A new class of hyper-bent functions and Kloosterman sums

This paper is devoted to the characterization of hyper-bent functions. Several classes of hyper-bent functions have been studied, such as Charpin and Gong’s ∑ r∈R Tr1 (arx r(2m−1)) and Mesnager’s ∑ r∈R Tr1 (arx r(2m−1)) + Tr1(bx 2n−1 3 ), where R is a set of representations of the cyclotomic cosets modulo 2 + 1 of full size n and ar ∈ F2m . In this paper, we generalize their results and consider a class of Boolean functions of the form ∑ r∈R ∑2 i=0 Tr n 1 (ar,ix r(2−1)+ 2 n−1 3 ) + Tr 1(bx 2n−1 3 ), where n = 2m, m is odd, b ∈ F4, and ar,i ∈ F2n . With the restriction of ar,i ∈ F2m , we present the characterization of hyper-bentness of these functions with character sums. Further, we reformulate this characterization in terms of the number of points on hyper-elliptic curves. For some special cases, with the help of Kloosterman sums and cubic sums, we determine the characterization for some hyper-bent functions including functions with four, six and ten traces terms. Evaluations of Kloosterman sums at three general points are used in the characterization. Actually, our results can generalized to the general case: ar,i ∈ F2n . And we explain this for characterizing binomial, trinomial and quadrinomial hyper-bent functions. Index Terms Bent function, hyper-bent functions, Walsh-Hadamard transform, Dickson polynomial, Kloosterman sums