S ∩ S | . Therefore,2 n − | S ∪ S | ≥ n − n − − n + d − d d − | S ∩ S | = (2 n − . . d − . d − d − 1) + | S ∩ S | + 1 ≥ | S ∩ S | + 1 , since 2 . d − . d − ≥ . λ ∈ F n always exists that satisfies Condition 2of Theorem 3.This proves the theorem.Below we characterize bent-negabent functions when n = 2 k . Proposition 5. Let n = 2 k and f : x T r ( λx k +1 ) be a quadratic functiondefined over F n . Then f is negabent if and only if λ + λ k = 1 . Moreover, f is bent-negabent if and only if λ + λ k / ∈ F .Proof. By Proposition 3 we have that f is negabent if and only if P ( x ) = λ n − k x n − k + λx k + x is a permutation, i.e. , P ( x ) k = λx + λ k x k + x k is a permutation. Since n = 2 k , therefore, P ( x ) k = ( λ + λ k ) x + x k .Now ( λ + λ k ) x + x k is permutation if and only if ( λ + λ k ) x + x k = 0, i.e. , λ + λ k = x k − , for all x ∈ F ∗ n . Note that λ + λ k ∈ F k for all λ ∈ F n and the mapping λ λ + λ k is onto. Let us consider the group G = { x k − | x ∈ F ∗ n } . The intersection of F k and G is { } . Therefore, T r ( λx k +1 ) is negabent if and only if λ + λ k = 1. We know that f is bent ifand only if λ = x k +1 for some x ∈ F n . Note that if λ = x k +1 then λ ∈ F k and λ + λ k = 0. Therefore, f is bent-negabent if and only if λ + λ k / ∈ F . Maiorana-McFarland is an important class of bent functions which was ex-tensively studied by Dillon [Dil74, pp. 90-95]. This class is usually calledthe class M of bent functions. Lemma 3. Let n = 2 t . Let us consider a Boolean function f defined by f : ( x, y ) ∈ F t × F t T r t ( xπ ( y ) + h ( y )) (12) where π is a function over F t and h is any function on F t . Then f is abent function if and only if π is a bijection. Theorem 5. Let f be a Maiorana-McFarland function as in Lemma 3.Then f is negabent if and only if for all a, b ∈ F ∗ t X y ∈ Y a,b ( − T r t ( aπ ( y ))+ h ( y )+ h ( y + b )+ by ) = 0 , (13) where Y a,b = { y ∈ F t | π ( y ) + π ( y + b ) = a } such that Y a,b is non empty. roof. From (5) we have, f ( x, y ) is negabent if and only if for all ( a, b ) ∈ F t × F t \ { (0 , } X ( x,y ) ∈ F t × F t ( − f ( x,y )+ f ( x + a,y + b )+ T r t ( ax )+ T r t ( by ) = 0 X ( x,y ) ∈ F t × F t ( − T r t ( x ( π ( y )+ π ( y + b )+ a ))+ T r t ( aπ ( y + b )+ h ( y )+ h ( y + b )+ by ) = 0 . Let S a,b = X ( x,y ) ∈ F t × F t ( − T r t ( x ( π ( y )+ π ( y + b )+ a ))+ T r t ( aπ ( y + b )+ h ( y )+ h ( y + b )+ by ) . We treat the sum S a,b in the following cases. CASE 1: For a = 0 and b = 0. Then S a,b = X ( x,y ) ∈ F t × F t ( − T r t ( ax )+ T r t ( aπ ( y )) = X x ∈ F t ( − T r t ( ax ) X y ∈ F t ( − T r t ( aπ ( y )) = 0 . CASE 2: For a = 0 and b = 0. Then S a,b = X ( x,y ) ∈ F t × F t ( − T r t ( x ( π ( y )+ π ( y + b )))+ T r t ( h ( y )+ h ( y + b )+ by ) = X y ∈ F t ( − T r t ( h ( y )+ h ( y + b )+ by ) X x ∈ F t ( − T r t ( x ( π ( y )+ π ( y + b ))) = X y ∈ F t ( − T r t ( h ( y )+ h ( y + b )+ by ) × π is a permutation, π ( y ) = π ( y + b )= 0 . CASE 3: For a = 0 and b = 0. Then S a,b = X ( x,y ) ∈ F t × F t ( − T r t ( aπ ( y + b )+ h ( y )+ h ( y + b )+ by )+ T r t ( x ( π ( y )+ π ( y + b )+ a )) = X y ∈ F t ( − T r t ( aπ ( y + b )+ h ( y )+ h ( y + b )+ by ) X x ∈ F t ( − T r t ( x ( π ( y )+ π ( y + b )+ a )) . 11f there exists some y such that y / ∈ Y a,b , i.e. , π ( y ) + π ( y + b ) = a , then X x ∈ F t ( − T r t ( x ( π ( y )+ π ( y + b )+ a )) = 0 . On the other hand if y ∈ Y a,b , i.e. , π ( y ) + π ( y + b ) = a , then X x ∈ F t ( − T r t ( x ( π ( y )+ π ( y + b )+ a )) = 2 t . Therefore, S a,b = 2 t X y ∈ Y a,b ( − T r t ( aπ ( y + b )+ h ( y )+ h ( y + b )+ by ) = 2 t X y ∈ Y a,b ( − T r t ( aπ ( y )+ a + h ( y )+ h ( y + b )+ by ) since π ( y + b ) = π ( y ) + a for y ∈ Y a,b = 2 t ( − T r ( a ) X y ∈ Y a,b ( − T r t ( aπ ( y )+ h ( y )+ h ( y + b )+ by ) . Therefore, S a,b = 0 if and only if X y ∈ Y a,b ( − T r t ( aπ ( y )+ h ( y )+ h ( y + b )+ by ) = 0 . Thus after discussing all the above cases it is clear that the Maiorana-McFarland bent function f is negabent if and only if X y ∈ Y a,b ( − T r t ( aπ ( y )+ h ( y )+ h ( y + b )+ by ) = 0 . This Theorem gives us the clue to construct negabent functions over thefinite fields that belong to the class of Maiorana-McFarland bent functions. Definition 2. A mapping F : F n → F n is called homomorphic if F ( x + y ) = F ( x ) + F ( y ) and F ( xy ) = F ( x ) F ( y ) for all x, y ∈ F n . The only possible homomorphic permutation over F n is of the form x x i . Note that T r n ( x ) = T r n ( x i ), therefore the mapping x T r n ( x )is invariant under the action of this permutation. Using this observation weshow an interesting consequence of Theorem 5, when the permutation π ischosen as π ( x ) = x i . 12 heorem 6. Let f : ( x, y ) ∈ F t × F t F be a Maiorana-McFarland bentfunction given by f ( x, y ) = T r t ( xy i + h ( y )) , (14) Then f is negabent if and only if T r t ( h ( y )) is a bent function over F t .Proof. Let π ( y ) = y i . Then π ( y ) is a homomorphic permutation polynomialover F t . From the linearity of π we have π ( y ) + π ( y + b ) = a if and only if π ( b ) = a . Then Y a,b = ( F t when π ( b ) = a empty when π ( b ) = a. Since π is a permutation, for each a there will be a b such that π ( b ) = a .For such a and b X y ∈ F t ( − T r t ( aπ ( y )+ h ( y )+ h ( y + b )+ by ) = X y ∈ F t ( − T r t ( π ( b ) π ( y )+ by + h ( y )+ h ( y + b )) = X y ∈ F t ( − T r t ( π ( by ))+ T r t ( by )+ T r t ( h ( y )+ h ( y + b )) . Note that T r t ( y ) = T r t ( π ( y )), for all y ∈ F t . So X y ∈ F t ( − T r t ( aπ ( y )+ h ( y )+ h ( y + b )+ by ) = X y ∈ F t ( − T r t ( h ( y )+ h ( y + b )) . Using Theorem 5, the function f is negabent if and only if X y ∈ F t ( − T r t ( h ( y ))+ T r t ( h ( y + b )) = 0 , for all b ∈ F ∗ t , i.e. , T r t ( h ( y )) is a bent function over F t .Thus the result follows.Similar kind of result was proved in [SGC + F n and the permutation was such that wt ( x + y ) = wt ( π ( x )+ π ( y )). However, the result of Theorem 6 is quite distinct as it is inthe domain of finite fields. Moreover, Theorem 5 is a general characterizationof bent-negabent Maiorana-McFarland functions and several constructionsof Maiorana-McFarland bent-negabent functions can be obtained from this.For instance, Theorem 6 allows us to construct bent-negabent Maiorana-McFarland function of degree n/ F n by choosing a bent function ofdegree n/ h , where n = 2 t . 13 Negabent functions from bent functions We show that given a negabent function over a finite field, one can constructa bent function, and vice versa. First we define Q : F n → F as Q ( x ) = n − X i =1 T r n ( x i +1 ) + T r n ( x n +1 ) (15)As mentioned earlier, for simplicity we write T r n ( x ) = T r ( x ).We also mention a result from [CC03] and [CCCF01] which will be use-ful in proving our result. These results were proved for Boolean functionsdefined over vector spaces, however, it is easy to see the equivalent resultswhen the Boolean function is defined by the trace representation. Lemma 4. [CC03, Corollary 1] Suppose H β = { x ∈ F n : T r ( βx ) = 0 } is ahyperplane. If f : F n → F is bent, then for any a / ∈ H β , X x ∈ F n ( − f ( x )+ f ( x + a )+ T r ( βx ) = 0 . (16) Lemma 5. [CCCF01, Theorem V.3] The Boolean function f : F n → F isbent if and only if there exists a hyperplane H such that f ( x ) + f ( x + a ) isbalanced for every nonzero a ∈ H . Theorem 7. Suppose f : F n → F , and Q is as defined in (15).1. if f is bent then f + Q is negabent.2. If f is negabent then f + Q is bent, roof. Suppose a ∈ F ∗ n , then Q ( x ) + Q ( x + a ) = n − X i =1 T r ( a i x + ax i ) + T r n ( ax n + a n x ) + constant= n − X i =1 T r (( a i + a n − i ) x ) + T r n ( a n x ) + constant= n − X i =1 T r (( a i + a n − i ) x ) + T r ( a n x ) + T r ( ax )+ T r ( ax ) + constant , = T r ( T r ( a ) x ) + T r ( ax ) + constant= T r ( a ) T r ( x ) + T r ( ax ) + constant . (17)Without any loss of generality we ignore the constant term in Q ( x )+ Q ( x + a ). Case 1: We prove that if f is bent, then f + Q is negabent, for which wehave to show that X x ∈ F n ( − f ( x )+ f ( x + a )+ Q ( x )+ Q ( x + a )+ T r ( ax ) = 0We have X x ∈ F n ( − f ( x )+ f ( x + a )+ Q ( x )+ Q ( x + a )+ T r ( ax ) = X x ∈ F n ( − f ( x )+ f ( x + a )+ T r ( a ) T r ( x ) . Subcase 1.1: If T r ( a ) = 0, then X x ∈ F n ( − f ( x )+ f ( x + a )+ T r ( a ) T r ( x ) = 0 , since f is bent . Subcase 1.2: If T r ( a ) = 1, then a does not belong to the hyperplane H = { x ∈ F n : T r (1 .x ) = 0 } . X x ∈ F n ( − f ( x )+ f ( x + a )+ T r (1 .x ) = 0 . So for any a ∈ F ∗ n , X x ∈ F n ( − f ( x )+ f ( x + a )+ Q ( x )+ Q ( x + a )+ T r ( ax ) = 0 . This implies that is f + Q is negabent. Case 2: Next we suppose that f is negabent and prove that g = f + Q isbent. For any a ∈ F ∗ n we have X x ∈ F n ( − g ( x )+ g ( x + a ) = X x ∈ F n ( − f ( x )+ f ( x + a )+ Q ( x )+ Q ( x + a ) = X x ∈ F n ( − f ( x )+ f ( x + a )+ T r ( a ) T r ( x )+ T r ( ax ) , by (17)= X x ∈ F n ( − f ( x )+ f ( x + a )+ T r ( ax ) , if a ∈ H , i.e.T r ( a ) = 0= 0 , since f is negabent . Therefore, we see that for any nonzero a that belongs to the hyperplane H , g ( x ) + g ( x + a ) is balanced. Hence by Lemma 5, we prove that g is bent.This theorem has interesting consequences. Corollary 1. The Boolean function f : F n → F is bent-negabent if andonly if both f and f + Q are bent. Corollary 2. The Boolean function f : F n → F is bent-negabent if andonly if f + Q is bent-negabent. Corollary 3. The function Q is bent but not negabent.Proof. It is easy to check that Q is bent by looking at its derivative givenin (17).Now on the contrary, assume that Q is negabent. Then g = Q + Q = 0is bent (by Theorem 7), which is a contradiction.16e now use the result of Theorem 7 to construct bent-negabent func-tions. Note that any two quadratic bent functions are affine equivalent. Itis clear that there is one-one correspondence between the bent function de-fined over F t and F t × F t . With abuse of notation we use Q ( x, y ) definedover F t × F t as the corresponding bent function for Q ( x ) which is definedin (15). That means the bent function G : ( x, y ) ∈ F t × F t F given by G ( x, y ) = T r t ( xy ) , (18)is affine equivalent to the bent function Q ( x, y ). This also means by Theorem7 that if f ( x, y ) is a bent function then f ( x, y ) + Q ( x, y ) is negabent andvice versa.Suppose G ( x, y ) and Q ( x, y ) are related by the relation Q ( x, y ) = G ( α x + α , α y + α ) + T r t ( βx ) + T r t ( γy ) + c, (19)for some α , α , α , α , β, γ in F t , c ∈ F . Theorem 8. Let f : ( x, y ) ∈ F t × F t F be a Maiorana-McFarland bentfunction given by f ( x, y ) = T r t ( xπ ( y )) + T r t ( h ( y )) , where π ( y ) is a complete mapping polynomial over F t , h ( y ) is any polyno-mial over F t , and G : ( x, y ) ∈ F t × F t F defined by G ( x, y ) = T r t ( xy ) .Then F ( x, y ) = f ( α x + α , α y + α ) + G ( α x + α , α y + α )+ T r t ( βx ) + T r t ( γy ) + c (20) is a bent-negabent function.Proof. We have f ( x, y ) + G ( x, y ) = T r t ( xπ ( y )) + T r t ( h ( y )) + T r t ( xy )= T r t ( x ( π ( y ) + y )) + T r t ( h ( y )) . Since π ( y ) is a complete mapping polynomial over F t , ( π ( y ) + y ) is a per-mutation polynomial, so f + G is a bent function. This also implies that F ( x, y ) = f ( α x + α , α y + α )+ G ( α x + α , α y + α )+ T r t ( βx )+ T r t ( γy )+ c is a bent function. We have 17 ( x, y )= f ( α x + α , α y + α ) + G ( α x + α , α y + α ) + T r t ( βx ) + T r t ( γy ) + c = f ( α x + α , α y + α ) + Q ( x, y ) , by19 . Note that F ( x, y ) + Q ( x, y ) = f ( α x + α , α y + α ) is also bent. So both F ( x, y ) and F ( x, y ) + Q ( x, y ) are bent. Therefore, by Corollary 1, F ( x, y )is bent-negabent.At this point, we would like to refer to [SGC + 12, Theorem 22], which alsostates a result that is similar to Theorem 8. In that result, the Boolean func-tion is defined over the vector space F n . Note that complete mapping polyno-mials are defined over finite fields, however, the proof of [SGC + 12, Theorem22] works in the vector space domain. They claim that π ( x , . . . , x t ) is apermutation of F t that corresponds to the permutation π ( x ) over the field F t , as well as π ( x , . . . , x t ) ⊕ ( x , . . . , x t ) is the permutation of F t that cor-responds to the permutation π ( x ) + x over the field F t . But it is not clearhow this correspondence is realized. On the other hand, Theorem 8 can di-rectly apply the complete mapping polynomials as the underlying Booleanfunction is defined over a finite filed.Now we construct infinite classes of n -variable bent-negabent functionwith the maximum degree n . Our construction is similar to that of Theorem5 of [SPT13]. Their proof works when there is a permutation polynomial p ( x ) over the vector space F n such that p ( x ) + x is also a permutationpolynomial over F n . However, this kind of permutation over the vectorspace F n is not characterized, on the other hand, these kind of permutationpolynomials (complete mapping polynomials) are well characterized overfinite field. Corollary 4. Suppose n = 2 t . Then the n -variable function F ( x, y ) definedin Theorem 8, where the polynomial h ( y ) has algebraic degree t = n is abent-negabent function of degree n .Proof. The algebraic degree of h ( y ) is t = n , which implies that the degreeof f ( x, y ) + G ( x, y ) is also n . That also implies that the degree F ( x, y ) = f ( α x + α , α y + α ) + G ( α x + α , α y + α ) + T r t ( βx ) + T r t ( γy ) + c is n .Two infinite classes of complete mapping polynomial are given in [LC07].18 heorem 9. [LC07, Theorem 4.3] Let p be a prime and m and ℓ are twopositive integers. Let k be the multiplicative order of p in Z m . Assume a ∈ F p kℓ is such that ( − a ) m = 1 . Then the polynomials π ( x ) = x ( x pkℓm − m + a ) , and π ( x ) = ax pkℓm − m +1 , are complete mapping polynomials over F p kℓm . Theorem 10. Suppose n = 2 kℓm and π ( y ) , π ( y ) are the complete mappingpolynomials as given in Theorem 9. Then the n -variable function F ( x, y ) asgiven in (20) is a bent-negabent with degree n , for f ( x, y ) = T r n ( xπ ( y )) + T r n ( y n − ) and f ( x, y ) = T r n ( xπ ( y )) + T r n ( y n − ) .Proof. This follows easily as the algebraic degree of T r n ( y n − ) is n . We have presented some characterizations of negabent functions over the fi-nite field. The analysis done here is useful in order to obtain further resultson negabent functions over finite fields. In this paper, we have characterizedquadratic negabent monomials. The characterization of negabent monomi-als of higher degree will be interesting. We also have characterized negabentfunctions which are Maiorana-McFarland bent. Moreover. we have pre-sented a construction of bent-negabent functions with optimal degree. Thisis the second known construction of such functions. However, it is interestingto see further classes of such functions. The author would like to thank Pascale Charpin who helped in provingCase 2 of Theorem 7. He is also thankful to Alexander Kholosha for helpfuldiscussions. 19 eferences [Car93] Claude Carlet. Two new classes of bent functions. In TorHelleseth, editor, EUROCRYPT , volume 765 of Lecture Notesin Computer Science , pages 77–101. Springer, 1993.[CC03] Anne Canteaut and Pascale Charpin. Decomposing bent func-tions. IEEE Transactions on Information Theory , 49(8):2004–2019, 2003.[CCCF01] Anne Canteaut, Claude Carlet, Pascale Charpin, and CarolineFontaine. On cryptographic properties of the cosets of R (1 , m ). IEEE Transactions on Information Theory , 47(4):1494–1513,2001.[Dil74] J. F. Dillon. Elementary Hadamard Difference sets . PhD thesis,University of Maryland, 1974.[DL04] Hans Dobbertin and Gregor Leander. A survey of some recentresults on bent functions. In Tor Helleseth, Dilip V. Sarwate,Hong-Yeop Song, and Kyeongcheol Yang, editors, SETA , volume3486 of Lecture Notes in Computer Science , pages 1–29. Springer,2004.[HK10] Tor Helleseth and Alexander Kholosha. x x + a and relatedaffine polynomials over GF (2 k ). Cryptography and Communica-tions , 2(1):85–109, 2010.[LC07] Yann Laigle-Chapuy. Permutation polynomials and applicationsto coding theory. Finite Fields and Their Applications , 13(1):58–70, 2007.[LHTK13] Nian Li, Tor Helleseth, Xiaohu Tang, and Alexander Kholosha.Several new classes of bent functions from Dillon exponents. IEEE Transactions on Information Theory , 59(3):1818–1831,2013.[Par00] Matthew G. Parker. Constabent properties of Golay-Davis-Jedwab sequences. In ISIT , page 302. IEEE, 2000.[PP07] Matthew G. Parker and Alexander Pott. On Boolean functionswhich are bent and negabent. In Solomon W. Golomb, GuangGong, Tor Helleseth, and Hong-Yeop Song, editors, SSC , volume20893 of Lecture Notes in Computer Science , pages 9–23. Springer,2007.[Rot76] O. S. Rothaus. On “Bent” functions. Journal of CombinatorialTheory, Series A , 20(3):300–305, 1976.[RP05] Constanza Riera and Matthew G. Parker. One and two-variableinterlace polynomials: A spectral interpretation. In Øyvind Ytre-hus, editor, WCC , volume 3969 of Lecture Notes in ComputerScience , pages 397–411. Springer, 2005.[Sar09] Sumanta Sarkar. On the symmetric negabent Boolean functions.In Bimal K. Roy and Nicolas Sendrier, editors, INDOCRYPT ,volume 5922 of Lecture Notes in Computer Science , pages 136–143. Springer, 2009.[Sar12] Sumanta Sarkar. Characterizing negabent Boolean functionsover finite fields. In Tor Helleseth and Jonathan Jedwab, edi-tors, SETA , volume 7280 of Lecture Notes in Computer Science ,pages 77–88. Springer, 2012.[SGC + 12] Pantelimon Stanica, Sugata Gangopadhyay, Ankita Chaturvedi,Aditi Kar Gangopadhyay, and Subhamoy Maitra. Investigationson bent and negabent functions via the nega-Hadamard trans-form. IEEE Transactions on Information Theory , 58(6):4064–4072, 2012.[SPP08] Kai-Uwe Schmidt, Matthew G. Parker, and Alexander Pott.Negabent functions in the Maiorana-McFarland class. InSolomon W. Golomb, Matthew G. Parker, Alexander Pott, andArne Winterhof, editors, SETA , volume 5203 of Lecture Notes inComputer Science , pages 390–402. Springer, 2008.[SPT13] Wei Su, Alexander Pott, and Xiaohu Tang. Characterization ofnegabent functions and construction of bent-negabent functionswith maximum algebraic degree.