Planarity of mappings x(Tr(x)-α/2x) on finite fields

Let q be a power of an odd prime, n >= 3 and Tr-n : F-qn -> F-q be the trace mapping. A mapping f = f (x) : F-qn -> F-qn is called planar (or perfect nonlinear) on F-qn if for any non-zero a is an element of F-qn, the difference mapping D (f,a) : F-qn -> F-qn is a permutation where for x is an element of F-qn, D (f,a)(x) = (x+a) - f(x). Kyureghyan and Ozbudak (2012) [8] considered the planarity of mappings f(n,alpha) (x) = x(Tr-n(x) - alpha/2x) on F-qn for alpha is an element of F-qn and proved that there is no planar f(n,alpha) 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 f(n,alpha) for n = 4, by using Kloosterman sums. Our proof also works for case n >= 5, so we present a new proof of the Kyureghyan-Ozbudak result. For case n = 3, we present an elementary proof of the first conjecture which says that there is no planar f(3,alpha) for alpha is an element of F-q\{2, 4}. (c) 2013 Elsevier Inc. All rights reserved.