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Article Complete characterization of a class of permutation trinomials in characteristic five(Cryptography and Communications, 2024-02-21) Grassl, Markus; Özbudak, Ferruh; Özkaya, Buket; Gülmez Temür, BurcuIn 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 .