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
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
(Σ
_i=0(
2
^k
) a
_i
x
ⁱ⁽
(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(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
(Σ
_i=0
^2k
a
_i
x
ⁱ⁽
(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.