Vectorial Hyperbent Trace Functions From the PSap Class-Their Exact Number and Specification

To identify and specify trace bent functions of the form Tr(P(x)), where P(x) is an element of F(2)n [x], has been an important research topic lately. We characterize a class of vectorial (hyper) bent functions of the form F(x) = Tr-k(n)(Sigma(2k)(i=0)a(i)x(i(2k-1))), where n = 2k, in terms of finding an explicit expression for the coefficients a(i) so that F is vectorial hyperbent. These coefficients only depend on the choice of the interpolating polynomial used in the Lagrange interpolation of the elements of U and some prespecified outputs, where U is the cyclic group of (2(n/2) + 1) th roots of unity in F(2)n. We show that these interpolation polynomials can be chosen in exactly (2(k) + 1)!2(k-1) ways and this is the exact number of vectorial hyperbent functions of the form Tr-k(n)(Sigma(2k)(i=0)a(i)x(i(2k-1))). Furthermore, a simple optimization method is proposed for selecting the interpolation polynomials that give rise to trace polynomials with a few nonzero coefficients.