In this paper, we show that the maximum number of bent component functions of a vectorial function F : GF(2)n→ GF(2)nis 2n- 2n/2. We also show that it is very easy to construct such functions. However, it is a much more challenging task to find such functions in polynomial form F ∈ GF(2n)[x], where F has only a few terms. The only known power functions having such a large number of bent components are xd, where d = 2n/2+ 1. In this paper, we show that the binomials Fi(x) = x2i(x + x(2n/2)) also have such a large number of bent components, and these binomials are inequivalent to the monomials x(2n/2+1) if 0n/2+1). We also determine the complete Walsh spectrum of our functions when n/2 is odd and gcd(i, n/2) = 1.

In this paper, we provide necessary and sufficient conditions for a function of the form F(x)=Trk2k(Σi=1taixri(2k-1)) to be bent. Three equivalent statements, all of them providing both the necessary and sufficient conditions, are derived. In particular, one characterization provides an interesting link between the bentness and the evaluation of F on the cyclic group of the (2k+1)th primitive roots of unity in GF(22k). More precisely, for this group of cardinality 2k+1 given by U={u ∈ GF(22k):u2k+1=1}, it is shown that the property of being vectorial bent implies that Im(F)=GF(2k)∪{0}, if F is evaluated on U, that is, F(u) takes all possible values of GF(2k)* exactly once and the zero value is taken twice when u ranges over U. This condition is then reformulated in terms of the evaluation of certain elementary symmetric polynomials related to F, which in turn gives some necessary conditions on the coefficients ai (for binomial trace functions) that can be stated explicitly. Finally, we show that a bent trace monomial of Dillon's type Trk2k(λxr(2k-1)) is never a vectorial bent function.

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.