Paper 2008/512

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.