Idempotent and p-potent quadratic functions: distribution of nonlinearity and codimension

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1 Downloaded from orbit.dtu.dk on: Oct 07, 2018 Idempotent and p-potent quadratic functions: distribution of nonlinearity and codimension Anbar Meidl, Nurdagül; Meidl, Wilfried Meidl; Topuzoglu, Alev Published in: Designs, Codes and Cryptography Link to article, DOI: /s Publication date: 2017 Document Version Peer reviewed version Link back to DTU Orbit Citation APA: Anbar Meidl, N., Meidl, W. M., & Topuzoglu, A Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension. Designs, Codes and Cryptography, 821, DOI: /s General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

2 Noname manuscript No. will be inserted by the editor Idempotent and p-potent quadratic functions: Distribution of nonlinearity and co-dimension This paper is dedicated to the memory of Tosun Terzioğlu Nurdagül Anbar Wilfried Meidl Alev Topuzoğlu Received: date / Accepted: date Abstract The Walsh transform Q of a quadratic function Q : F p n F p satisfies Qb {0, p n+s 2 } for all b F p n, where 0 s n 1 is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class C 1 is defined for arbitrary n as C 1 = {Qx = Tr n n 1/2 a i x 2i +1 : a i F 2 }, and the larger class C 2 is defined for even n as C 2 = {Qx = Tr n n/2 1 a i x 2i +1 + Tr n/2 a n/2 x 2n/2 +1 : a i F 2 }. For an odd prime p, the subclass D of all p-ary quadratic functions is defined as D = {Qx = Tr n n/2 i=0 a i x pi +1 : a i F p }. We determine the generating function for the distribution of the parameter s for C 1, C 2 and D. As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case p > 2, the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed-Muller codes corresponding to C 1 and C 2 in terms of a generating function. Nurdagül Anbar Technical University of Denmark, Matematiktorvet, Building 303B, DK-2800, Lyngby, Denmark nurdagulanbar2@gmail.com Wilfried Meidl Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria meidlwilfried Alev Topuzoğlu Sabancı University, MDBF, Orhanlı, Tuzla, İstanbul, Turkey. alev@sabanciuniv.edu

3 2 Nurdagül Anbar et al. Keywords Quadratic functions plateaued functions bent functions Walsh transform idempotent functions rotation symmetric Reed-Muller code Mathematics Subject Classification T06 11T71 11Z05 1 Introduction Let p be a prime and let f be a function from F p n to F p. Then we can write f in polynomial form, for instance as the trace of a polynomial of degree at most p n. Clearly such a representation is not unique. Recall that the algebraic degree of f is the largest p-ary weight of an exponent in a polynomial representation of f. In this article we deal with functions Q from F p n to F p of algebraic degree 2, which we call quadratic functions. The degree of a function on F p n will always refer to the polynomial degree, i.e. to the largest exponent of its unique representation as a polynomial of degree at most p n. Omitting the linear and constant terms, a quadratic function Q from F p n to F p, for a prime p, can be expressed in trace form as n/2 Qx = Tr n a i x pi +1, a i F p n, 1 i=0 where Tr n denotes the absolute trace from F p n to F p. When n is odd, this representation is unique. For even n the coefficient a n/2 is taken modulo the additive subgroup G = {x F p n : Tr n n/2x = 0} of F p n, where Tr n n/2 denotes the relative trace from F p n to F p n/2. Furthermore, if p = 2, then a 0 = 0, hence i in the summation in 1 ranges over 1 i n/2, since a 0 x 2 is a linear term. The Walsh transform f of a function f : F p n F p is the function from F p n into the set of complex numbers defined as fb = x F p n ɛ fx Trnbx p, where ɛ p = e 2πi/p is a complex p-th root of unity. Quadratic functions belong to the class of plateaued functions, for which for every b F p n, the Walsh transform fb vanishes or has absolute value p n+s/2 for some fixed integer 0 s n. Accordingly we call f s-plateaued. Note that if p = 2, then ɛ p = 1, and fb is an integer. Hence for any s- plateaued function from F 2 n to F 2, n and s must be of the same parity. Recall that a 0-plateaued function from F p n to F p is called bent. Clearly a Boolean bent function can exist only when n is even. When p is odd, a 1-plateaued function is called semi-bent. A Boolean function f is called semi-bent, if f is 1 or 2-plateaued, depending on the parity of n.

4 Title Suppressed Due to Excessive Length 3 The nonlinearity N f of a function f : F p n F p is defined to be the smallest Hamming distance of f to any affine function, i.e. N f = min {x F p n : fx Tr n ux + v}. u F p n,v F p The nonlinearity of a Boolean function f can be expressed in terms of the Walsh transform as N f = 2 n 1 2 max fb. 2 b F 2 n By Parseval s identity we have b F 2 n fb 2 = 2 2n for any Boolean function f. As a consequence, Boolean bent functions are the Boolean functions attaining the highest possible nonlinearity. Since high nonlinearity is crucial for cryptographic applications, Boolean bent functions are of particular interest. Recall that the rth order Reed-Muller code Rr, n of length 2 n is defined as Rr, n = {fα 1, fα 2,, fα 2 n f P r }, where P r is the set of all polynomials from F 2 n to F 2 or from F n 2 to F 2 of algebraic degree at most r, and α 1, α 2,..., α 2 n are the elements of F 2 n or F n 2 in some fixed order. The set of quadratic Boolean functions together with the constant and affine functions form the second order Reed-Muller codes. In this work we focus on the subclasses of the set of quadratic functions given in Equation 1, obtained by restricting the coefficients to the prime subfield. Namely, for p = 2 we consider C 1 and the larger class C 2 defined as n 1/2 C 1 = Qx = Tr n a i x 2i +1 n : a i F 2, 1 i 2, and C 2 = Qx = Tr n n/2 1 a i x 2i +1 + Tr n/2 a n/2 x 2n/2 +1 : a i F 2, 1 i n/2 }. Note that C 1 is defined for arbitrary n, while C 2 is defined for even n only. These two classes of Boolean functions have attracted significant attention in the last decade, see the articles [5,6,9,11 13,15 17]. For p > 2 we put n/2 D = Qx = Tr n a i x pi +1 : a i F p. i=0 The class D has also been studied previously, see [11,12,14 16]. The study of the Walsh spectrum of quadratic functions in C 1 and D has been initiated in [12], where the authors determine all n for which all such

5 4 Nurdagül Anbar et al. quadratic functions are semi-bent. This result was extended in [5] for C 1. In the articles [9,17] bent functions in C 2 are constructed. Enumeration of functions in C 1, C 2 and D for particular values of s has been a challenging problem. In [17] the number of bent functions in C 2 was obtained for some special classes of n. Counting results on Boolean quadratic functions in C 1 with a large value of s have been obtained in the paper [6]. Far reaching enumeration results for the sets C 1 and D have been obtained in [11, 15,16] by the use of methods originally employed in the analysis of the linear complexity of periodic sequences, such as the method in [7]. Let us denote the number of s-plateaued quadratic functions in C 1 and D by N n s and N n p s, respectively. In [15], the number N n s has been determined for n = 2 m, m 1, and all possible values of s, with combinatorial methods. In [16], N n s and N n p s are described by the use of generating polynomials G n z and G n p z, defined by G n z = n t=0 N n n tz t and G p n z = n t=0 N p n n tz t. In the cases of odd n and n = 2m for an odd m, the polynomial G n z is determined as a product of polynomials, see [16]. Similarly G n p z is obtained for n with gcdn, p = 1 in [4,16]. In particular, explicit formulas for the number of bent functions in D and of semi-bent functions in C 1 are given for such n. For a result on the number of semi-bent functions in C 2 we may refer to the recent article [13]. The average behaviour of the Walsh transform for many classes of functions in C 1 and D is analysed in [11]. We remark that unlike C 2, the set C 1 does not contain bent functions. We note that the main results in [16] are obtained by using the discrete Fourier transform. This method is only applicable when gcdn, p = 1. Hence the study of the case gcdn, p > 1 necessitates different methods. Moreover the discrete Fourier transform can not be used for the analysis of the the important class C 2. In order to obtain G n z when n = 2m for odd m, a modification of the method that was used to analyse linear complexity of sequences in [7] was first employed in [16]. However the arguments in [16] heavily rely on the property, specific to the case studied in that paper, that the multiplicity of 1 as a root of gcdx n + 1, Ax is exactly 2, where Ax is defined as in 3. In this work we present the solution of the enumeration problem in full generality. We first extend the above results to functions in C 1 and C 2 for arbitrary n and all possible s, by determining, i the generating polynomial G n z for any even number n, ii the generating polynomial H n z = n t=0 M nn tz t concerning the number M n s of s-plateaued functions in C 2, for any even number n. Finally we adapt our method to the considerably more complicated case of functions in D, and determine iii the generating polynomial G p n z for any number n, gcdn, p > 1.

6 Title Suppressed Due to Excessive Length 5 We therefore completely describe the distribution of the parameter s in the set C 2, and in the sets C 1 and D in all remaining cases. This also yields the distribution of the nonlinearity in C 1 and C 2 in full generality. In particular, one can obtain the number of bent functions in the sets C 2 and D, or the number of semi-bent functions in C 1, C 2 and D for arbitrary integers n. Remark 1 The classes C 1, C 2 and D have additional properties that we explain below. A Boolean function f, defined over F 2 n is idempotent if it satisfies fx 2 = fx for all x F 2 n. The set of idempotent quadratic functions coincides with C 1 when n is odd, and it coincides with C 2 when n is even. For an odd prime p, one can similarly define a p-potent function as a function f from F p n to F p that satisfies fx p = fx for all x F p n. As one would expect, the set of p-potent quadratic functions coincides with D. It is observed in [3] that there is a nonlinearity preserving one-to-one correspondence between the set of idempotent quadratic functions from F 2 n to F 2 and the set of rotation symmetric quadratic functions from F n 2 to F 2. Note that only even n is considered in [3], but the same applies to the case of odd n. Following the arguments of [3], one can show that this property extends to any prime p 2 and hence there is a one-to-one correspondence between the set D and the set of rotation symmetric quadratic functions from F n p to F p, which preserves the parameter s. Hence many results on idempotent and p-potent quadratic functions also yield results on rotation symmetric quadratic functions, which add to the interest in the classes C 1, C 2 and D. Having obtained the distribution of s in C 1, C 2, D, we are able to give the distribution of the nonlinearity of rotation symmetric quadratic functions from F n 2 to F 2, and the distribution of the co-dimension of rotation symmetric quadratic functions from F n p to F p, for odd p. We also analyse the subcodes of the second order Reed-Muller code obtained from C 1 and C 2, and present the weight distribution for both subcodes of R2, n. 2 Preliminaries In this section we summarize basic tools that we use to obtain our results. We essentially follow the notation of [15, 16]. For technical reasons we include the 0-function, for which all coefficients a i are zero, in all sets C 1, C 2 and D. Being constant, the zero function is n-plateaued. Let Qx be in C 1, i.e. Qx = Tr n n 1/2 a i x 2i +1, a i F 2. We associate to Q, the polynomial Ax = n 1/2 a i x i + a i x n i 3 of degree at most n. For even n we consider Qx C 2, i.e. Qx = Tr n n/2 1 a i x 2i +1 + Tr n/2 a n/2 x 2n/2 +1, a i F 2, and the associated poly-

7 6 Nurdagül Anbar et al. nomial Ax = n/2 1 a i x i + a i x n i + a n/2 x n/2 of degree at most n. When p is odd and n is arbitrary we consider Qx D, i.e. the quadratic function of the form Qx = Tr n n/2 i=0 a i x pi +1, a i F p, and the associated polynomial Ax = n/2 a i x i + a i x n i i=0 of degree at most n. With the standard Welch squaring method of the Walsh transform, one can easily see that in all three cases, the quadratic function Q is s-plateaued if and only if s = deg gcdx n, Ax, see also [8,12,14,17]. We observe that Ax = x d hx, where d is a non-negative integer and h is a self-reciprocal polynomial of degree n 2d. Note that d is a positive integer in the case p = 2. When Qx C 1 or Qx C 2, and hence Q is a Boolean function, the polynomial gcdx n + 1, Ax is also self-reciprocal, and Ax can be written as Ax = x d fxgx, where f is a self-reciprocal divisor of x n +1 of degree s, and g is a self-reciprocal polynomial with degree smaller than n s, satisfying gcdg, x n + 1/f = 1. When p is odd, then x n = x ψx, where ψx = 1 + x + + x n 1 F p [x] is self-reciprocal. Hence gcdx n, Ax = x ɛ fx, ɛ {0, 1}, for a self-reciprocal divisor f of x n. Thus Ax can be written as Ax = x d x ɛ fxgx, where g satisfies gcdg, x n /x ɛ f = 1. Obviously the factorization of x n +1 and ψx into self-reciprocal factors plays an important role. In accordance with [15,16], for a prime power q, we call a self-reciprocal polynomial f F q [x] prime self-reciprocal if i f is irreducible over F q, or ii f = ugg, where g is irreducible over F q, the polynomial g g is the reciprocal of g and u F q is a constant. To analyse the factorization of x n + 1 and x n /x into prime selfreciprocal polynomials, we recall the canonical factorization of x n into irreducible polynomials. Since x n = x m pv if n = mp v, gcdm, p = 1, we can assume that n and p are relatively prime. Let α be a primitive nth root of unity in an extension field of F p, and let C j = {jp k mod n : k N} be the

8 Title Suppressed Due to Excessive Length 7 cyclotomic coset of j modulo n relative to powers of p. Then x n F p [x] can be factorized into irreducible polynomials as x n = h f t x with f t x = t=1 i C jt x α i, where C j1,..., C jh are the distinct cyclotomic cosets modulo n. It is observed in [15,16] that, when p is odd, an irreducible factor f t x = i C jt x α i of x n, different from x, is self-reciprocal if and only if C jt contains with i, its additive inverse i modulo n. Otherwise there exists a cyclotomic coset C jt, which consists of the additive inverses of the elements of C jt, and the polynomial ft x = i C jt x α i, which is the reciprocal of f t. In this case f t ft is a prime self-reciprocal divisor of x n. Most of our results are expressed in terms of the degrees of the prime selfreciprocal factors of x n. We remark that by Lemma 2 in [15], the cardinalities of the cyclotomic cosets modulo n, and the degrees of the prime self-reciprocal divisors of x n can be obtained directly from the factorization of n. As explained above our aim is to determine the generating polynomials G n z and H n z for even n, and G n p z for n = p v m, v > 0. This enables us to solve the problem of enumerating quadratic functions in the sets C 1, C 2 and D with prescribed s. We adapt the number theoretical approach used in [16, Section V], by which the generating polynomial G n z is obtained to yield the number of s-plateaued quadratic functions in C 1 for the two cases; odd n and n = 2m, where m is odd. We start by giving some definitions and lemmas. For a self-reciprocal polynomial f F p [x] we define Cf := {g F p [x] g is self-reciprocal, degg is even, and degg < degf}, Kf := {g Cf gcdgx, fx = 1}, and φ p f := Kf. Let f be a monic self-reciprocal polynomial in F p [x] with even degree. Let f = r e1 1 re k k be the factorization of f into distinct monic self-reciprocal polynomials all of even degree, i.e. either r 1 = x and r j, 2 j k, are prime self-reciprocal polynomials, or r j is prime self-reciprocal for all 1 j k. Then we define the following variant of the Möbius function µ p as { 1 µ p f := k if e 1 = = e k = 1, 0 otherwise. As for the classical Möbius function on the set of positive integers, we have { 1 if f = 1, µ p d = 0 otherwise, d f where the summation is over all monic self-reciprocal divisors d of f of even degree.

9 8 Nurdagül Anbar et al. Remark 2 Compare µ p with the Möbius function used in [16], which is defined on the set of all self-reciprocal polynomials. Here considering the Möbius function on the set of self-reciprocal polynomials of even degree, where the set of the prime elements is the union of {x } and the set of prime self-reciprocal polynomials of even degree, proved to be advantageous and has facilitated obtaining G n z, H n z and G n p z in full generality. The next lemma is Lemma 8 in [16], except that the condition on a selfreciprocal polynomial not to be divisible by x+1 is replaced by the condition that it is of even degree. The proof is similar to the one in [16] with this slight modification and hence we omit it. Lemma 1 [16, Lemma 8] Let f F p [x] be a monic self-reciprocal polynomial whose degree is a positive even integer. Then φ p d = p degf 2, d f where the sum runs over all monic self-reciprocal divisors d of f of even degree. The next lemma is again similar to Lemma 9 in [16], except that polynomials involved are of even degree. We give the proof for the convenience of the reader. Lemma 2 [16, Lemma 9] Let f, f 1, f 2 F p [x] be monic self-reciprocal polynomials whose degrees are positive even integers. i We have φ p f = d f µ p dp degf degd 2, where the sum runs over all monic self-reciprocal divisors d of f of even degree. ii If f = f 1 f 2 such that gcdf 1, f 2 = 1, then φ p f = φ p f 1 φ p f 2. Proof. Taking the summation over all monic self-reciprocal divisors d of f of even degree, by Lemma 1 we get µ p d p degf degd 2 = f µ p p degd 2 d d f d f = f µ p φ p d 1 = φ p d 1 µ p g = φ p f, d d f d 1 d d 1 f g f d 1 where in the last step we use the fact that the inner sum is 1 for d 1 = f and 0 otherwise. With φ p f = µ p d p degf degd 2 d f = d f µ p dp degf degd 2 µ p d = µ p dp degf degd 2, d f d f

10 Title Suppressed Due to Excessive Length 9 we finish the proof for i. If gcdf 1, f 2 = 1, then µ p f 1 f 2 = µ p f 1 µ p f 2. Taking the summation over all monic self-reciprocal divisors d i of f i with degd i is even, i = 1, 2, we then have φ p f 1 φ p f 2 = µ p d 1 p degf 1 degd 1 2 d 1 f 1 which shows ii. d 2 f 2 µ p d 2 p degf 2 degd 2 2 = degf 1 f 2 degd 1 d 2 µ p d 1 d 2 p 2 = µ p dp degf degd 2, d1 f1 d f d 2 f 2 For an integer t, we define N n f; t by 1 if t = 0 N n f; t = d f,degd=t φ pd if t > 0 is even 0 otherwise, where the above sum runs over all monic self-reciprocal divisors d of f of degree t. Then we define the generating function G n f; z for the distribution of the values N n f; t by G n f; z = N n f; tz t. We note that G n f; z is a polynomial by definition of N n f; t. The next lemma and its proof resemble Lemma 10 in [16] for p = 2. Lemma 3 [16, Lemma 10] Let f = f 1 f 2 for two self-reciprocal polynomials f 1, f 2 of even degree. If gcdf 1, f 2 = 1, then G n f; z = G n f 1 ; zg n f 2 ; z. Proof. We have to show that for each t 0, N n f; t = N n f 1 ; t 1 N n f 2 ; t t 1. 0 t 1 t For t = 0, t > degf and t odd this is trivial. Assume that 1 t degf is even. Taking into account that φ p 1 = 0, by Lemma 2ii, and the definition of N n f; t we obtain N n f 1 ; t 1 N n f 2 ; t t 1 = N n f 1 ; 0N n f 2 ; t + N n f 1 ; tn n f 2 ; 0 0 t 1 t + φ p d 1 φ p d 2 1<t 1 <t 1 d 1 f 1 d 2 f 2 even degd 1 =t 1 degd 2 =t t 1 = N n f 2 ; t + N n f 1 ; t + 1<t 1 <t 1 even = N n f 2 ; t + N n f 1 ; t + d f degd=t = N n f; t. φ p d 1 d 2 d 1 f 1 ;degd 1 =t 1 d 2 f 2 ;degd 2 =t t 1 φ p d 1 d1 f1 degd 1 =t φ p d d2 f2 degd 2 =t φ p d 2 4

11 10 Nurdagül Anbar et al. Lemma 4 Let r be a prime self-reciprocal polynomial of even degree. Then G n r k ; z = 1 + For an even integer k we have k p degrj 2 p degrj degr 2 z jdegr. G n x + 1 k ; z = G n x k ; z = 1 + k 2 p j 1 p z 2j. Proof. If r is a prime self-reciprocal polynomial of even degree, then G n r k ; z = N n r k ; tz t = 1 + φ p d z t t: even = 1 + = 1 + k φ p r j z jdegr = 1 + k t: even, t>0 d r k ; degd=t k µ p dp degrj degd 2 z jdegr d r j p degrj 2 p degrj degr 2 z jdegr. The expression for G n x 2 k/2 ; z follows as a special case, and G n x k/2 ; z is determined in the same way since the sum in 4 is over selfreciprocal polynomials d of even degree dividing f, and N f; t = 0 for odd t. 3 Distribution of the nonlinearity in C 1 and C 2 Our aim in this section is to determine both G n z and H n z for all even integers n. By Remark 1, this does not only enable us to completely describe the distribution of the nonlinearity for the set of idempotent quadratic Boolean functions, but also for the set of rotation symmetric quadratic Boolean functions. We first express our counting functions N n s and M n s in terms of N f; t. Proposition 1 Let n be even and let N n s and M n s be the number of s-plateaued quadratic functions in C 1 and C 2, respectively. Then x n + 1 N n s = N n x ; n s and M n s = N n x n + 1; n s.

12 Title Suppressed Due to Excessive Length 11 Proof. The statement is clear when n s is zero or odd. Suppose n s > 0 is even. First we consider quadratic functions Q C 1. For the corresponding associate polynomial Ax we have gcdax, x n + 1 = x f 1 x for some self-reciprocal divisor f 1 of x n + 1/x of degree s 2, i.e. Ax = x c x f 1 xgx for an integer c 1 and a self-reciprocal polynomial g of even degree less than x n s, which is relatively prime to dx = n +1 x+1 2 f 1x. In other words, g is any of the φ 2 d polynomials in Kd. To determine the number N n s we consider all divisors x f 1 x of x n + 1 of degree s, or equivalently, all divisors dx of x n + 1/x of degree n s. Hence we obtain N n s as N n s = φ 2 d. d xn +1 x+1 2 and degd=n s If Q C 2, then the associated polynomial Ax satisfies gcdax, x n + 1 = f 1 x, where f 1 is a self-reciprocal polynomial of degree s, i.e. Ax = x c f 1 xgx for an integer c 1 and a self-reciprocal polynomial g of even degree less than n s with gcd g, x n + 1/f 1 x = 1. Therefore g Kd. As a consequence, the number of s-plateaued quadratic functions in C 2 is M n s = φ 2 d, which finishes the proof. d x n +1 and degd=n s The following is the main theorem of this section. Theorem 1 Let n = 2 v m, m odd, v > 0, and let x n +1 = x+1 2v r 2v 1 r 2v k, where r 1,..., r k are prime self-reciprocal polynomials of even degree. We set G i z := v 2 j degr i 2 2 j 1 degr i 2 z j degri. Then the generating polynomial G n z = n t=0 N nn tz t is given by 2 v 1 1 k G n z = j 1 z 2j G i z, and the generating polynomial H n z = n t=0 M nn tz t is given by 2 v 1 k H n z = j 1 z 2j G i z.

13 12 Nurdagül Anbar et al. Proof. By Proposition 1 and Lemma 3 we have and G n z = G n xn + 1 x ; z = G nx v 1 1 ; z H n z = G n x n + 1; z = G n x v 1 ; z k k G n r 2v i ; z, G n r 2v i ; z. For a prime self-reciprocal polynomial r of even degree, Lemma 4 for p = 2 gives G n r 2v ; z = v G n x + 1 2e ; z = jdegr 2 2 j 1degr 2 z jdegr, and e 2 j 1 z 2j. Combining those formulas yields the assertion. Remark 3 Putting v = 1 and m > 1 in Theorem 1, one obtains G n z in Theorem 5ii of [16]. Note that the Theorem 5ii in [16] contains an additional factor 2, since a quadratic function there may also have a linear term. Similarly the expression for G n z with m = 1 gives Theorem 6 in [15]. As a corollary of Theorem 1 we obtain the number M n 0 of bent functions in the set C 2 as the coefficient of z n in H n z for arbitrary even integers n. This complements the results of [9,17], where M n 0 has been presented for the special cases n = 2 v p r, where p is a prime such that the order of 2 modulo p is p or p /2. Corollary 1 Let n = 2 v m, m odd, v > 0, and let x n +1 = x+1 2v r 2v 1... r 2v k, where r 1,, r k are prime self-reciprocal polynomials of even degree. Then the number of bent functions in C 2 is M n 0 = 2 2v 1 k 2 2 v degri 2 2 2v 1 degr i 2 which is also the number of rotation symmetric quadratic bent functions in n variables. Similarly one may obtain M n s and N n s for other small values of s. The number of semi-bent function in C 1 for even n, and in C 2 is presented in the next corollary. Note that the number of semi-bent functions in C 1 when n is odd is given in [16, Corollary 7].,

14 Title Suppressed Due to Excessive Length 13 Corollary 2 Let n = 2 v m, m odd, v > 0, and let x n +1 = x+1 2v r 2v 1... r 2v k, where r 1,, r k are prime self-reciprocal polynomials of even degree. The number of semi-bent functions in C 1 is N n 2 = 2 2v 1 2 k The number of semi-bent functions in C 2 is 2 2 v degri 2 2 2v 1 degr i 2. - M n 2 = N n 2 if 3 does not divide n, - M n 2 = N n v v 2 k r i x 2 +x+1 2 2v degr i 2 2 2v 1 degr i 2 if 3 divides n. Proof. The corollary follows from Theorem 1 with the observation that x 2 +x+1 divides x n if and only if 3 divides n. Remark 4 In [13, Theorem 12], for n 0 mod 3 an expression for the number K of semi-bent functions in C 2 \ C 1 is given, which involves a Möbius function on the set of monic self-reciprocal polynomials. Our formula is more explicit. K = 2 2v v 2 k r i x 2 +x v degri 2 2 2v 1 degr i 2 Remark 5 To completely describe the nonlinearity distribution in C 1 and C 2 it is inevitable to consider the generating polynomials. Otherwise, in order to determine N n s or M n s for a specific s, one would first have to find all possible ways of expressing s as a sum of degrees of polynomials in the prime self-reciprocal factorization of x n + 1, which for general n is illusive. 4 Weight distribution of subcodes of second order Reed-Muller codes Let Q be a set of quadratic functions, which do not contain linear or constant terms. Assume that Q is closed under addition. Denote the set of affine functions from F 2 n to F 2 by A = {Tr n bx + c : b F 2 n, c F 2 }. Then the set Q A = {Qx + lx : Q Q, l A} gives rise to a linear subcode R Q of the second order Reed-Muller code R2, n, which contains the first order Reed-Muller code R1, n as a subcode. Clearly, we can write Q A as the union Q A = Q Q Q + A of disjoint cosets of A. The weight distribution of other subcodes of R2, n containing R1, n has been investigated in [1, 10]. More precisely, the authors determine the weight distribution of some extended cyclic codes of length 2 n with special generator

15 14 Nurdagül Anbar et al. polynomials in terms of Gaussian coefficients. In our work, we are interested in the classes of subcodes generated by the cosets of R1, n containing rotation symmetric functions. We emphasize that these classes of subcodes are different than the ones investigated in [1, 10]. For example, the punctured code the reverse operation of extending code of our corresponding subcodes is not cyclic. To obtain the weight distribution of the code R Q, it is sufficient to know the weight distribution for each of these cosets. It can be seen easily that the weight of the codeword c Q of a quadratic function Q can be expressed in terms of the Walsh transform as wtc Q = 2 n 1 2 Q0. For a quadratic function Q we define Q b,c x = Qx + Tr n bx + c. Using Q b,c 0 = 1 Trnc Qb one can show that the weight distribution of the coset Q + A for an s-plateaued quadratic function Q is as follows. There are - 2 n s codewords of weight 2 n n+s 2 1, - 2 n s codewords of weight 2 n 1 2 n+s 2 1, and - 2 n+1 2 n s+1 codewords of weight 2 n 1. Hence, if one knows the number of s-plateaued quadratic functions in Q for every s, one can determine the weight distribution of R Q. If Q is the set of all quadratic functions, then R Q = R2, n. The weight distribution of R2, n is completely described in [2] by explicit, quite involved formulas. Here we focus on the subcodes of R2, n, obtained from the sets C 1 and C 2, in other words from the set of idempotent quadratic functions. Putting k = n s which is even, the observations above imply that the only weights that can occur are 2 n 1 and 2 n 1 ± 2 n 1 k 2, 0 k n. Moreover, codewords of the weights 2 n n 1 k 2 and 2 n 1 2 n 1 k 2 appear the same number of times. Hence to describe the weight distribution of the codes R C we may consider the polynomial W C z = n k=0 AC k zk, where A C k is the number of codewords in R C of weight 2 n 1 ± 2 n 1 k 2. Again by the above observations, A C1 k = N nn k2 k and A C2 k = M nn k2 k. Consequently, W C1 z = W C2 z = n k=0 n k=0 A C1 k zk = A C2 k zk = n N n n k2 k z k = G n 2z, k=0 n M n n k2 k z k = H n 2z. 5 k=0 For the number A C1 of codewords in R C1 of weight 2 n 1 we have A C1 = n N n n k2 n+1 2 k+1 = 2 n+1 k=0 = 2 n+1 G n 1 2G n 2. n k=0 N n n k 2 n N n n k2 k k=0

16 Title Suppressed Due to Excessive Length 15 Similarly, A C2 = 2 n+1 H n 1 2H n 2. The following theorem describes the weight distribution of the codes R C. Theorem 2 Let n = 2 t m, m odd, and let x n + 1 = x + 1 2t r1 2t rl 2t for prime self-reciprocal polynomials r 1,..., r l of even degree. Then for even n where and W C2 z = n k=0 A C2 k zk = j 1 z 2j 2 t 1 l W i z, 2 t W i z = j degr i 2 2 3j 1 degr i 2 z j degri, l 2 t A C2 = 2 n+1+2t j degr i 2 2 j 1 degr i 2 232t l 2 t j degr i 2 2 3j 1 degr i 2. When t = 0, i.e., n is odd, we have n l [ W C1 z = A C1 k zk = degri/2 2 degri z degri], and 6 k=0 A C1 = 2 3n l degri/2 2 degri Proof. By using 5, the formulas for W C1 z and W C2 z follow from the generating function G n z = l [ degr i/2 z degri] when n is odd see Theorem 5i in [16] and Theorem 1. The formulas for A C1 and A C2 are obtained by expanding 2 n+1 G n 1 2G n 2 and 2 n+1 H n 1 2H n 2. Remark 6 When n is odd, the code R C1 has 2 3n+1/2 codewords, i.e. dim R C1 = 3n + 1/2. Observing that the coefficient of z k in 6 is not zero if and only if k = r i {r 1,...,r l } degr i, for r = min{degr i } l we conclude that R C1 is a [2 n, 3n + 1/2, 2 n 1 2 n 1 r 2 ] code. Remark 7 We emphasize that in order to describe the weight distribution of the codes that we study, it is not possible to avoid the use of generating polynomials. Note that every explicit formula for the number of codewords with a given weight should depend on all possible ways of expressing the corresponding parameter s as a sum of degrees of polynomials in the prime self-reciprocal factorization of x n + 1. Hence such formulas for general n cannot be given unless the degrees of these divisors follow a particular pattern. Description of the weight distribution via a generating polynomial is for these codes the best one can expect..

17 16 Nurdagül Anbar et al. 5 Enumeration of s-plateaued quadratic functions for odd characteristic In this section we apply our method to quadratic functions in D. Recall that results on D translate to results on the set of rotation symmetric functions from F n p to F p. As we will see, determining the generating function in odd characteristic is considerably more involved. We can restrict ourselves to the case gcdn, p > 1 since the case gcdn, p = 1 is dealt with in [16] with a different method which employs discrete Fourier transform. We emphasize that the method of [16] is not applicable to the case gcdn, p > 1. n/2 Recall that to Qx = Tr n i=0 a i x pi +1 D, the associate is Ax = n/2 a i x i + a i x n i, i=0 which is a polynomial of degree at most n. We treat the cases of odd and even n separately. The case of odd n In this case the factorization of x n into prime self-reciprocal divisors is given by x n = x pv r pv 1 rpv k. Firstly we note that x n is not divisible by x + 1 as n is an odd integer, and hence each r i, 1 i k, is a prime self-reciprocal polynomial of even degree. Write Ax as n/2 Ax = a i x i + a i x n i = x c hx, i=0 where hx is a self-reciprocal polynomial of degree n 2c. Since n is odd, also the degree of h is odd, and hence x + 1 is a factor of Ax. In fact, x + 1 must appear as an odd power in the factorization of Ax. In particular, Ax = x c x + 1ϱx for a self-reciprocal polynomial ϱx of even degree n 2c 1. As a consequence, gcdax, x n = gcdϱx, x n = x ɛ f 1 x, for a self-reciprocal polynomial f 1 of even degree, and some ɛ {0, 1}. That is, ϱx = x ɛ x n f 1 xg 1 x with gcd x ɛ f 1 x, g 1x = 1. We will frequently use the following observation, which immediately follows from the definition of φ p.

18 Title Suppressed Due to Excessive Length 17 Proposition 2 For a self-reciprocal polynomial d of even degree we have φ p x ± 1d = φ p x ± 1 2 d. We now represent the counting function N n p s in terms of the function N n f; t in 4. The arguments are more complicated than in the even characteristic case, for instance, here the distinction between odd and even n is required. As we see in the proposition below, for odd n we also need to consider odd and even s separately. In the next subsection, where n is even, we have to treat four cases. Proposition 3 Let n be an odd integer such that the p-adic valuation of n is v p n = v. The number N n p s of quadratic s-plateaued functions in D is given by N n p s = N n xn 1 x 1 pv ; n s N nx x n ; n s + 1 N n xn 1 x 1 pv ; n s + 1 if s is odd, if s is even. Proof Suppose that gcdax, x n = x f 1 x, i.e. ɛ = 1. This holds if only if s is odd. In this case, degg 1 is odd and g 1 x is divisible by x 1 since ϱx is a self-reciprocal polynomial of even degree, i.e. g 1 x = x gx for some self-reciprocal polynomial g of even degree. Furthermore we know that x n 1 x n 1 x 1f 1x. gcd x 1f, g 1x 1x = 1, and therefore x is not a factor of Consequently, x pv 1 must divide f 1 x. Hence hx can be expressed as hx = x + 1x pv +1 x n fxgx with gcd x pv fx, gx = 1, where g is a self-reciprocal polynomial of even degree smaller than n s, and x hence g K n 1 x. Furthermore, each divisor d of n 1 of degree x 1 pv fx x 1 pv n s uniquely determines fx. As a result, there exist x n ; n s x pv φ p d = N n x d n 1 x 1 pv ; degd=n s s-plateaued quadratic functions in D if s is odd. Now we consider the case of even s, i.e. ɛ = 0 and gcdax, x n = fx, where the multiplicity of x in f is even. As a result, xn 1 fx is divisible by x, which implies that gx is not divisible by x. In this case, hx = x + 1fxgx with x n gcd fx, gx = 1, i.e. g K x n 1 fx = K x xn 1 fx by Proposition 2. Furthermore for any self-reciprocal divisor d of x n 1 of degree n s 1, the factor x 1d uniquely

19 18 Nurdagül Anbar et al. determines f. Hence there are x 1d x n 1; degd=n s 1 = φ p x d x 1 2 d x 1x n 1; degd=n s 1 φ p x 2 d s-plateaued functions in D in the case of even s, where d runs over all monic self-reciprocal divisors of x n 1. In order to determine the last sum, we separate the set S = {d : d is monic, self-reciprocal, d x x n, degd = n s + 1} into two disjoint subsets. The set S 1 consists of elements of S which are divisible by x, and S 2 is the complement, i.e. Then S 1 = {d : d = x 2 k S, k is self-reciprocal, degk = n s }, { x n } S 2 = d : d S, d. x pv or equivalently φ p d = φ p d φ p d, d S 1 d S d S 2 x 1 2 d x 1x n 1; degd=n s 1 = d x 1x n 1; degd=n s+1 φ p x 2 d φ p d φ p d x d n 1 x 1 pv ; degd=n s+1 = N n x x n ; n s + 1 N n xn x pv ; n s + 1. Remark 8 Note that from the proof of Proposition 3, we see that s p v if s is odd. Theorem 3 Let n be an odd integer and let x n = x pv r pv 1 rpv k be the factorization of x n 1 into distinct prime self-reciprocal polynomials. Then G p n where = n t=0 N n p n tz t is given by p v +1 2 k 1 + p j 1 p z 2j 1 G i z, G p n z = G i z = 1 + p v p degr j i 2 p degrj i degr i 2 z jdegri.

20 Title Suppressed Due to Excessive Length 19 Proof We can express the generating function as G p n z = t: even = t: even = t: even = t: even N n p n tz t + N n x n x pv ; t N n x n x pv ; t N n p n tz t t: odd z t + t: odd z t + 1 z t: odd [ x n N n ; t z t + 1 x pv z = G n x n x pv ; z = k + 1 z N nx x n ; t + 1 N n x n ; t + 1 x pv N nx x n ; t + 1 N n x n x pv ; t + 1 x N nx x n n ] ; t N n ; t z t x pv t: even [ x G nx x n n ; z G n ; z x pv G nr pv i ; z 1 z + 1 z Gnx pv +1 ; z. Note that in the fourth equality we added and subtracted N n x x n x 1; 0 = N n 1 n x 1 ; 0 = 1. In the last equality we used Lemma 3. We then pv obtain the claimed formula by Lemma 4. ] z t z t+1 The case of even n In this case the factorization of x n is given by x n = x pv x + 1 pv r pv 1 rpv k, for some distinct prime self-reciprocal polynomials r i, 1 i k, of even degree. Now n/2 Ax = a i x i + a i x n i = x c hx, i=0 where hx is self-reciprocal, and of even degree n 2c. For the gcdax, x n 1 we have gcdax, x n = gcdhx, x n = x ɛ x + 1 δ f 1 x, 7 for a self-reciprocal polynomial f 1 of even degree, and elements ɛ, δ {0, 1}. Consequently, hx = x 1 ɛ x+1 δ x n f 1 xg 1 x, with gcd x ɛ x + 1 δ f 1 x, g 1x We now have to distinguish four cases depending on the values of ɛ and δ in 7. = 1.

21 20 Nurdagül Anbar et al. i Let ɛ = δ = 1, i.e., gcdax, x n = gcdhx, x n = x x + 1f 1 x. In this case, s is even and the self-reciprocal polynomial h of even degree is given by hx = x x + 1f 1 xx x + 1gx x n with gcd x x + 1gx, = 1, x x + 1f 1 where gx is a self-reciprocal polynomial of degree n s 2. As a result, f 1 x = x pv 1 x + 1 pv 1 fx for a self-reciprocal polynomial f of even degree which of course is a product of some r i s. As a consequence, hx = x pv +1 x + 1 pv +1 fxgx, for some self-reciprocal polynomial g. Note that degg is an even integer smaller than n s and gcd g, n 1 x = 1. Recalling that x 1 pv x+1 pv fx deg = n s, we see that g K. x n 1 x 1 pv x+1 pv fx of degree n s uniquely de- Furthermore each divisor d of termines f, and hence there exist x n 1 x 1 pv x+1 pv φ p d x d n 1 x 1 pv x+1 pv ; degd=n s x n 1 x 1 pv x+1 pv fx such polynomials h. ii Let ɛ = 1 and δ = 0, i.e., gcdax, x n = gcdhx, x n = x 1f 1 x. In this case, s is odd and the self-reciprocal polynomial h of even degree is given by x n hx = x f 1 xx gx with gcd x gx, = 1, x f 1 for some self-reciprocal polynomial g,where degg is even, and smaller than n s. Consequently, f 1 x = x pv 1 fx for some self-reciprocal polynomial f of even degree, and therefore hx = x pv +1 fxgx, x where gcd g, n 1 = 1. Since xn 1 x 1 pv fx x 1f 1 contains the factor x + 1, the polynomial g is not divisible by x+1. In particular, g Kx+1d, where d = is a self-reciprocal polynomial of even degree n s 1. x n 1 x 1 pv x+1fx x n 1 x 1 pv x+1 As above, each self-reciprocal divisor d of uniquely determines f. Consequently there exist such polynomials h. φ p x + 1d x d n 1 x 1 pv ; degd=n s 1 x+1 of degree n s

22 Title Suppressed Due to Excessive Length 21 iii The case ɛ = 0 and δ = 1 is very similar. With the same argument as in ii we get x n hx = x + 1f 1 xx + 1gx with gcd x + 1gx, = 1, x + 1f 1 8 for some self-reciprocal polynomial g, where degg is even, smaller than n s, and φ p x d x d n 1 x+1 pv ; degd=n s 1 x 1 is the number of polynomials h of the form 8. iv Let ɛ = δ = 0, i.e., gcdax, x n = gcdhx, x n = fx for some self-reciprocal polynomial f of even degree. In this case we see that the self-reciprocal polynomial h is given by hx = fxgx with gcd gx, xn = 1, fx where gx is a self-reciprocal polynomial of degree n s. Note that xn 1 fx is divisible by both x and x + 1, so gx is not divisible by neither. Therefore hx = fxgx with gcd gx, x x + 1 xn = 1, 9 fx where gx is a self-reciprocal polynomial of degree smaller than n s + 2. As for each self-reciprocal divisor d of x n of degree n s, f is uniquely determined, we can again express the number of polynomials h of the form 9 in terms of φ p as φ p x x + 1d. x 1x+1d x 1x+1x n 1; degd=n s Remark 9 From the above observations we conclude that again s p v if s is odd. Theorem 4 Let n be an even integer and let x n 1 = x 1 pv x+1 pv r pv 1 rpv k be the factorization of x n into distinct prime self-reciprocal polynomials. Then the generating function G n p is given by G p n z = 1 + p v +1 2 p j 1 p z 2j 1 2 k G i z, where G i z = 1 + p v p degr j i 2 p degrj i degr i 2 z jdegri.

23 22 Nurdagül Anbar et al. Proof First let s be odd, which applies in the cases ii and iii above. The discussion of these cases imply for N n p s that N p n s = d x n 1 x 1 pv x+1 ; degd=n s 1 φ px + 1d + d x n 1 x+1 pv x 1 ; degd=n s 1 φ px d d x n 1 x 1 pv x+1 ; degd=n s 1 d x n 1 x+1 pv x 1 ; degd=n s 1 = φ px d + φ px 2 d. To determine the sum 10 we separate the set { x n } S = h : h is monic, self-reciprocal, h, degh = n s + 1 x pv into the two disjoint subsets S 1 = {h : h S and gcdh, x + 1 = 1}, and S 2 = {h : h S and x + 1 h}. Note that being self-reciprocal and of even degree, the polynomials h in the set S 2 are divisible by x Then for the first sum in 10 we obtain φ p x d = φ p h x d n 1 x 1 pv ; degd=n s 1 h S2 x+1 = φ p h φ p h h x+1xn 1 x 1 pv ; degh=n s+1 h S 1 = φ p h φ p h h x+1xn 1 x 1 pv ; degh=n s+1 x h n 1 x 1 pv x+1 pv ; degh=n s+1 x + 1x n x n = N n ; n s + 1 N n ; n s + 1. x pv x pv x + 1 pv In a similar way, for the second sum in 10 we get φ p x 2 d x d n 1 x+1 pv ; degd=n s 1 x 1 x x n x n N n ; n s + 1 N n ; n s + 1. x + 1 pv x pv x + 1 pv Combining, we obtain x + 1x N n p n s = N n ; n s + 1 x pv x n 2N n ; n s + 1 x pv x + 1 pv ; n s + 1 x + 1 pv + N n x x n 10

24 Title Suppressed Due to Excessive Length 23 for an odd integer s. Now we consider the case of even s, which corresponds to i and iv. By the above observations, N n p s = φ p d x d n 1 x 1 pv x+1 pv ; degd=n s + x 1x+1d x 1x+1x n 1; degd=n s x n = N n ; n s x pv x + 1 pv + x 1x+1d x 1x+1x n 1; degd=n s φ p x x + 1d φ p x x + 1d. We set T := x 1x+1d x 1x+1x n 1; degd=n s φ px x + 1d and in order to determine T we consider the set S = {hx : hx is self-reciprocal, h x 1x+1x n 1, degh = n s+2} and its subsets S 1 = {hx : h S, gcdh, x = 1}, S 2 = {hx : h S, gcdh, x + 1 = 1}, S 3 = {hx : h S, gcdh, x 2 = 1}. Note that since h is a self-reciprocal polynomial of even degree, h is divisible by x 1 2 if it is divisible by x 1. As a consequence, T = φ p h φ p h φ p h + φ p h. h S 1 h S 2 h S 3 h x 1x+1x n 1; degh=n s+2 We also note that for h in the set S, h S 1 if and only if h x + 1 xn 1 x 1, pv h S 2 if and only if h x 1 xn 1 x+1 and h S x pv 3 if and only if h n 1 x 1 pv x+1. pv Therefore we can express the sum T as T = φ p h φ p h h x 1x+1x n 1; degh=n s+2 h x+1 xn 1 x 1 pv ; degh=n s+2 φ p h + φ p h. h x 1 xn 1 x+1 pv ; degh=n s+2 x h n 1 x 1 pv x+1 pv ; degh=n s+2 Hence, for an even integer s N p n s = x n Nn x pv x + 1 ; n s + N pv n x x + 1x n ; n s + 2 x + 1x n x x n N n ; n s + 2 N x pv n ; n s + 2 x + 1 pv x n +N n x pv x + 1 ; n s + 2. pv

25 24 Nurdagül Anbar et al. From the above equalities we get n G p n z = N p n n tzt = N p n n tzt + N p n n tzt t=0 t: odd t: even = [ x + 1x n x x n x n ] N n ; t N x pv n ; t + 1 2N x + 1 pv n x pv x + 1 ; t + 1 z t pv t: odd + x [N n n x t: even pv x + 1 ; t + N pv n x x + 1x n ; t + 2 x + 1x n N n ; t + 2 x pv x x n x n ] N n ; t N x + 1 pv n x pv x + 1 ; t + 2 z t pv [ x + 1x n x x n N n ; t N x pv n ; t + 1 2N x + 1 pv n = 1 z t: odd + x n N n x t: even pv x + 1 ; t pv x + 1x n N n ; t + 2 x pv z t + 1 z 2 t: even N n x x n x + 1 pv ; t + 2 [N n x x + 1x n ; t N n x n x pv x + 1 pv ; t + 1 x n ] x pv x + 1 ; t + 2 z t+2 pv ] z t+1 [ x + 1x n x x n N n ; t + N x pv n ; t 2N x + 1 pv n = 1 z t: even + x n N n x t: even pv x + 1 ; t pv x + 1x n N n ; t x pv z t + 1 z 2 t: even N n x x n x + 1 pv ; t [ Nn x x + 1x n ; t x n x pv x + 1 pv ; t x n ] + N n x pv x + 1 ; t z t pv ] z t [ x + 1x n x x n G n ; z + G x pv n = 1 z x n + G n x pv x + 1 ; z pv G n x + 1x n x pv ; z ; z 2G x + 1 pv n + 1 z 2 [ G n x x + 1x n ; z G n x x n x + 1 pv ; z + G n x n x pv x + 1 pv ; z x n ] x pv x + 1 ; z pv ] = k 1 G nr pv v +1 i ; z Gnx + 1p ; z + 1 z z Gnx pv +1 ; z 2 z Gnx p z2 By Lemma 4 we get G p n z = k v +1 x + 1 pv +1 ; z Gnx + 1p z2 v +1 ; z v +1 Gnx p ; z + 1. z2 z 2 2 G n r pv i ; z z G nx p v +1 ; z 2 z z 2 G nx pv +1 ; z 2 2 z 2 G nx pv +1 ; z + 1 z 2.

26 Title Suppressed Due to Excessive Length 25 Putting G i := G n r pv i ; z, and A := this yields G p n z = = p v +1 2 p j 1 p z 2j = G n x pv +1 ; z k [ 2 G i z 1 + A 2 z z A2 2 z A + 1 ] z A z 2 k G i. Simplifying and using Lemma 4 we get the desired result. As an immediate corollary of Theorems 3 and 4 we obtain the number of bent functions in the set D. Note that the case of gcdn, p = 1 is covered in [16, Corollary 7]. Corollary 3 Let the factorization of x n into distinct prime self-reciprocal polynomials be x n = x pv r pv 1 rpv k when n is odd, and x n = x 1 pv x+1 pv r pv 1 rpv k when n is even. Then the number of bent functions in D is if n is odd, and if n is even. N n 0 = p p pv 1 2 k p p v degri 2 p pv 1 degr i 2 N n 0 = p p pv 1 k 2 p 2 p v degri 2 p pv 1 degr i 2,, 6 Acknowledgements The first author gratefully acknowledges the support from The Danish Council for Independent Research Grant No. DFF and H.C. Ørsted COFUND Post-doc Fellowship from the project Algebraic curves with many rational points. The second author is supported by the Austrian Science Fund FWF Project no. M 1767-N26.

27 26 Nurdagül Anbar et al. References 1. Berlekamp, E.R.: The weight enumerators for certain subcodes of the second order binary Reed-Muller codes. Information and Control 17, Berlekamp, E.R., Sloane, N.: The weight enumerator of second-order Reed-Muller codes. IEEE Trans. Inform. Theory IT-16, Carlet, C., Gao, G., Liu, W.: A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions. J. Combin. Theory, Series A 127, Çeşmelioğlu, A., Meidl, W.: Non-weakly regular bent polynomials from vectorial quadratic functions. In: Pott, A., et.al. eds Proceedings of the 11th international conference on Finite Fields and their Applications, Contemporary Mathematics, 2015, Charpin, P., Pasalic, E., Tavernier, C.: On bent and semi-bent quadratic Boolean functions. IEEE Trans. Inform. Theory 51, Fitzgerald, R.W.: Trace forms over finite fields of characteristic 2 with prescribed invariants. Finite Fields Appl. 15, Fu, F. W., Niederreiter, H., Özbudak, F.: Joint linear complexity of multisequences consisting of linear recurring sequences. Cryptogr. Commun. 1, Helleseth, T., Kholosha, A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inform. Theory 52, Hu, H., Feng, D.: On Quadratic bent functions in polynomial forms. IEEE Trans. Inform. Theory 53, Kasami T.: The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes. Information and Control 18, Kaşıkcı, C., Meidl, W., Topuzoğlu, A: Spectra of quadratic functions: Average behaviour and counting functions. Cryptogr. Commun. DOI /s Khoo, K., Gong, G., Stinson, D.: A new characterization of semi-bent and bent functions on finite fields. Designs, Codes, Cryptogr. 38, Kocak, N., Kocak, O., Özbudak, F., Saygi, Z.: Characterization and enumeration of a class of semi-bent Boolean functions. Int. J. Inform. Coding Theory 3, Li, S., Hu, L., Zeng, X.: Constructions of p-ary quadratic bent functions. Acta Appl. Math. 100, Meidl, W., Topuzoğlu, A.: Quadratic functions with prescribed spectra. Designs, Codes, Cryptogr. 66, Meidl, W., Roy, S., Topuzoğlu, A.: Enumeration of quadratic functions with prescribed Walsh spectrum. IEEE Trans. Inform. Theory 60, Yu, N.Y., Gong, G.: Constructions of quadratic bent functions in polynomial forms. IEEE Trans. Inform. Theory 52,

28 Title Suppressed Due to Excessive Length 27 7 Appendix We give some examples of generating function for p = 3. Example n = 9 13: In this case, x = x 1 9 r 9 1r 9 2 for prime self-reciprocal polynomials r 1, r 2 both of degree 6. By Theorem 3, G z = j 1 z 2j j 1 z 6j 2. Expanding this polynomial we obtain G z = z + 6 z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z 117