Vectorial Hyperbent Trace Functions From the $\mathcal {PS}_{\rm ap}$ Class—Their Exact Number and Specification
To identify and specify trace bent functions of the form Tr(P(x)), where P(x) ∈ F(2<sup>n</sup>)[x], has been an important research topic lately. We characterize a class of vectorial (hyper)bent functions of the form F(x) = Tr<sub>k</sub><sup>n</sup> (Σ<sub>i=0(</sub>2<sup>k</sup>) a<sub>i</sub>x<sup>i(</sup>(2<sup>k</sup>)<sup>-1)</sup>), where n = 2k, in terms of finding an explicit expression for the coefficients a<sub>i</sub> 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<sup>n/2</sup> + 1)th roots of unity in F(2<sup>n</sup>). We show that these interpolation polynomials can be chosen in exactly (2<sup>k</sup> + 1)!2<sup>k-1</sup> ways and this is the exact number of vectorial hyperbent functions of the form Tr<sub>k</sub><sup>n</sup> (Σ<sub>i=0</sub><sup>2k</sup> a<sub>i</sub>x<sup>i(</sup>(2<sup>k</sup>)<sup>-1)</sup>). Furthermore, a simple optimization method is proposed for selecting the interpolation polynomials that give rise to trace polynomials with a few nonzero coefficients.