Paper 2013/405

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

Chunming Tang, Yanfeng Qi, and Maozhi Xu

Abstract

In this paper, we first discuss the bentness of a large class of quadratic Boolean functions in polynomial form $f(x)=\sum _{i=1}^{\frac{n}{2}-1}T{r}_1^n(c_i{x}^{1+2^i})+T{r}_1^{n/2}(c_{n/2}{x}^{1+2^{n/2}})$, where $c_i\in GF(2^n)$ for $1\le i\le \frac{n}{2}-1$ 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 $$ , where $$, $$ and $$ 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: $$ and $$, we present the enumeration of quadratic bent functions.