Vectorial Bent Functions From Multiple Terms Trace Functions
In this paper, we provide necessary and sufficient conditions for a function of the form F(x)=Trk<sup>2k</sup>(Σi=1<sup>t</sup>aix<sup>ri(2k</sup>-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 (2<sup>k</sup>+1)th primitive roots of unity in GF(2<sup>2k</sup>). More precisely, for this group of cardinality 2<sup>k</sup>+1 given by U={u ∈ GF(2<sup>2k</sup>):u<sup>2k</sup>+1=1}, it is shown that the property of being vectorial bent implies that Im(F)=GF(2<sup>k</sup>)∪{0}, if F is evaluated on U, that is, F(u) takes all possible values of GF(2<sup>k</sup>)* 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 Trk<sup>2k</sup>(λx<sup>r(2k</sup>-1)) is never a vectorial bent function.