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 $$f(x)=\sum _{i=1}^{{n}/{2}-1}\mathrm {Tr}^n_1(c_ix^{1+2^i})+ \mathrm {Tr}_1^{n/2}(c_{n/2}x^{1+2^{n/2}})$$f(x)=∑i=1n/2-1Tr1n(cix1+2i)+Tr1n/2(cn/2x1+2n/2), where n is even, $$c_i\in \mathrm {GF}(2^n)$$ci∈GF(2n) for $$1\le i \le {n}/{2}-1$$1≤i≤n/2-1 and $$c_{n/2}\in \mathrm {GF}(2^{n/2})$$cn/2∈GF(2n/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 $$f(x)=\sum _{i=1}^{{m}/{2}-1}\mathrm {Tr}^n_1(c_ix^{1+2^{ei}})+ \mathrm {Tr}_1^{n/2}(c_{m/2}x^{1+2^{n/2}})$$f(x)=∑i=1m/2-1Tr1n(cix1+2ei)+Tr1n/2(cm/2x1+2n/2), where $$n=em$$n=em, m is even, and $$c_i\in \mathrm {GF}(2^e)$$ci∈GF(2e). The bentness of these functions is characterized and some methods for deriving new quadratic bent functions are given. Finally, when m and e satisfy some conditions, we determine the number of these quadratic bent functions.