Complete characterization of a class of permutation trinomials in characteristic five


In this paper, we address an open problem posed by Bai and Xia in [2].We study polynomials of the form f (x) = x^4q+1 + λ_1x^5q + λ_2x^q+4 over the finite field F_5^k , which are not quasimultiplicative equivalent to any of the known permutation polynomials in the literature. We find necessary and sufficient conditions on λ1, λ2 ∈ F_5^k so that f (x) is a permutation monomial, binomial, or trinomial of F_5^2k .


Open Access; Published by Cryptography and Communications;; Markus Grassl, International Centre for Theory of Quantum Technologies, University of Gdansk, Gdansk, Poland; Ferruh Özbudak, FENS, Sabancı University, Ístanbul, Turkey; Buket Özkaya, 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