**A new class of Bent functions in Polynomial Forms**

*Sihem Mesnager*

**Abstract: **This paper is a contribution to the construction of bent functions
having the form $f(x) = \tr {o(s_1)} (a x^ {s_1}) +
\tr {o(s_2)} (b x^{s_2})$ where $o(s_i$) denotes the cardinality of the cyclotomic
class of 2 modulo $2^n-1$ which contains $i$ and whose coefficients
$a$ and $b$ are, respectively in $F_{2^{o(s_1)}}$ and
$F_{2^{o(s_2)}}$. Many constructions of monomial bent functions are
presented in the literature but very few are known even in the
binomial case.

We prove that the exponents $s_1=2^{\frac n2}-1$ and $s_2={\frac {2^n-1}3}$, where $a\in\GF{n}$ and $ b\in\GF[4]{}$ provide the construction of new infinite class of bent functions over $\GF{n}$ with maximum algebraic degree. For $m$ odd, we give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums of the corresponding coefficients. For $m$ even, we give a necessary condition in terms of these Kloosterman sums.

**Category / Keywords: **Boolean function, Bent functions, Maximum nonlinearity, Walsh-Hadamard transformation, Kloosterman sums.

**Date: **received 3 Dec 2008, last revised 5 Dec 2008

**Contact author: **mesnager at math jussieu fr

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | BibTeX Citation

**Version: **20081209:072502 (All versions of this report)

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