A construction of bent functions from plateaued functions

A construction of bent functions from lateaued functions Ayça Çeşmelioğlu, Wilfried Meidl Sabancı University, MDBF, Orhanlı, 34956 Tuzla, İstanbul, Turkey. Abstract In this resentation, a technique for constructing bent functions from lateaued functions is introduced and analysed. This generalizes earlier techniques for constructing bent from near-bent functions. Using this construction, we obtain a big variety of inequivalent bent functions, some weakly regular and some non-weakly regular. Classes of bent function with some additional roerties that enable the construction of strongly regular grahs are constructed, and exlicit exressions for bent functions with maximal degree are resented. 1 Introduction For a rime, let f be a function from F n to F. The Fourier transform of f is then defined to be the comlex valued function f on F n f(b) = x F n ɛ f(x) b x where ɛ = e 2πi/ and b x denotes the conventional dot roduct in F n. The Fourier sectrum sec(f) of f is the set of all values of f. We remark that one can equivalently consider functions from an arbitrary n-dimensional vector sace over F to F, and substitute the dot roduct with any (nondegenerate) inner roduct. Frequently the finite field F n with the inner roduct Tr n (bx) is used, where Tr n (z) denotes the absolute trace of z F n. The function f is called a bent function if f(b) 2 = n for all b F n. The normalized Fourier coefficient of f at b F n is defined by n/2 f(b). For a bent function the normalized Fourier coefficients are obviously ±1 when = 2, and for > 2 we always have (cf. [7]) { n/2 ±ɛ f(b) f (b) = : n even or n odd and 1 mod 4 ±iɛ f (b) (1) : n odd and 3 mod 4 where f is a function from F n to F. 1

A bent function f : F n F is called regular if for all b F n n/2 f(b) = ɛ f (b). When = 2, a bent function is trivially regular, and as can be seen from (1), for > 2 a regular bent function can only exist for even n and for odd n when 1 mod 4. A function f : F n F is called weakly regular if, for all b F n, we have n/2 f(b) = ζ ɛ f (b) for some comlex number ζ with ζ = 1. By (1), ζ can only be ±1 or ±i. Note that regular imlies weakly regular. A function f : F n F is called lateaued if f(b) 2 = A or 0 for all b F n. By Parseval s identity, we obtain that A = n+s for an integer s with 0 s n. Moreover, suort of f defined by su( f) = {b F n f(b) 0} has cardinality n s. We will call a lateaued function with f(b) 2 = n+s or 0 an s-lateaued function. The case s = 0 corresonds to bent functions by definition. For 1-lateaued functions the term nearbent function is common (see [3, 9]), binary 1-lateaued and 2-lateaued functions are referred to as semi-bent functions in [5]. We resent a technique for constructing bent functions from lateaued functions which generalizes earlier constructions of bent functions from nearbent functions. Though the technique also works for = 2, we assume in the following that is odd, as we are mainly interested in this tye of functions, which we also will call -ary functions. In Section 2 we analyse the Fourier sectrum of quadratic functions and the effect of equivalence transformations to the Fourier sectrum. In articular, we show under which conditions the multilication of a -ary function with a constant changes the signs in the Fourier sectrum. The rocedure for constructing bent functions from s-lateaued functions is resented in Section 3. In Section 4 we oint out that the construction delivers a large variety of rovable inequivalent bent functions, and we give some simle examles of weakly regular and non-weakly regular bent functions. Bent functions with some additional roerties can be used to construct strongly regular grahs (see [6, 11, 12]). We will show how to obtain a large variety of such bent functions. Finally, we resent simle exlicit exressions for bent functions in odd dimension with maximal ossible degree. 2

2 Fourier sectrum Two functions f and g from F n to F are called extended affine equivalent (EA-equivalent) if g(x) = cf(l(x) + u) + v x + e for some c F, e F, u, v F n, and a linear ermutation L : F n F n. As well known, EA-equivalence reserves the main characteristics of the Fourier sectrum, in articular if f is bent then also g is bent. More recisely we have the following roerties which can be verified straightforward. (i) (f + e)(b) = ɛ e f(b), (ii) if f v (x) = f(x) + v x then f v (b) = f(b v), (iii) f(x + u)(b) = ɛ b u f(b), (iv) if L(x) = Ax for A GL(F ) then f(l(x))(b) = f((a 1 ) T b), where A T denotes the transose of the matrix A. We note that these transformations do not only reserve the absolute value of the Fourier coefficients but also their sign is not changed. This is different if f is multilied by a constant c F. Before we analyse the effect of this transformation, we give an analysis of the Fourier transform of quadratic functions. Using the roerties (i),(ii), we omit the affine art and consider quadratic functions f(x) = 1 i j n a ijx i x j from F n to F, where we ut x = (x 1,..., x n ). Every such quadratic function f can be associated with a quadratic form f(x) = x T Ax where x T denotes the transose of the vector x, and A is a symmetric matrix with entries in F. By [10, Theorem 6.21] any quadratic form can be transformed to a diagonal quadratic form by a coordinate transformation, i.e. D = C T AC for a nonsingular (even orthogonal) matrix C over F and a diagonal matrix D = diag(d 1,..., d n ), and it is sufficient to describe the Fourier sectrum of a quadratic form f(x) = d 1 x 2 1 + + d n s x 2 n s := Q d n,n s(x) for some 0 s n 1 and d = (d 1,..., d n s ). We may assume that the nonzero elements of the matrix D are d 1,..., d n s. The following roosition 3

describing Fourier sectrum of Q d n,n s(x) was resented in [3], where bent functions have been constructed from near-bent functions. For convenience we will include the roof. Proosition 1 [3] For the quadratic function Q d n,n s(x) = d 1 x 2 1 + + d n s x 2 n s from F n to F, let = n s i=1 d i, and let η denote the quadratic character of F. The Fourier sectrum of Q d n,n s is given by { 0, η( ) n+s sec(q d 2 ɛ f (b) b su( Q } d n,n s ) : 1 mod 4, n,n s) = { 0, η( )i n s n+s 2 ɛ f (b) b su( Q } d n,n s ) : 3 mod 4, if s > 0, where f (x) is a function from su( Q d n,n s ) to F, and { } η( ) n sec(q d 2 ɛ f (b) b F n : 1 mod 4, n,n) = { } η( )i n n 2 ɛ f (b) b F n : 3 mod 4, where f (x) is a function from F n to F. Proof: We start with two facts which are simle to verify. For two functions f : F n F and g : F m F, we define the function f g from F n F m by (f g)(x, y) = f(x) + g(y). Then (see also [1]) (f g)(u, v) = f(u)ĝ(v). (2) For a function f : F m F let f be the function from F m+n = F m F n to F defined by f(x, y) = f(x). Then (see also [3, Lemma 2], and comare with Lemma 1 in Section 3) f(b, c) = { n f(b) : c = 0, 0 : else. (3) We first consider Q d 1,1 (x) = dx2 and note that by [10, Theorem 5.33] Q d 1,1 (0) = x F ɛ dx2 = η(d)g(η, χ 1 ) where χ 1 is the canonical additive character of F and G(η, χ 1 ) is the associated Gaussian sum. Consequently Q d 1,1 (b) = ɛ dx2 bx = ɛ d(x b/(2d))2 b 2 /(4d) = ɛ b2 /(4d) η(d)g(η, χ 1 ). x F x F 4

With [10, Theorem 5.15] we then obtain Q d 1,1 (b) = η( ) 1 2 ɛ b2 /(4d) : 1 mod 4, η( )i 1 2 ɛ b2 /(4d) : 3 mod 4. With equation (3) we get the assertion for Q d n,1 for arbitrary n. The general assertion then follows with induction from equation (2). Remark 1 Since the multilication of a quadratic function f by a constant c F causes a multilication by c of every element in the associated diagonal matrix, the Fourier sectra of the functions f and cf is identical if and only if n s is even or n s is odd and c is a square in F. If n s is odd and c is a nonsquare, then the Fourier coefficients of f and cf have oosite sign. Remark 2 Let Q d n,n s(x) = d 1 x 2 1 + + d n sx 2 and Q d n,n s(x) = d 1 x2 1 + + d n sx 2 be two quadratic s-lateaued functions from F n to F, then one can be obtained from the other by a coordinate transformation if and only if η( ) = η( ), where = n s i=1 d i and = n s i=1 d i (see e.g. [10, Exercise 6.24]). If n s is odd we can also change the character of by multilying the s-lateaued function by a nonsquare. Consequently every quadratic s- lateaued function from F n to F is EA-equivalent to x 2 1 + x2 2 + + x2 n s if n s is odd. If n s is even then there are two EA-inequivalent classes of quadratic s-lateaued functions in dimension n. We will show next that Remark 1 is a secial case of a much more general theorem. For a function f : F n F and b F n, let N b (j) = {x F n : f(x) b x = j} for each j = 0,..., 1. Then and for any c F we have ĉf(cb) = x F n ɛ cf(x) (cb) x 1 f(b) = N b (j)ɛ j, = x F n 1 ɛ c(f(x) b x) = N b (j)ɛ cj. (4) 5

Suose that f(b) = (n+s)/2, then we have [7,.2019] 1 N b (j)ɛ j (n+s)/2 ɛ f (b) = 0 (5) when n s is even, and 1 ( ( j f ) ) (b) N b (j) (n+s 1)/2 ɛ j = 0 (6) when n s is odd, where 0 f (b) 1 is an integer deending on b. If n s is even, then using the automorhism σ c of Q(ɛ ) that fixes Q and σ c (ɛ ) = ɛ c, equation (5) imlies 1 N b (j)ɛ cj Consequently using equation (4) 1 ĉf(cb) = N b (j)ɛ cj = ± (n+s)/2 ɛ cf (b). = ± (n+s)/2 ɛ cf (b). If n s is odd, with the automorhism σ c and equation (6) we get 1 ( j f N b (j)ɛ cj ) (b) (n+s 1)/2 ɛ cj = 0. Hence equation (4) can be written as ĉf(cb) = ± (n+s 1)/2 ( j f (b) ) ɛ cj. Relacing j first by j + f (b) and then by c 1 j, we obtain 1 ( ĉf(cb) = ± (n+s 1)/2 ɛ cf (b) c 1 ) j ɛ j ( ) c 1 = ± ( c = ɛ (c 1)f (b) ) ɛ (c 1)f (b) f(b). We have shown the following theorem. 1 (n+s 1)/2 ɛ f (b) ( ) j ɛ j 6

Theorem 1 For an element b F n and a function f : F n F suose that f(b) 2 = n+s for some s 0. If n s is even or n s is odd and c is a square in F, then ĉf(cb) = ɛk f(b) for some integer k. If n s is odd and c is a nonsquare in F, then ĉf(cb) = ɛk f(b) for some integer k. 3 The construction In this section we describe the rocedure to construct -ary bent functions from F n+s to F from s-lateaued functions from F n to F. The construction seen in the framework of finite fields F n has been used in [4] to show the existence of ternary bent functions attaining the uer bound on algebraic degree given by Hou [8]. The construction can be seen as a generalization of the constructions in [5, 3, 9] where s = 1. Theorem 2 For each u = (u 1, u 2,, u s ) F s, let f u (x) : F n F be an s-lateaued function. If su( f u ) su( f v ) = for u, v F s, u v, then the function F (x, y 1, y 2,, y s ) from F n+s to F defined by F (x, y 1, y 2,, y s ) = u F s is bent. Proof: ( 1) s s i=1 y i(y i 1) (y i ( 1)) f u (x) (y 1 u 1 ) (y s u s ) For a F n, b F s, and utting y = (y 1,..., y s ), the Fourier transform F of F at (a, b) is F (a, b) = x F n,y Fs = y F s ɛ b y ɛ F (x,y) a x b y x F n ɛ fy(x) a x = y F s = y F s ɛ b y ɛ b y x F n f y (a). ɛ F (x,y) a x As each a F n belongs to the suort of exactly one f y, y F s, for this y we have F (a, b) = ɛ b y fy (a) = n+s 2. Given s-lateaued functions, there are various ossible aroaches to roduce a set of s-lateaued functions with Fourier transforms with airwise disjoint suort. We suggest a simle one using the following lemma. 7

Lemma 1 For some integers n and s < n, let f : F n s F be a bent function and u = (u n s+1,..., u n ) F s. Then the function in dimension n is s-lateaued with f u (x 1..., x n ) = f(x 1,..., x n s ) + n i=n s+1 u i x i su( f u ) = {(b 1,..., b n s, u n s+1,..., u n ) b i F, 1 i n s}. Proof: For b = (b 1,..., b n ) f u (b) = x F n = ɛ fu(x) b x n i=n s+1 ɛ (u i b i )x i ɛ f(x1,...,xn s) x n s+1,...,x n F x 1,...,x n s F Since f is a bent function in the variables x 1,..., x n s, we have n s i=1 b ix i = (n s)/2 and thus x 1,...,x n s F ɛ f(x1,...,xn s) { f (n+s)/2 if b u (b) = i = u i, n s + 1 i n, 0 else. n s i=1 b ix i As their Fourier sectrum is comletely known we will aly Lemma 1 to quadratic functions x 2 1 + + x2 n s. Corollary 1 For u F s let d u (F ) n s. Then d u x 2 1. x 2 n s + u x n s+1. x n, u F s is a set of s-lateaued functions with Fourier transforms having airwise disjoint suort. 8

We remark that this rocedure of searating the suorts of the Fourier transforms can be alied to any set of bent functions in n s variables which by Lemma 1 can be seen as a set of s-lateaued functions in n variables. Examle 1 For = 3, n = 2, s = 1 we may choose f 0 (x) = x 2 1, f 1(x) = 2x 2 1 + x 2, f 2 (x) = 2x 2 1 + 2x 2. Writing x 3 for y, with Theorem 2 we obtain the bent function F (x) = x 2 1x 2 3 + x 2 1 + x 2 x 3 in dimension 3 and algebraic degree 4. 4 Alications Inequivalent bent functions With the construction in Theorem 2, a large variety of inequivalent bent functions, weakly regular as well as non-weakly regular ones, can be obtained. As it is well known, EA-equivalent functions have always the same algebraic degree. By Theorem 1, EA-equivalence does not change the sign of the Fourier coefficients of bent functions in even dimension. If the dimension is odd, then an equivalence transformation either does not change the sign of any Fourier coefficient of a bent function, or the signs of all Fourier coefficients are altered. In articular a weakly regular bent function and a non-weakly regular bent function are never EA-equivalent. Using the construction in Theorem 2 we can design inequivalent bent functions of the same algebraic degree. Examle 2. Consider the 2-lateaued functions from F 4 3 to F 3 g 0 (x 1, x 2, x 3, x 4 ) = x 2 1 + x 2 2, g 1 (x 1, x 2, x 3, x 4 ) = 2x 2 1 + x 2 2. Choosing f 0j (x 1, x 2, x 3, x 4 ) = g 0 (x 1, x 2, x 3, x 4 )+jx 4 and f ij (x 1, x 2, x 3, x 4 ) = g 1 (x 1, x 2, x 3, x 4 ) + ix 3 + jx 4 for i = 1, 2 and j = 0, 1, 2, and alying the construction in Theorem 2, we get the bent function in dimension 6 F (x 1, x 2, x 3, x 4, y 1, y 2 ) = x 2 1y 2 1 + x 2 1 + x 2 2 + x 3 y 1 + x 4 y 2. Examle 3. With the same 2-lateaued functions g 0 and g 1 from F 4 3 to F 3 as in Examle 2, we choose f 00 (x 1, x 2, x 3, x 4 ) = g 0 (x 1, x 2, x 3, x 4 ) and for 0 i, j 2 and (i, j) (0, 0) we choose f ij (x 1, x 2, x 3, x 4 ) = g 1 (x 1, x 2, x 3, x 4 ) + ix 3 + jx 4. Then the construction in Theorem 2 yields the bent function F (x 1, x 2, x 3, x 4, y 1, y 2 ) = 2x 2 1y 2 1y 2 2 + x 2 1y 2 1 + x 2 1y 2 2 + x 2 1 + x 2 2 + x 3 y 1 + x 4 y 2. 9

Let be defined as in Proosition 1, then for g 0 we have = 1, a square, and for g 1 we have = 2, a nonsquare in F 3. Therefore the functions in Examles 2 and 3 are non-weakly regular. In the construction of Examle 2 the function g 0 is used 3 times, g 1 is used 6 times. From the descrition of the Fourier sectrum of a quadratic function given in Proosition 1 we conclude that 3 3 4 Fourier coefficients have negative sign, and 6 3 4 Fourier coefficients have ositive sign. In fact the Fourier sectrum of the bent function in Examle 2 is { 27 63, 27 162, (27ɛ 2 3 )162, 27ɛ 90 3, 27ɛ162 3, ( 27ɛ 2 3 )90 }, where the integer in the exonent denotes the multilicity of the corresonding Fourier coefficient. For constructing Examle 3, g 0 is used only once, 8 times g 1 is used. Consequently 3 4 Fourier coefficients have negative sign, 8 3 4 have ositive sign. In fact the Fourier sectrum of the bent function in Examle 3 is { 27 9, 27 216, (27ɛ 2 3 )216, 27ɛ 36 3, 27ɛ216 3, ( 27ɛ 2 3 )36 }. By Theorem 1 the two bent functions of algebraic degree 6 are inequivalent. Bent functions and strongly regular grahs In [2, 6, 11] it is shown that artial difference sets and strongly regular grahs can be obtained from some classes of -ary bent functions: Let n be an even integer and f : F n F be a bent function with the additional roerties that (a) f is weakly regular (b) for a constant k with gcd(k 1, 1) = 1 we have for all t F Then the sets D 0, D R, D N defined by f(tx) = t k f(x). D 0 = {x F n f(x) = 0}, D N = {x F n f(x) is a nonsquare of F }, D R = {x F n f(x) is a nonzero square of F } are artial difference sets of F n. Their Cayley grahs are strongly regular. There are a few examles of bent functions known that satisfy the above conditions, some of them yielding new strongly regular grahs, see [11]. Also note that every -ary quadratic function f which does not have a linear term satisfies f(tx) = t 2 f(x) for all t F. But in general, a bent function does not satisfy those conditions. In the following we relate our construction of bent functions to the construction of strongly regular grahs. We resent a general formula for a class 10

of bent functions which enable the construction of strongly regular grahs. As we will see, this class of bent functions is interesting from different oint of views. For each j {1,, s}, let σ j reresent the elementary symmetric function σ j (y 1,, y s ) := y i1 y ij 1 i 1 < <i j s with s indeterminates over F. We define a function G(y 1,, y s ) from F s to F by s G(y 1,, y s ) = ( 1) j σ j (y 1 1,, ys 1 ). Lemma 2 j=1 { 0 if (y1,, y G(y 1,, y s ) = s ) = (0,, 0) 1 otherwise. Proof: If y 1, y 2,, y s 0 then ( ) ( ) ( ) ( ) s s s s G(y 1,, y s ) = ( 1) s +( 1) s 1 +( 1) s 2 + +( 1) = 1. s s 1 s 2 1 If some y i1,, y ir = 0 then G(y 1,, y s ) will be reduced to the sum of ( 1)st owers elementary symmetric functions in s r variables and we will have ( ) ( ) ( ) s r s r s r G(y 1,, y s ) = ( 1) s r +( 1) s r 1 + +( 1) = 1. s r s r 1 1 Theorem 3 Let g 0, g 1 be two distinct bent functions from F n s to F satisfying g i (tx 1,..., tx n s ) = t 2 g i (x 1,..., x n s ) for all t F. We interret g 0, g 1 as s-lateaued functions in n variables, and define a function F from F n F s = F n+s to F by F (x, y 1,, y s ) = G(y 1,, y s )(g 0 (x) g 1 (x))+x n s+1 y 1 + +x n y s +g 0 (x) Then F is a bent function of degree s( 1) + d, where d is the degree of g 0 g 1, that satisfies F (tx, ty 1,, ty s ) = t 2 F (x, y 1,, y s ) for all t F. 11

Proof: For each nonzero vector (y 1,, y s ) F s, we define the function f y1,,y s (x) = g 1 (x) + x n s+1 y 1 + + x n y s from F n to F. By Corollary 1 and the remarks thereafter {g 0, f y1,,y s (y 1,, y s ) F s \ {(0,..., 0)}} is a set of s-lateaued functions with Fourier transforms with airwise disjoint suort. Then = x F n = x F n = ĝ 0 (a) + = ĝ 0 (a) + F (a, b 1,, b s ) = x F n ɛ g 0(x) a.x ɛ g 0(x) a.x (y 1,,y s) F s 1 + (y 1,,y s) F s \{(0,,0)} (y 1,,y s) F s \{(0,,0)} ɛ F (x,y 1,,y s) a.x b 1 y 1 b sy s ɛ (g 0(x) g 1 (x))g(y 1,,y s)+(x n s+1 b 1 )y 1 + +(x n b s)y s (y 1,,y s) F s \{(0,,0)} ɛ b 1y 1 b sy s ɛ b 1y 1 b sy s ɛ (g 1(x) g 0 (x))+(x n s+1 b 1 )y 1 + +(x n b s)y s x F n ɛ g 1(x)+x n s+1 y 1 + +x ny s a.x f y1,y 2,,y s (a) Remark 3 The bent function of Theorem 3 is obtained with the construction in Theorem 2 taking f 0 = g 0 and f u = g 1, u 0, and by searating the suorts of the Fourier transforms similarly as in Corollary 1 for quadratic functions. Examle 4. For the 1-lateaued quadratic functions g 0, g 1 : F 3 3 F 3 given by g 0 (x 1, x 2, x 3 ) = 2x 2 1 + x2 2 and g 1(x 1, x 2, x 3 ) = x 2 1 + 2x2 2 with Theorem 3 (and utting y = x 4 ) we obtain the bent function of algebraic degree 4 F 1 (x 1, x 2, x 3, x 4 ) = 2x 1 x 2 4 + x 2 2x 2 4 + x 3 x 4 + 2x 2 1 + x 2 2. Examle 5. Alying Theorem 3 to the 2-lateaued quadratic functions g 0 (x 1, x 2, x 3, x 4 ) = x 2 1 +x2 2, g 1(x 1, x 2, x 3, x 4 ) = 4x 2 1 +x2 2 from F4 5 to F 5, yields the bent function of algebraic degree 6 F 2 (x 1, x 2, x 3, x 4, x 5, x 6 ) = 2x 2 1x 4 5x 4 6 + 3x 2 1x 4 5 + 3x 2 1x 4 6 + x 3 x 5 + x 4 x 6 + x 2 1 + x 2 2. The functions in Examles 2 and 3 are all in even dimension and satisfy the conditions (a),(b), therefore corresond to strongly regular grahs. According to the signs of the Fourier coefficients, the grah corresonding to F 1 is of negative Latin square tye, and the grah corresonding to F 2 is of Latin square tye (see [11]). 12

Examles of bent functions with maximal degree A -ary bent function f in dimension n can have algebraic degree at most ( 1)n/2 + 1, see Hou [8]. The bent function f must then be non-weakly regular. In a first aroach, in [4] the construction described in Theorem 2 is used with maximal ossible s = n 1 to show the existence of ternary bent functions in odd dimension attaining the Hou s uer bound on the algebraic degree. Constructions of such functions from F 3 n F s 3 to F 3 are described. We here resent some simle exlicit exressions for such bent functions. We first use Theorem 3 to obtain a closed formula for arbitrary odd dimension. The functions g 0 (x) = x 2 1, g 1(x) = 2x 2 1 from Fn 3 to F 3 are (n 1)- lateaued. Alying Theorem 3 to these functions yields a ternary bent function in dimension n+s and algebraic degree 2n. As obvious this function attains Hou s bound on the algebraic degree, and we have the following corollary. Corollary 2 For an arbitrary integer n let x = (x 1,..., x n ), y = (y 1,..., y n 1 ). The function F from F n 3 Fs 3 = Fn+s 3 to F 3 F (x, y) = 2x 2 1G(y 1,..., y n 1 ) + x 2 1 + x 2 y 1 + + x n y n 1 (7) is a bent function with maximal ossible algebraic degree. Examle 6. For n = 2 the ternary function (7) of maximal ossible algebraic degree 4 is the function in Examle 1 F (x) = x 2 1x 2 3 + x 2 1 + x 2 x 3, for n = 3, function (7) is the ternary function F (x 1, x 2, x 3, y 1, y 2 ) = 2x 2 1y 2 1y 2 2 + x 2 1y 2 1 + x 2 1y 2 2 + x 2 y 1 + x 3 y 2 + x 2 1 of algebraic degree 6. Corollary 2 gives one exlicit formula for a ternary bent function with maximal algebraic degree in arbitrary odd dimension. A large number of ternary bent functions with maximal degree can be obtained using sets of s-lateaued functions described as in Corollary 1 with s = n 1. Examle 7. Using the notation of Corollary 1 for n = 3 thus s = 2 we may choose d 00 = d 02 = d 20 = d 11 = d 22 = 1 and d 01 = d 10 = d 12 = d 21 = 2. Alying the construction of Theorem 2, yields the ternary bent function F (x 1, x 2, x 3, y 1, y 2 ) = x 2 1y 2 1y 2 2 + x 2 1y 2 1y 2 + 2x 2 1y 2 1 + x 2 1y 1 y 2 2 + x 2 1y 1 y 2 + 2x 2 1y 1 + 2x 2 1y 2 2 + 2x 2 1y 2 + x 2 1 + x 2 y 1 + x 3 y 2 13

of degree 6. By Remark 3 the bent function in dimension 5 in Examle 6 is obtained by choosing d 00 = 1 and d u = 2 for u 00. By Proosition 1 the roortion of Fourier coefficients with ositive sign is smaller than for the function in Examle 7. Consequently these two functions of maximal algebraic degree are inequivalent. Remark 4 One may hoe that this construction of ternary bent functions of maximal degree can be generalized to the case > 3. One may start with bent functions from F to F, i.e. in dimension 1, with algebraic degree (+1)/2, interret this functions as (n 1)-lateaued functions in dimension n and roceed as above for the case = 3 to construct a bent function of maximal degree ( 1)(2n 1)/2+1 in dimension 2n 1. But by [8, Theorem 4.6] bent functions from F to F are always quadratic. Consequently this rocedure is only alicable for = 3. References [1] C. Carlet, H. Dobbertin, G. Leander, Normal extensions of bent functions. IEEE Trans. Inform. Theory 50 (2004), 2880 2885. [2] Y.M. Chee, Y. Tan, X.D. Zhang, Strongly regular grahs constructed from -ary bent functions, rerint 2010. [3] A. Çeşmelioğlu, G. McGuire, W. Meidl, A construction of weakly and non-weakly regular bent functions, rerint 2010. [4] A. Çeşmelioğlu, W. Meidl, Bent functions of maximal degree, rerint 2010. [5] P. Charin, E. Pasalic, C. Tavernier, On bent and semi-bent quadratic Boolean functions. IEEE Trans. Inform. Theory 51 (2005), 4286 4298. [6] T. Feng, B. Wen, Q. Xiang, J. Yin, Partial difference sets from -ary weakly regular bent functions and quadratic forms, rerint 2010. [7] T. Helleseth, A. Kholosha, Monomial and quadratic bent functions over the finite field of odd characteristic. IEEE Trans. Inform. Theory 52 (2006), 2018 2032. [8] X.D. Hou, -ary and q-ary versions of certain results about bent functions and resilient functions. Finite Fields Al. 10 (2004), 566 582. 14

[9] G. Leander, G. McGuire, Construction of bent functions from near-bent functions. Journal of Combinatorial Theory, Series A 116 (2009), 960 970. [10] R. Lidl, H. Niederreiter, Finite Fields, 2nd ed., Encycloedia Math. Al., vol. 20, Cambridge Univ. Press, Cambridge, 1997. [11] Y. Tan, A. Pott, T. Feng, Strongly regular grahs associated with ternary bent functions. Journal of Combinatorial Theory, Series A 117 (2010), 668 682. [12] Y. Tan, J. Yang, X. Zhang, A recursive aroach to construct -ary bent functions which are not weakly regular. In: Proceedings of IEEE International Conference on Information Theory and Information Security, Beijing, 2010, to aear. 15