Permutation polynomials of the form <i>X</i>+γTr(<i>X<sp>k</sp></i>)

Gohar Kyureghyan and Michael E. Zieve:
Permutation polynomials of the form X+γTr(X^k).
Contemporary Developments in Finite Fields and Their Applications, to appear.

(Prior to publication, this paper should be cited as arXiv:1603.01175.)

We exhibit nine sequences of permutation polynomials of the form X+γTr(X^k over the finite field F of order qⁿ, where γ ∈ F and Tr denotes the trace map from F to its order-q subfield. Computation of all permutation polynomials of this form over fields of size less than 5000 suggests that our examples comprise all examples except for some trivial cases and a few sporadic examples. Intriguingly, our proofs of the different sequences use various different methods, including a new variant of Dobbertin's method among others.

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