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) ∈ F(2n)[x], has been an important research topic lately. We characterize a class of vectorial (hyper)bent functions of the form F(x) = Trkn(Σi=0(2k) aixi((2k)-1)), where n = 2k, in terms of finding an explicit expression for the coefficients aiso 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 (2n/2+ 1)th roots of unity in F(2n). We show that these interpolation polynomials can be chosen in exactly (2k+ 1)!2k-1ways and this is the exact number of vectorial hyperbent functions of the form Trkn(Σi=02kaixi((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.