5. Results on permutation polynomials of shape xt + γTrq n/q(xd)
Daniel Gerike, Gohar Kyureghyan · 2019
The maps of shapeT(x) = x t + γTr q n /q (x d ) combine in an interesting way the additive and multiplicative structures of F qn and serve as a source for maps with special properties required in different areas of applications. In this paper, we briefly survey known results on such permutations and continue their study. We prove that if T(x) is bijective on F q n then necessarily gcd(t, q n −1) = 1. We show that F(x) = x q2+q−1 + Tr q3/q (x) has very special properties on F q 3 by determining explicitly its iterates, the inverse map, the set of fixed points and its cycle structure.