Planarity of mappings x(Tr(x)−α2x) on finite fields

Let q be a power of an odd prime, n⩾3 and Trn:Fqn→Fq be the trace mapping. A mapping f=f(x):Fqn→Fqn is called planar (or perfect nonlinear) on Fqn if for any non-zero a∈Fqn, the difference mapping Df,a:Fqn→Fqn is a permutation where for x∈Fqn, Df,a(x)=f(x+a)−f(x). Kyureghyan and Özbudak (2012) [8] considered the planarity of mappings fn,α(x)=x(Trn(x)−α2x) on Fqn for α∈Fqn and proved that there is no planar fn,α for n⩾5. For the case n=3 and n=4, they raised three conjectures. In this paper we prove the third conjecture which says that there is no planar fn,α for n=4, by using Kloosterman sums. Our proof also works for case n⩾5, so we present a new proof of the Kyureghyan-Özbudak result. For case n=3, we present an elementary proof of the first conjecture which says that there is no planar f3,α for α∈Fq\{2,4}.