A 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 family Sigma(r is an element of R) Tr-1(n) (a(r)x(r(2m -1))) and Mesnager's family Sigma(r is an element of R) Tr-1(n) (a(r)x(r(2m -1))) + Tr-1(2) (bx(2n -1/3)). In this paper, we generalize these results by considering the following class of Boolean functions over F-2n : Sigma(r is an element of R) Sigma(2)(i=0) T r(1)(n) (a(r,i)x(r(2m-1)+2n-1/3i)) + T r(1)(2) (bx (2n-1/3)), where n = 2m, m is odd, b is an element of F-4, and a(r,i) is an element of F-2n. With the restriction of a(r,i) is an element of F-2m, we present a characterization of hyper-bentness of these functions in terms of crucial exponential sums. For some special cases, we provide explicit characterizations for some hyper-bent functions in terms of Kloosterman sums and cubic sums. Finally, we explain how our results on binomial, trinomial and quadrinomial hyper-bent functions can be generalized to the general case where the coefficients a(r,i) belong to the whole field F-2n.