Vectorial Hyperbent Trace Functions From the 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) = Tr_kⁿ (Σ_i=0(2^k) a_ixⁱ⁽(2^k)^-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(2ⁿ). 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_kⁿ (Σ_i=0^2k a_ixⁱ⁽(2^k)^-1)). Furthermore, a simple optimization method is proposed for selecting the interpolation polynomials that give rise to trace polynomials with a few nonzero coefficients.