New Quadratic Bent Functions in Polynomial Forms with Coefficients in Extension Fields

In this paper, we first discuss the bentness of a large class of quadratic Boolean functions in polynomial form f(x) = ∑n 2 −1 i=1 Tr n 1 (cix 1+2i) + Tr n/2 1 (cn/2x 1+2n/2), where ci ∈ GF (2) for 1 ≤ i ≤ n 2 − 1 and cn/2 ∈ GF (2). The bentness of these functions can be connected with linearized permutation polynomials. Hence, methods for constructing quadratic bent functions are given. Further, we consider a subclass of quadratic Boolean functions of the form f(x) = ∑m 2 −1 i=1 Tr n 1 (cix 1+2ei) + Tr n/2 1 (cm/2x 1+2n/2) , where ci ∈ GF (2), n = em and m is even. The bentness of these functions are characterized and some methods for constructing new quadratic bent functions are given. Finally, for a special case: m = 20p and gcd(e, p− 1) = 1, we present the enumeration of quadratic bent functions.