New quadratic bent functions in polynomial forms with coefficients in extension fields

Abstract

In this paper, we first discuss the bentness of a large class of quadratic Boolean functions in polynomial form $$f(x)=\sum _{i=1}^{{n}/{2}-1}\mathrm {Tr}^n_1(c_ix^{1+2^i})+ \mathrm {Tr}_1^{n/2}(c_{n/2}x^{1+2^{n/2}})$$ , where n is even, $$c_i\in \mathrm {GF}(2^n)$$ for $$1\le i \le {n}/{2}-1$$ and $$c_{n/2}\in \mathrm {GF}(2^{n/2})$$ . 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 $$f(x)=\sum _{i=1}^{{m}/{2}-1}\mathrm {Tr}^n_1(c_ix^{1+2^{ei}})+ \mathrm {Tr}_1^{n/2}(c_{m/2}x^{1+2^{n/2}})$$ , where $$n=em$$ , m is even, and $$c_i\in \mathrm {GF}(2^e)$$ . 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.