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 $f(x)=\sum_{i=1}^{\frac{n}{2}-1}Tr^n_1(c_ix^{1+2^i})+ Tr_1^{n/2}(c_{n/2}x^{1+2^{n/2}})$, where $c_i\in GF(2^n)$ for $1\leq i \leq \frac{n}{2}-1$ and $c_{n/2}\in GF(2^{n/2})$. 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 $f(x)=\sum_{i=1}^{\frac{m}{2}-1}Tr^n_1(c_ix^{1+2^{ei}})+ Tr_1^{n/2}(c_{m/2}x^{1+2^{n/2}})$ , where $c_i\in GF(2^e)$, $n=em$ 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: $m=2^{v_0}p^r$ and $gcd(e,p-1)=1$, we present the enumeration of quadratic bent functions