Permutations on finite fields with invariant cycle structure on lines

Abstract

We study the cycle structure of permutations F(x) = x+γ f (x) onFqn,where f : Fqn → Fq . We show that for a 1-homogeneous function f the cycle structure of F can be determined by calculating the cycle structure of certain induced mappings on parallel lines of γ Fq. Using this observation we describe explicitly the cycle structure of two families of permutations over Fq2 : x + γ Tr(x2q−1), where q ≡ −1 (mod 3) and γ ∈ Fq2 , with γ 3 = − 1 27 and x + γ Tr

x 22s−1+3·2s−1+1 3

, where q = 2s , s odd and γ ∈ Fq2 , with γ (q+1)/3 = 1.