On the normality of p-ary bent functions

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1 Noname manuscrit No. (will be inserted by the editor) On the normality of -ary bent functions Ayça Çeşmelioğlu Wilfried Meidl Alexander Pott Received: date / Acceted: date Abstract In this work, the normality of bent functions in odd characteristic is analysed. It turns out that differently to Boolean bent functions, many - also quadratic - bent functions in odd characteristic and even dimension are not normal. It is shown that regular Coulter-Matthews bent functions are normal. Keywords Bent Functions Normality Coulter-Matthews bent functions 1 Introduction For a rime, let f be a function from an n-dimensional vector sace V n over F to F. The Walsh transform of f is then defined to be the comlex valued function f on V n f(b) = ɛ f(x) <b,x> x V n where ɛ = e 2πi/ and < b, x > denotes a (nondegenerate) inner roduct on V n. The classical frameworks are V n = F n and < b, x > is the conventional dot roduct denoted by, and V n = F n and < b, x >= Tr n (bx), where Tr n (z) denotes the absolute trace of z F n. In this contribution we will consider examles in both frameworks, but general definitions and results will be formulated in the framework of V n = F n. A. Çeşmelioğlu is suorted by Tübitak BİDEB 2219 Scholarshi Programme. W.Meidl is suorted by Tübitak Project no.111t234. Ayça Çeşmelioğlu Alexander Pott Otto-von-Guericke-University, Faculty of Mathematics, Magdeburg, Germany cesmelio@ovgu.de, alexander.ott@ovgu.de Wilfried Meidl Sabancı University, MDBF, Orhanlı, Tuzla, İstanbul, Turkey wmeidl@sabanciuniv.edu

2 2 Ayça Çeşmelioğlu et al. The function f is called a bent function if f(b) = n/2 for all b F n. If f(b) {0, (n+1)/2 } for all b F n, then we call f near-bent (for = 2 the term semi-bent is common), and more generally f is called s-lateaued for an integer 0 s n if f(b) {0, (n+s)/2 } for all b F n. We remark that for = 2 the Walsh transform yields an integer. Hence if f is s-lateaued, then n and s must have the same arity. In articular, binary bent functions only exist for n even. For odd, bent functions exist for n even and for n odd. For the Walsh coefficient f(b) we always have (cf. [8]) n/2 f(b) = { ±ɛ f (b) : n even or n odd and 1 mod 4 ±iɛ f (b) : n odd and 3 mod 4, (1) where f is a function from F n to F. 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, otherwise it is called not weakly regular. By (1), ζ can only be ±1 or ±i. Note that regular imlies weakly regular. The classical examle for a bent function is the Maiorana-McFarland bent function from F m F m = F 2m to F defined by f(x, y) = x π(y) + σ(y) for a ermutation π of F m and an arbitrary function σ : F m F. We remark that the condition that π is a ermutation is necessary and sufficient for f being bent. The Maiorana-McFarland function is always a regular bent function. Moreover f(x, π 1 (0)) = σ(π 1 (0)) is constant for all x F m, hence a Maiorana-McFarland function in an examle of a normal function, which is defined as follows. For an even integer n = 2m, a function f : F n F is called normal if there is an affine subsace of dimension m = n/2 on which the function is constant, f is called weakly normal if there is an affine subsace of dimension m = n/2 on which the function is affine, see [11]. The notion of normal Boolean functions was introduced in [6]. By counting arguments one can show that nearly all Boolean functions are non-normal, however almost all known Boolean bent functions are normal, see [11].

3 On the normality of -ary bent functions 3 2 Normality of -ary bent functions Normality for Boolean bent functions was investigated in the articles [2, 3, 5, 11]. Recently, in [10] a -ary bent function has been shown to be normal. In this section we further investigate normality for -ary bent functions. As easily observed, normality of functions from F n to F is invariant under coordinate transformation. However, the addition of an affine function alters a normal to a weakly normal function. Hence normality is not reserved under EA-equivalence transformations. Consequently in relation to bentness, which is invariant under EA-equivalence, in view of the following lemma the concet of weak normality is more natural (see also [5]). Lemma 1 A function f : F n F is weakly normal if and only if there exists an a F n such that f(x) a x is normal. Proof For an even integer n = 2m let f : F n F be weakly normal. Then there exists a subsace E of dimension m such that f(x) is affine on b + E for some b F n. Assume that f(x) = d x+c, c F, d F n, on b+e and choose any a d + E, and consider the function f(x) a x. On b + E, the values of the function are given as f(x) a x = (d a) x + c = (d a) b + c since d a E. Hence f(x) a x is normal. The converse follows similarly. The following theorem establishes a relationshi between regularity and normality of -ary bent functions, and it analyses the behaviour of -ary (weakly) normal bent functions on the cosets of the subsace E wherever f is affine. Theorem 1 For n = 2m let f : F n F be a bent function. (i) If f is weakly regular but not regular, then f is not (weakly) normal. (ii) If f is normal, hence constant on E + b for an m-dimensional subsace E and b F n, then f is balanced on the remaining cosets. The dual f of f is (weakly) normal. Proof Let E be an arbitrary subsace of F n, let b F n and let E be the orthogonal comlement of E in F n. Then f(u) = = ɛ b u ɛ b u x F n = E x b+e ɛ f(x) u x x F n ɛ f(x) ɛ (b x) u ɛ f(x). (2) Let f : F n F, n = 2m, be a normal bent function, let E be an m- dimensional subsace of F n and b F n such that f(x) = c F for all x b + E. Since n is even, the ossible Walsh transform values are f(u) = ± m ɛ f (u) for any u F n. With Equation (2) we then obtain m ( 1) ju ɛ f (u)+b u = n ɛ c, i.e. ( 1) ju ɛ f (u)+b u = m ɛ c,

4 4 Ayça Çeşmelioğlu et al. where j u {0, 1} for all u E. Consequently, we require that for all u E we have j u = 0 and f (u) + b u = c. The first condition imlies that f(u) = m ɛ f (u) for all u E. Hence the normal bent function f must be either regular or not weakly regular with Walsh coefficients with a ositive sign on E. Since for a weakly regular (but not regular) bent function f(x) also f(x) a x is weakly regular (but not regular), by Lemma 1 a weakly regular (but not regular) bent function cannot be weakly normal as well, which finishes the roof for (i). The second condition, f (u) = c b u for all u E, imlies that the dual f of a normal bent function f is weakly normal. Let b F n and b b + E. Then with Equation (2) we get Consequently, x b +E ɛ b u ɛ f(x) = m ɛ f (u) = m ɛ f (u)+b u x b +E = ɛ c ɛ f(x). ɛ (b b) u = 0. The last equality follows since f (u) = c b u on E and b b / E. Hence the function f is balanced on each coset of E in F n excet b+e, which finishes the roof. We remark that weakly normal functions f in dimension n are affine on an n/2- dimensional subsace E, whereas the comleted Maiorana-McFarland class (the set of all bent functions EA-equivalent to a Maiorana-McFarland function) is characterized by the much stronger condition that f is also affine on every coset of E, see [2, Lemma 33]. Hence the comleted Maiorana-McFarland class is trivially weakly normal. Using that there exist exactly two EA-inequivalent classes of quadratic bent functions, one with solely regular bent functions and one with weakly regular bent functions only, Theorem 1 confirms results in [1] according to which not all quadratic bent functions in odd characteristic are in the comleted Maiorana-McFarland class. More generally, we can show the following theorem. Theorem 2 Let f be a quadratic bent function in odd characteristic and dimension n. Then - f is in the comleted Maiorana-McFarland class if n is even and f is regular, - f is affine on all cosets of an s = (n 2)/2-dimensional subsace E if n > 2 is even and f is weakly regular, - f is affine on all cosets of an s = (n 1)/2-dimensional subsace E if n > 1 is odd. Combining Theorem 2 with [8, Corollary 6] on the signs of the Walsh coefficients for quadratic bent monomials, we obtain the subsequent corollary, which generalizes Theorem 4 in [1].

5 On the normality of -ary bent functions 5 Corollary 1 For an even integer n, the monomial bent function f(x) = Tr n (ax j +1 n ), 1 j n, gcd(j,n) odd, from F n to F is in the comleted Maiorana-McFarland class if and only if - 1 mod 4 and a is a nonsquare in F n, or - 3 mod 4, n 2 mod 4, and a is a square in F n, or - 3 mod 4, n 0 mod 4, and a is a nonsquare in F n. By Theorem 1, not weakly regular bent functions, from which one may exect a more chaotic behaviour than from weakly regular bent functions, still may be normal. In the following examle we show the normality of a not weakly regular bent function resented in [9]. Examle 1 Let ω be a root of the irreducible olynomial x 4 + x + 2 F 3 [x], which is a rimitive element of F 3 4. Then f : F 3 4 F 3 given by f(x) = Tr 4 (ω 10 x 22 + x 4 ) is not weakly regular bent, see [9]. With Magma we observe that f(x) = 0 for all x in the 2-dimensional subsace E := san{ω, ω 3 + ω 2 }. Hence f is normal. From Theorem 1 and its roof we know that then the dual f is affine on the orthogonal comlement E = san{1, ω 2 + 2ω} (with resect to Tr n (xy)) and f is weakly normal. Looking at the Walsh coefficients we observe that f(u) = 9 for all u E, hence f (u) = 0 for all u E and f is normal as well. Remark 1 Differently to (weakly) regular bent functions, the dual of a not weakly regular bent function need not be bent. The function f(x) = Tr 4 (ω 10 x 22 + x 4 ) is an examle of a not weakly regular bent function for which the dual is not bent, see [4]. The most famous non-quadratic bent functions in odd characteristic are the coordinate functions f α : F 3 n F 3 f α (x) = Tr n (αx 3k +1 2 ), α F 3 n, of the Coulter-Matthews erfect nonlinear function f : F 3 n F 3 n f(x) = x 3k +1 2, k odd and gcd(n, k) = 1, which is the only known non-quadratic erfect nonlinear function. We close this section with an analysis of the normality of this family of bent functions. Proosition 1 Let n = 2m with m 1. Then for each α F 3n, the Walsh transform f α of the weakly regular bent function f α (x) = Tr n (αx 3k +1 2 ), k is odd, gcd(n, k) = 1, satisfies { η(α)3 f α (β) = m ɛ f α (β) 3 n 0 mod 4, η(α)3 m ɛ f α (β) 3 n 2 mod 4, for all β F 3 n, where η reresents the quadratic character on F 3 n.

6 6 Ayça Çeşmelioğlu et al. Proof For each α F 3 n and β F 3n, with Lemma 3 of [7] we have f α (β) = ɛ α,β i n 3 m and ɛ α,0 { 1, 1} and ɛ α,β ɛ α,0 {1, ɛ 3, ɛ 2 3}. Hence for each α F 3 n, one needs to find ɛ α,0 for determining when f α is regular bent or not. With the following argument in the roof of Lemma 3 of [7], the Walsh transform of f α is reduced to the quadratic case: f α (0) = x F 3 n since gcd( 3k +1 2, 3 n 1) = 2. Therefore ɛ Trn(αx 3 k +1 2 ) 3 = x F 3 n f α (0) = η(α)( 1) n 1 i n 3 m. ɛ Trn(αx2 ) 3 The last ste follows from Theorem 5.33, 5.15 in [12], i.e. the result on Gaussian sums. Remark 2 For even n, half of the coordinate functions of the Coulter-Matthews olynomial are regular and the rest are weakly regular (but not regular) bent. The conditions in the revious roosition exlicitly describe the Walsh transform values. Namely, the bent functions f α (x) = Tr n (αx 3k +1 2 ) are regular bent if n 0 mod 4 and α is a nonsquare in F 3 n or n 2 mod 4 and α is a square in F 3 n and weakly regular (but not regular) bent if n 0 mod 4 and α is a square in F 3 n or n 2 mod 4 and α is a nonsquare in F 3 n. Theorem 3 The regular Coulter-Matthews bent functions are normal. Proof Assume ω F 3 n is a rimitive element of F 3 n. Let n = 2m and f ɛ (x) = Tr n (ω ɛ x 3k +1 2 ) be the Coulter-Matthews bent function from F 3 n to F 3 where 0 ɛ 3 n 2 and ɛ is even if m is odd and ɛ odd is if m is even. Any x F 3 n can be reresented in the form x = ω l(3m +1)+j, 0 l 3 m 2, 0 j 3 m. This reresentation corresonds to the artition F 3 = n 3m j=0 ωj F 3 m. Then, f ɛ (x) = Tr n (ω ɛ x 3k +1 2 ) = Tr n (ω ɛ+ 3k +1 2 j ω l(3m +1) 3k +1 2 ). We want to find j, 0 j 3 m, such that Tr n (w ɛ+ 3k +1 2 j ω l(3m +1) 3k +1 2 ) = 0 for all 0 l 3 m 2. For a fixed j, let α = ω ɛ+ 3k +1 2 j and z = ω l(3m +1) 3k Then α F 3 n is fixed and we want to have 0 = Tr n (αz) = αz (αz) 3m 1 + (αz) 3m + (αz) 3m (αz) 32m 1 = (α + α 3m )z + (α 3 + α 3m+1 )z (α 3m 1 + α 32m 1 )z 3m 1 = Tr m ((α + α 3m )z)

7 On the normality of -ary bent functions 7 for all z F 3 m which is of the form z = ω l(3m +1) 3k +1 2, 0 l 3 m 2. If we can find j such that α + α 3m = 0, then f ɛ (x) = Tr n (ω ɛ x 3k +1 2 ) = 0 on ω j F 3 m. Since ω j F 3 m is a subsace of dimension m, this imlies then the normality of f ɛ (x). We have α + α 3m = 0 ω 3m (ɛ+ 3k +1 2 j ) = ω ɛ+ 3k +1 2 j ω 3m (ɛ+ 3k +1 2 j ) = ω ɛ+ 3k +1 2 j+ 32m 1 2 ω (3m 1)(ɛ+ 3k +1 2 j) 32m 1 2 = 1 3 2m 1 (3 m 1)(ɛ + 3k + 1 j) 32m m + 1 ɛ + 3k + 1 j 3m k + 1 j 3m + 1 ɛ mod 3 m Since gcd(3 m + 1, 3k +1 2 ) = 2 and 3m +1 2 ɛ is even, this congruence always has exactly two solutions. Hence, there are at least two subsaces of dimension n/2 = m on which the Coulter-Matthews regular bent functions take the value 0. 3 Conclusion We analyse normality of -ary bent functions, show that normal bent functions must be regular or not weakly regular. We give a characterization of the quadratic monomials which belong to the comleted Maiorana-McFarland class and hence obtain a generalization of Theorem 4 in [1]. We show that the regular Coulter-Matthews bent functions, which do not belong to the comleted Maiorana-McFarland class, are though normal. References 1. L. Budaghyan, C. Carlet, T. Helleseth, A. Kholosha, Generalized bent functions and their relation to Maiorana-McFarland class. Proceedings IEEE Int. Sym. on Inform. Theory 2012, A. Canteaut, M. Daum, H. Dobbertin, G. Leander, Finding nonnormal bent functions. Discr. Al. Math. 154 (2006), C. Carlet, H. Dobbertin, G. Leander, Normal extensions of bent functions. IEEE Trans. Inform. Theory 50 (2004), A. Çeşmelioğlu, W. Meidl, A. Pott, On the dual of (non)-weakly regular bent functions and self-dual bent functions. Prerint P. Charin, Normal Boolean functions, J. Comlexity 20 (2004), H. Dobbertin, Construction of bent functions and balanced boolean functions with high nonlinearity, in: Proceedings of Fast Software Encrytion (B. Preneel Ed.), Leuven 1994, Lecture Notes Comut. Sci. 1008, Sringer 1995,

8 8 Ayça Çeşmelioğlu et al. 7. K. Feng, J. Luo, Value distributions of exonential sums from erfect nonlinear functions and their alications. IEEE Trans. Inform. Theory 53 (2007), no. 9, T. Helleseth, A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inform. Theory 52 (2006), T. Helleseth, A. Kholosha, New binomial bent functions over the finite fields of odd characteristics. IEEE Trans. Inform. Theory 56 (2010), W. Jia, X. Zeng, T. Helleseth, C. Li, A class of binomial bent functions over the finite fields of odd characteristic. IEEE Trans. Inform. Theory 58 (2012), 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, 1997.