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 $f(x)=\text{tr}o(s_1)(a{x}^{s_1})+\text{tr}o(s_2)(b{x}^{s_2})$ 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 $$ and $$, where $$ and $$ provide the construction of new infinite class of bent functions over $$ with maximum algebraic degree. For $$ odd, we give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums of the corresponding coefficients. For $$ even, we give a necessary condition in terms of these Kloosterman sums.