**A new class of semi-bent quadratic Boolean functions**

*Chunming Tang and Yanfeng Qi*

**Abstract: **In this paper, we present a new class of semi-bent quadratic Boolean functions of the form $f(x)=\sum_{i=1}^{\lfloor\frac{m-1}{2}\rfloor}Tr^n_1(c_ix^{1+4^{i}})$ $~(c_i\in \mathbb{F}_4$,$n=2m)$. We first characterize the semi-bentness of these quadratic Boolean functions. There exists semi-bent functions only when $m$ is odd. For the case: $m=p^r$, where $p$ is an odd prime with some conditions, we enumerate the semi-bent functions. Further, we give a simple characterization of semi-bentness for these functions with linear properties of $c_i$. In particular, for
a special case of $p$, any quadratic Boolean function $f(x)=\sum_{i=1}^{\frac{p-1}{2}}Tr^{2p}_1(c_ix^{1+4^{i}})$ over $\mathbb{F}_{2^{2p}}$ is a semi-bent function.

**Category / Keywords: **foundations / Semi-bent function, Boolean function, m-sequence, cyclotomic polynomial, bent function

**Date: **received 13 Aug 2013

**Contact author: **tangchunmingmath at 163 com

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

**Version: **20130815:072347 (All versions of this report)

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