Classification of some quadrinomials over finite fields of odd characteristic

Özbudak, Ferruh
Gülmez Temür, Burcu
Journal Title
Journal ISSN
Volume Title
Finite Fields and Their Applications
In this paper, we completely determine all necessary and su cient conditions such that the polynomial f (x) = x3+axq+2+bx2q+1+cx3q , where a, b, c ∈ F∗ q , is a permutation quadrinomial of Fq2 over any nite eld of odd characteristic. This quadrinomial has been studied rst in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coe cients that results with new permutation quadrinomials, where char(Fq ) = 2 and nally, in [16], Li, Qu, Li and Chen proved that the su cient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial x3 +axq+2 +bx2q+1 +cx3q , where char(Fq ) = 3, 5 and a, b, c ∈ F∗q and proposed some new classes of permutation quadrinomials of Fq2 . In particular, in this paper we classify all ermutation polynomials of Fq2 of the form f (x) = x3 + axq+2 + bx2q+1 + cx3q , where a, b, c ∈ F∗q , over all nite elds of odd characteristic and obtain several new classes of such permutation quadrinomials.
Published by Finite Fields and Their Applications;; Ferruh Özbudak, Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey; Burcu Gülmez Temür, Department of Mathematics, Atılım University, Ankara, Turkey.
Permutation polynomials; Finite fields; Absolutely irreducible MSC 11T06; 11T71; 12E10