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

*Chunming Tang, Yanfeng Qi, Maozhi Xu*

**Abstract: **In this paper, we first discuss the bentness of a large class of quadratic Boolean functions in polynomial form
$f(x)=\sum_{i=1}^{\frac{n}{2}-1}Tr^n_1(c_ix^{1+2^i})+ Tr_1^{n/2}(c_{n/2}x^{1+2^{n/2}})$, 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.

**Category / Keywords: **Bent function, Boolean function, linearized permutation polynomial,cyclotomic polynomial, semi-bent function

**Date: **received 19 Jun 2013

**Contact author: **tangchunmingmath at 163 com

**Available format(s): **PDF | BibTeX Citation

**Version: **20130620:112506 (All versions of this report)

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