Paper 2013/493

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 $f(x)=\sum _{i=1}^{\lfloor \frac{m-1}{2}\rfloor }T{r}_1^n(c_i{x}^{1+4^i})$ $(c_i\in {\mathbb{F}}_4$,$n=2m)$. 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 $f(x)=\sum _{i=1}^{\frac{p-1}{2}}T{r}_1^{2p}(c_i{x}^{1+4^i})$ over ${\mathbb{F}}_{2^{2p}}$ is a semi-bent function.