A construction of bent functions from plateaued functions

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1 A construction of bent functions from lateaued functions Ayca Cesmelioglu, Wilfried Meidl To cite this version: Ayca Cesmelioglu, Wilfried Meidl. A construction of bent functions from lateaued functions. WCC Worksho on coding and crytograhy, Ar 2011, Paris, France , <inria > HAL Id: inria htts://hal.inria.fr/inria Submitted on 11 Jul 2011 HAL is a multi-discilinary oen access archive for the deosit and dissemination of scientific research documents, whether they are ublished or not. The documents may come from teaching and research institutions in France or abroad, or from ublic or rivate research centers. L archive ouverte luridiscilinaire HAL, est destinée au déôt et à la diffusion de documents scientifiques de niveau recherche, ubliés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires ublics ou rivés.

2 A construction of bent functions from lateaued functions Ayça Çeşmelioğlu and Wilfried Meidl Sabancı University, MDBF, Orhanlı, Tuzla, İstanbul, Turkey Abstract. In this resentation a technique for constructing bent functions from lateaued functions is introduced. This generalizes earlier techniques for constructing bent from near-bent functions. Analysing the Fourier sectrum of quadratic functions we then can construct weakly regular as well as non-weakly regular bent functions both in even and odd dimension. This tye of functions yield the first known infinite classes of non-weakly regular bent functions. Weakly regular bent functions with certain additional roerties can be used to construct artial difference sets and strongly regular grahs. We show how to obtain aroriate bent functions using our construction. Keywords: Bent functions, weakly regular bent, non-weakly regular bent, quadratic functions, lateaued functions. 1 Introduction Let be a rime, and let V n be any n-dimensional vector sace over F and f be a function from V n to F. If = 2 we call f a binary or Boolean function, if is an odd rime we call f a -ary function. The Fourier transform of f is the comlex valued function f on V n given by f(b) = ǫ f(x) b,x x V n where ǫ = e 2πi/ and, denotes any inner roduct on V n. The function f is called a bent function if f(b) 2 = n for all b V n. The normalized Fourier coefficient at b V n of a function from V n to F is defined by n/2 f(b). A binary bent function clearly must have normalized Fourier coefficients ±1, and for the -ary case we always have (cf. [6]) n/2 f(b) = { where f is a function from V n to F. ±ǫ f (b) : n even or n odd and 1 mod 4 ±iǫ f (b) : n odd and 3 mod 4 (1)

3 144 A -ary bent function f is called regular if for all b V n n/2 f(b) = ǫ f (b). As easily seen from (1), a -ary regular bent function can only exist for even n and for odd n when 1 mod 4. A -ary bent function f is called weakly regular if, for all b V n, we have n/2 f(b) = ζ ǫ f (b) for some comlex number ζ with absolute value 1. By (1), ζ can only be ±1 or ±i. Note that regular imlies weakly regular. A function f from V n to F is called lateaued if f(b) 2 = A or 0 for all b V n. By Parseval s identity we obtain that A = n+s for an integer s with 0 s n. 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 near-bent function is common (see [3, 7]), binary 1-lateaued and 2-lateaued functions are referred to as semi-bent functions in [4]. We resent a technique for constructing bent functions from lateaued functions which generalizes earlier constructions of bent functions from near-bent functions. Emloying quadratic lateaued functions, families of bent functions with interesting roerties are obtained. Using lateaued instead of only quadratic near-bent functions, we can obtain bent functions of various algebraic degree. With our construction, we obtain the first known infinite classes of non-weakly regular bent functions in odd dimension. Using direct sums of bent functions we also can construct non-weakly regular bent functions in even dimension. Weakly regular bent functions with certain additional roerties can be used to construct strongly regular grahs. With our construction, we can obtain various classes of bent functions in any characteristic that can be used to construct strongly regular grahs (of Latin square and of negative Latin square tye). Until now, only a few such bent functions were known, though already yielding new strongly regular grahs, [9]. 2 Quadratic functions If we associate V n with the finite field F n, we use the inner roduct x, y = Tr n (xy), where Tr n (z) denotes the absolute trace of z F n. The Fourier transform of a function f from F n to F is then the comlex valued function on F n given by f(b) = ǫ f(x) Trn(bx). x F n A function f from F n to F of the form f(x) = Tr n ( l i=0 a i x i +1 ) (2)

4 145 is called quadratic, its algebraic degree is two, see [4, 6]. Every quadratic function from F n to F is s-lateaued, where s is the dimension of the kernel of the linear transformation on F n defined by L(x) = l i=0 (a l i xl+i + a l i i x l i). (3) Choosing and fixing a basis {α 1,...,α n } of F n over F, we corresond x = n i=1 x iα i to the vector x = (x 1,...,x n ). Then we can associate a quadratic function f(x) = Tr n ( l i=0 a ix i +1 ) with a quadratic form f(x) = x T Ax, where x T denotes the transose to the vector x, and the matrix A has entries in F. Any quadratic form is equivalent to a diagonal quadratic form, i.e. D = C T AC for a nonsingular matrix C over F and a diagonal matrix D = diag(d 1,...,d n ). The following roosition is obtained by generalizing the related result in [6], which corresonds to the case that f(x) is nondegenerate, using [8, Theorem 6.27, Theorem 5.15] if n s odd and [8, Theorem 6.26] if n s is even. Proosition 1. Let f be an s-lateaued quadratic function from F n to F and f(x) = x T Ax be the associated quadratic form. Then a corresonding diagonal matrix D has n s (not necessarily distinct) nonzero entries d 1,...,d n s, and the Fourier sectrum of f is given by { } 0, η( ) n+s 2 ǫ J(b) : 1 mod 4, { 0,( 1) n s 1 2 η( )i n+s { 0,( 1) n s 2 η( ) n+s 2 ǫ J(b) 2 ǫ J(b) } } : 3 mod 4 and n s odd, : 3 mod 4 and n s even, where J(x) is a function from the suort su( f) := {b F n f(b) 0} of f to F, = n s i=1 d i, and η denotes the quadratic character in F. 3 Construction of bent from lateaued functions Theorem 1. For each a = (a 1, a 2,, a s ) F s, let f a (x) be an s-lateaued function from F n to F. If su( f a ) su( f b ) = for a,b F s,a b, then the function F(x, y 1, y 2,, y s ) from F n F s to F defined by is bent. F(x, y 1, y 2,, y s ) = a F s ( 1) s s i=1 y i(y i 1) (y i ( 1)) f a (x) (y 1 a 1 ) (y s a s )

5 146 Sketch of the roof The Fourier transform F of F at (α,a) F n F s is F(α,a) = ǫ F(x,y1,,ys) Trn(αx) a y = x F n,y 1,,y s F ǫ a y y 1,,y s F f y (α). As each α F n belongs to the suort of exactly one f y, y F s, for this y we have F(α,a) = ǫ a y f y (α) = n+s 2. The following theorem shows how to construct a set of s-lateaued functions such that the suorts of the Fourier transforms are airwise disjoint. Theorem 2. For each a = (a 1,, a s ) F s, let g a be a quadratic s-lateaued function from F n to F and L a be the corresonding linearized olynomial. For all a F s, if L a has the same kernel in F n with basis {β 1,, β s } and γ a F n is such that g a (β j ) + Tr n (γ a β j ) = g 0 (β j ) + a j, (4) for all j = 1,, s, then the s-lateaued function f a defined by f a (x) = g a (x) + Tr n (γ a x) satisfies su( f a ) su( f b ) = for b F s,a b. Proof We have to show that α su( f b ) imlies α su( f a ) for a b. Suose α su( f b ), i.e. g b (β j ) + Tr n (γ b β j ) + Tr n (αβ j ) = g 0 (β j ) + b j + Tr n (αβ j ) = 0, for each j = 1,, s. Let a b and suose that a j b j for some 1 j s. Then f a (β j )+Tr n (αβ j ) = g a (β j )+Tr n (γ a β j )+Tr n (αβ j ) = g 0 (β j )+a j +Tr n (αβ j ) 0. Remark 1. By the linear indeendence of β 1,...,β s, the existence of γ a satisfying equation (4) for all j = 1,...,s, is guaranteed. In fact, if {δ 1, δ 2,...,δ n } and {ρ 1, ρ 2,...,ρ n } are dual bases of F n over F, and β j = b j1 δ 1 +b j2 δ 2 + +b jn δ n, then the coefficients x i of γ a = x 1 ρ 1 + x 2 ρ x n ρ n satisfy Tr n (γ a β j ) = b j1 x 1 + b j2 x b jn x n. A value for γ a is then obtained with a solution of the linear system Bx = c over F for the (s n)-matrix B = (b jk ) and c = (c 1,, c s ) T with c j = g 0 (β j ) + a j g a (β j ), j = 1,2,...,s. We can consequently construct bent functions starting with any quadratic function f. In a first ste we determine the value of s for the function f. Then we can

6 147 choose s constants in F and the linearized olynomials have the same kernel in F n for all of the form cf, c F, hence we can aly Theorem 2 and Theorem 1. For some simle quadratic functions we know the value s in advance: I: The quadratic monomial f(x) = Tr n (ax r +1 ) F n[x] is s-lateaued for some a F n if and only if n is even, s is an even divisor of n and ν(s) = ν(r)+1, where ν denotes the 2-adic valuation on integers. The aroriate elements a F n can be determined and for the remaining a, f(x) is bent. ) II: Binomials. Let c F, then the function f(x) = Tr n (cx r +1 cx t +1 from F n to F is near-bent if and only if gcd(n, r +t) = gcd(n, r t) = gcd(n, ) = 1. The kernel of the corresonding linearized olynomial is F. Moreover if n is odd, then defined as in Proosition 1 is indeendent of the choice of r, t satisfying the above condition. Hence the Fourier sectrum is indeendent of the choice of r, t. ) For c F the function f = Tr n (cx r +1 + cx t +1 from F n to F is near-bent if and only if gcd(n, 2(r+t)) = gcd(n, 2(r t)) = 2, r t is odd, and gcd(n, ) = 1. The kernel of the corresonding linearized olynomial consists of the solutions of x + x. 4 Non-weakly regular bent functions, weakly regular bent functions and strongly regular grahs All classical examles of bent functions are weakly regular. However, in general, weak regularity is not easy to rove. With the subsequent corollaries of Proosition 1 and our construction resented in Section 3 we can design weakly regular and non-weakly regular bent functions. This also yields the first infinite classes of non-weakly regular bent functions. In order to construct non-weakly regular bent functions, we need to change the signs of the Fourier coefficients of the s-lateaued functions. Proosition 1 suggests that if n s is odd, we can change signs by multilying the functions with nonsquare elements of F. If n s is even, we only obtain weakly regular bent functions for any choice of the coefficients. The following two corollaries can be seen as results of Proosition 1 and Theorem 1, 2. For the Corollaries 1,2 we fix the following notation: For integers n, s G = {g a a F s } is a set of s quadratic s-lateaued functions from F n to F such that the corresonding linearized olynomials L a have all the same kernel in F n; C = {c a a F s } is a set of s nonzero elements of F ; {γ a a F s } is a set of elements of F n such that the set F = {f a a F s } of s quadratic s-lateaued functions from F n to F defined by f a (x) = c a g a (x) + Tr n (γ a x) satisfies su( f a ) su( f b ) = for a,b F s,a b. Corollary 1. Let F be the bent function in dimension n+s given as in Theorem 1 constructed with the set F. If n s is odd, then F is weakly regular for (

7 148 1) s /2 s 1 choices for C, and non-weakly regular for the remaining (2 s 1 1)( 1) s /2 s 1 choices for C. Corollary 2. Let g a = g for all a F s and let F be the bent function in dimension n + s given as in Theorem 1 constructed with the set F. If n s is even, then F is weakly regular. If n s is odd, then F is weakly regular if and only if all elements of C have the same quadratic character in F. Remark 2. Versions of the above corollaries for the secial case that s = 1 were used in our work [3] to resent the first infinite classes of non-weakly regular bent functions. These bent functions are in odd dimension, their algebraic degree is + 1. With s > 1 non-weakly regular bent functions of higher degree can be constructed. We now resent the first construction of non-weakly regular bent functions in even dimension. Let the vector sace V m+n of dimension m + n over F be the direct sum of the vector saces V m and V n of dimension m and n, resectively, and let f 1 be a -ary function on V m and f 2 be a -ary function on V n. Then the direct sum f 1 f 2 of f 1 and f 2 is defined as the -ary function on V m+n = V m V n given by (f 1 f 2 )(x 1 + x 2 ) = f 1 (x 1 ) + f 2 (x 2 ). As ointed out in [1], we have f 1 f 2 = f 1 f2, thus f 1 f 2 is bent if f 1 and f 2 are bent. In articular if exactly one f 1 or f 2 is non-weakly regular, then f 1 f 2 is non-weakly regular. The latter argument was recently used in [10] to give a recursive aroach to construct non-weakly regular bent functions - once a nonweakly regular bent function is obtained. Taking for f 1 a non-weakly regular bent function in odd dimension (Corollaries 1,2) and for f 2 a weakly regular bent function in odd dimension (e.g. quadratic bent function, Corollaries 1,2), then f 1 f 2 is non-weakly regular in even dimension. Examles Examle 1. The monomials g 0 (x) = Tr 4 (x 4 ), g 1 (x) = Tr 4 (x 28 ) have the same corresonding linearized olynomial L(x) = x + x 32 with a kernel of dimension 2 in F 3 4. A basis for this kernel is {β,β 3 } where β is a root of the olynomial x 4 + x Since we have g 0 (β) = g 0 (β 3 ) = g 1 (β) = g 1 (β 3 ) = 0 for each a = (a 1, a 2 ) F 3 F 3, we choose γ a F 3 4 such that Tr 4 (γ a β) = a 1,Tr 4 (γ a β 3 ) = a 2. For the nine required 2-lateaued functions with airwise disjoint suorts of their Fourier transforms we then can choose f (0,0) (x) = Tr 4 (x 4 + x), f (0,2) (x) = Tr 4 (x 4 + (2β 3 + 2)x), f (1,1) (x) = Tr 4 (x 4 + (β 3 + β)x), f (2,0) (x) = Tr 4 (x 28 + (2β + 1)x), f (2,2) (x) = Tr 4 (x 28 + (2β 3 + 2β)x). f (0,1) (x) = Tr 4 (x 4 + (β 3 + 1)x), f (1,0) (x) = Tr 4 (x 4 + βx), f (1,2) (x) = Tr 4 (x 28 + (2β 3 + β + 1)x), f (2,1) (x) = Tr 4 (x 28 + (β 3 + 2β)x),

8 149 The ternary function yz(y 1)(z 1)(y 2)(z 2) F(x, y, z) = f a (x) (y a 1 )(z a 2 ) a=(a 1,a 2) F 3 F 3 is a weakly regular bent function on F 3 4 F 3 F 3 of algebraic degree 6. Examle 2. Let = 3, n = 8. The quadratic binomials g 0 (x) = Tr 8 (x 10 +x 4 ) and g 1 (x) = Tr 8 (x x ) on F 3 8 are near bent. The kernel of the corresonding linearized olynomial consists of the solutions of x 3 + x. Let β F 3 2 be a root of the olynomial x F 3 [x]. Then, the function f 1 (x, y) = 2y 2 Tr 8 (x x 4 +x)+ytr 8 (x x 4 +βx)+tr 8 (x 10 +x 4 +x) from F 3 8 F 3 to F 3 is a weakly regular bent function of algebraic degree 4. Examle 3. With g 0 (x), g 1 (x) in Examle 2 and using different coefficients, we obtain the function f 2 (u, w) = 2w 2 Tr 8 (v v v) + wtr 8 (v v v 10 + v 4 + βv) + Tr 8 (v 10 + v 4 + v) from F 3 8 F 3 to F 3 which is a non-weakly regular bent function of algebraic degree 4 in odd dimension 9. Examle 4. The direct sum of f 1, f 2 is the ternary function H(x, y,v, w) = f 1 (x, y) + f 2 (v,w) on (F 3 8 F 3 ) (F 3 8 F 3 ) which is a non-weakly regular bent function of algebraic degree 4 in even dimension 18. Examle 5. The direct sum of the weakly regular bent function F(x, y,z) in Examle 1 and the non-weakly regular bent function f 2 (v, w) of Examle 3 is the ternary function yz(y 1)(z 1)(y 2)(z 2) F(x, y,z) + f 2 (v, w) = f a (x) (y a 1 )(z a 2 ) a=(a 1,a 2) F 3 F 3 + 2w 2 Tr 8 (v v v) + wtr 8 (v v v 10 + v 4 + βv) + Tr 8 (v 10 + v 4 + v) on (F 3 4 F 3 F 3 ) (F 3 8 F 3 ) which is a non-weakly regular bent function of algebraic degree 6. In [2,5, 9] 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

9 150 (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. Some of the classical bent functions (e.g. quadratic and Coulter-Matthews) satisfy condition (b). In fact, using Coulter-Matthews functions, some new strongly regular grahs have been discovered in [9]. But in general, condition (b) is not satisfied. With an aroriate choice of near bent functions, the construction of Theorem 1 yields bent functions satisfying (b): Theorem 3. Let g 0, g 1 be distinct quadratic near-bent functions from F n to F such that the corresonding linearized olynomials L 0, L 1 have the same (1- dimensional) kernel with basis {β}, and suose that g 0 (β) = g 1 (β) = 0. Let γ F n such that Tr n (γβ) 0. Then the function F(x, y) = y 1 (g 1 (x) g 0 (x)) + ytr n (γx) + g 0 (x) is a bent function of degree +1 that satisfies F(tx, ty) = t 2 F(x, y) for all t F. Examles In the following art, we construct examles of weakly regular bent functions in even dimension satisfying (b). Examle 6. = 3, n = 7: Let g 0 (x) = Tr 7 (x 10 x 4 ), g 1 (x) = Tr 7 (x 82 x 28 ). Using Theorem 3, we construct the bent function F(x, y) = y 2 Tr 7 (x 82 2x 10 + x 4 ) + ytr 7 (x) + Tr 7 (x 10 x 4 ) in dimension 8 and of algebraic degree 4 on F 3 7 F 3. The normalized Fourier sectrum of F(x, y) with multilicities is ( 1) 2133,( ǫ 3 ) 2214,( ǫ 2 3) 2214 where ǫ 3 is a comlex rimitive third root of unity. Examle 7. = 3, n = 8: g 0 (x) = Tr 8 (x x ), g 1 (x) = Tr 8 (x 10 + x 4 ). By using Theorem 3, we construct the ternary bent function F(x, y) = y 2 Tr 8 (x x x x 4 ) + ytr 8 (γx) + Tr 8 (x 10 + x 4 ) of degree 4 in odd dimension 9. The quadratic monomial f(x) = Tr 5 (x 10 ) on F 3 5 is weakly regular bent and if we form the direct sum of F(x, y) with f(x), the resulting ternary function on F 3 8 F 3 F 3 5 will be regular bent i.e. the normalized Fourier sectrum will be {1, ǫ 3, ǫ 2 3}. We remark that this direct sum reserves (b). We further remark that using 2g 0 (x),2g 1 (x) instead of g 0 (x), g 1 (x), changes the sign in the Fourier sectrum. The resulting strongly regular grah will then be of negative latin square tye, see [9].

10 151 Examle 8. = 5 and n = 7: g 0 (x) = Tr 7 (x x 2 ), g 1 (x) = Tr 7 (x x 5+1 ) be two functions from F 5 7 to F 5. Then we construct the following bent function of degree 6 on F 5 7 F 5 : F(x, y) = y 4 Tr 7 (x x x 5+1 x 2 ) + ytr 7 (x) + Tr 7 (x x 2 ). Examle 9. = 5, n = 4: g 0 (x) = Tr 4 (x x ), g 1 (x) = Tr 4 (4x x ). We construct the following bent function on F 5 4 F 5 : F(x, y) = y 4 Tr 4 (3x x ) + ytr 4 (βx) + Tr 4 (x x ). The quadratic monomial f(z) = Tr 5 (z 26 ) is a bent function on F 5 5 and by taking the direct sum, we obtain G(x, y,z) = F(x, y) + g(z) = y 4 Tr 4 (3x x ) + ytr 4 (βx) + Tr 4 (x x ) + Tr 5 (z 26 ), which is a bent function of degree 6 defined on (F 5 4 F 5 ) F 5 5. References 1. C. Carlet, H. Dobbertin, G. Leander, Normal extensions of bent functions. IEEE Trans. Inform. Theory 50 (2004), Y.M. Chee, Y. Tan, X.D. Zhang, Strongly regular grahs constructed from -ary bent functions, rerint A. Çeşmelioğlu, G. McGuire, W. Meidl, A construction of weakly and non-weakly regular bent functions, rerint P. Charin, E. Pasalic, C. Tavernier, On bent and semi-bent quadratic Boolean functions. IEEE Trans. Inform. Theory 51 (2005), T. Feng, B. Wen, Q. Xiang, J. Yin, Partial difference sets from -ary weakly regular bent functions and quadratic forms, rerint T. Helleseth, A. Kholosha, Monomial and quadratic bent functions over the finite field of odd characteristic. IEEE Trans. Inform. Theory 52 (2006), G. Leander, G. McGuire, Construction of bent functions from near-bent functions. Journal of Combinatorial Theory, Series A 116 (2009), R. Lidl, H. Niederreiter, Finite Fields, 2nd ed., Encycloedia Math. Al., vol. 20, Cambridge Univ. Press, Cambridge, Y. Tan, A. Pott, T. Feng, Strongly regular grahs associated with ternary bent functions. Journal of Combinatorial Theory, Series A 117 (2010), 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.

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