1 New Quadratic Bent Functions in Polynomial Forms with Coefficients in Extension Fields Chunming Tang, Yanfeng Qi, Maozhi Xu n n Abstract—In this paper, we first discuss the bentness of a large where n is odd, T r1 (x) is the trace function from GF (2 ) to class of quadratic Boolean functions in polynomial form f(x) = GF (2) and ci 2 GF (2). They proved that f(x) is semi-bent P n −1 i n=2 2 n 1+2 n=2 1+2 2 n i=1 T r1 (cix ) + T r1 (cn=2x ), where ci GF (2 ) if and only if ≤ ≤ n − 2 n=2 for 1 i 2 1 and cn=2 GF (2 ). The bentness of these functions can be connected with linearized permutation gcd(c(x); xn + 1) = x + 1; polynomials. Hence, methods for constructing quadratic bent P n−1 functions are given. Further, we consider a subclass of quadratic 2 i n−i P m − ei where c(x) = i=1 ci(x + x ). 2 1 n 1+2 Boolean functions of the form f(x) = i=1 T r1 (cix ) + Charpin, Pasalic and Tavernier [6] generalized Khoo et al.’s n=2 1+2n=2 2 e T r1 (cm=2x ) , where ci GF (2 ), n = em and m is results to even n and considered quadratic functions of the even. The bentness of these functions are characterized and some form methods for constructing new quadratic bent functions are given. b n−1 c 2 v0 r − X Finally, for a special case: m = 2 p and gcd(e; p 1) = 1, we n 1+2i 2 present the enumeration of quadratic bent functions. f(x) = ciT r1 (x ); ci GF (2): i=1 Index Terms—Bent function, Boolean function, linearized per- mutation polynomial, cyclotomic polynomial, semi-bent function When n is even, they proved that f(x) is semi-bent if and only if gcd(c(x); xn + 1) = x2 + 1; I. INTRODUCTION P n−2 2 i n−i A bent function, whose Hamming distance to the set where c(x) = i=1 ci(x + x ). For odd n, they investi- n−1 n −1 of all affine Boolean functions equals 2 ± 2 2 , is a gated the conditions for the semi-bent functions of f(x) with Boolean function with even n variables from GF (2n) to three and four trace terms. GF (2). Further, it has maximum nonlinearity and the absolute For further generalization, Ma, Lee and Zhang [17] applied value of its Walsh transform has a constant magnitude [22]. techniques from [14] and considered the quadratic Boolean Nonlinearity is an important property for a boolean function functions of the form in cryptographic applications. Much research has been paid n−2 X2 n on bent functions [3], [4], [5], [6], [7], [10], [14], [17], i n=2 2 f(x) = c T rn(x1+2 ) + T r (x1+2 ); (1) [25]. Since bent functions with maximal nonlinearity have i 1 1 i=1 a close relationship with sequences, bent functions are often 2 n=2 used in the construction of sequences with maximally linear where ci GF (2) and T r1 (x) is the trace function from n complexity and low correlation[2], [8], [9], [15], [16], [21], GF (2 2 ) to GF (2). They proved that f(x) is a bent function [23]. Further, many applications of bent functions can be found if and only if in coding theory [18] and combinatorial design. gcd(c(x); xn + 1) = 1; As another class of Boolean functions, semi-bent functions P n−2 2 i n−i n=2 are also highly nonlinear. For an even integer n, the Walsh where c(x) = i=1 ci(x + x ) + x . For some spe- n spectra of bent functions with n variables has the value ±2 2 cial cases of n, Yu and Gong [25] considered the concrete while the Walsh spectra of semi-bent functions belongs to constructions of bent functions of the form (1) and presented n+2 f0; ±2 2 g. For an odd integer n, the Walsh spectra of some enumeration results. n+1 semi-bent functions belongs to f0; ±2 2 g. Khoo, Gong and Hu and Feng [10] generalized results of Ma, Lee and Zhang Stinson [13], [14] considered the quadratic Boolean function [17] and studied the quadratic Boolean functions of the form − of the form − m 2 n 1 X2 X2 ei n i n 1+2 n=2 1+2 2 n 1+2 f(x) = ciT r (βx ) + T r (βx ); (2) f(x) = ciT r1 (x ); 1 1 i=1 i=1 2 2 e C. Tang is with School of Mathematics and Information, China West where ci GF (2), n = em, m is even and β GF (2 ). Normal University, Sichuan Nanchong, 637002, China. e-mail: tangchunming- They obtained that f(x) is bent if and only if [email protected] Y. Qi is with LMAM, School of Mathematical Sciences, Peking University, gcd(c(x); xm + 1) = 1; Beijing, 100871, and Aisino corporation Inc., Beijing, 100097, China P m−2 M. Xu are with LMAM, School of Mathematical Sciences, Peking Univer- 2 i m−i m=2 sity, Beijing, 100871, China where c(x) = i=1 ci(x +x )+x . Further, they pre- sented the enumerations of bent functions for some specified 2 e−1 ei e m. Note that β 2 GF (2e), then (β2 )1+2 = β2 = β. The When n is odd, f(x) can be represented by function f(x) of the form (2) satisfies that n−1 X2 i m−2 n 1+2 2 f(x) = T r (cix ); (6) X − − n 1 e 1 ei n=2 e 1 2 n 2 1+2 2 1+2 i=0 f(x) = ciT r1 ((β x) ) + T r1 ((β x) ); i=1 n where ci 2 GF (2 ). − n 2e 1 For a Boolean function f(x) over GF (2 ), the Hadamard where ci 2 GF (2). From the transformation x 7−! β x, a transform is defined by bent function of the form (2) is changed into a bent function X f(x)+T rn(λx) n of the form (1). Actually, (2) does not introduce new bent f^(λ) = (−1) 2 ; λ 2 GF (2 ): functions. x2GF (2n) In this paper, we first consider quadratic Boolean functions of the form For a quadratic Boolean function f(x) of the form (5) or (6), the distribution of the Hadamard transform can be described n −1 X2 n 1+2i n=2 1+2n=2 by the bilinear form f(x) = T r1 (cix ) + T r1 (cn=2x ); (3) i=1 Qf (x; y) = f(x + y) + f(x) + f(y): (7) 2 n ≤ ≤ n − 2 n=2 where ci GF (2 ) for 1 i 2 1 and cn=2 GF (2 ). For the bilinear form Qf , define And we study the bentness of these functions from some f 2 n 8 2 n g specific linearized polynomials. Further, we generalize results Kf = x GF (2 ): Qf (x; y) = 0; y GF (2 ) (8) in [10], [17] and study the bentness of quadratic Boolean and kf = dimGF (2)(Kf ). Then 2j(n − kf ). The distribution functions of the form of the Hadamard transform values of f^(λ) is given in the m −1 X2 following theorem [11]. n 1+2ei n=2 1+2n=2 f(x) = T r1 (cix ) + T r1 (cm=2x ); (4) Theorem 2.1: Let f(x) be a quadratic Boolean function of i=1 the form (5) or (6) and kf = dimGF (2)(Kf ), where Kf is e defined in (8). The distribution of the Hadamard transform where ci 2 GF (2 ). Further, we gives some examples of new values of f(x) is given by bent functions. And we construct new quadratic bent functions 8 from known quadratic bent functions. Finally, we presents > n n−kf <>0; 2 − 2 times − enumerations of bent functions of the form (4) for the case n+kf n kf ^ n−kf −1 −1 v0 r f(λ) = 2 2 ; 2 + 2 2 times m = 2 p and gcd(e; p−1) = 1, where v0 > 0, r > 0, p is an > > n+k n−k : f n−k −1 f −1 odd prime satisfying ordp(2) = p − 1 or ordp(2) = (p − 1)=2 −2 2 ; 2 f − 2 2 times: ((p − 1)=2 is odd). The rest of the paper is organized as follows: Section 2 Bent functions as an important class of Boolean functions are introduces some notations and backgrounds. Section 3 gives defined below. the description of bentness of quadratic Boolean functions Definition Let f(x) be a Boolean function from GF (2n) to considered in this paper and methods of constructing new bent GF (2). Then f(x) is called a bent function if for any λ 2 n n n functions. Section 4 enumerates the number of quadratic bent GF (2 ), f^(λ) 2 f2 2 ; −2 2 g. functions for special n. Finally, Section 5 makes a conclusion Bent functions only exist in the case for even n. From Theorem for this paper. 2.1, the following result on bent functions is given below. Corollary 2.2: Let f(x) be a quadratic function of the form II. PRELIMINARIES n (5) over GF (2 ), then f(x) is bent if and only if Kf = f0g, n In this section, some notations are given first. Let GF (2 ) where Kf is defined in (8).