**A Family of $p$-ary Binomial Bent Functions**

*Dabin Zheng and Xiangyong Zeng and Lei Hu*

**Abstract: **For a prime $p$ with $p\equiv 3\,({\rm mod}\, 4)$ and an odd number
$m$, the Bentness of the $p$-ary binomial function $f_{a,b}(x)={\rm
Tr}_{1}^n(ax^{p^m-1})+{\rm Tr}_{1}^2(bx^{\frac{p^n-1}{4}})$ is
characterized, where $n=2m$, $a\in \bF_{p^n}^*$, and $b\in
\bF_{p^2}^*$. The necessary and sufficient conditions of
$f_{a,b}(x)$ being Bent are established respectively by an
exponential sum and two sequences related to $a$ and $b$. For the
special case of $p=3$, we further characterize the Bentness of the
ternary function $f_{a,b}(x)$ by the Hamming weight of a sequence.

**Category / Keywords: **foundations /

**Date: **received 17 Nov 2009

**Contact author: **dbzheng at gucas ac cn

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

**Version: **20091123:133356 (All versions of this report)

**Short URL: **ia.cr/2009/563

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