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**2008/512**

**Title:**

A new class of Bent functions in Polynomial Forms

**Authors:**
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.

**ePrint:**
https://eprint.iacr.org/2008/512

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