Large classes of permutation polynomials over F q 2 .

Permutation polynomials (PPs) of the form (x − x + c) q2−1 3 +1 + x over Fq2 were presented by Li, Helleseth and Tang [Finite Fields Appl. 22 (2013) 16–23]. More recently, we have constructed PPs of the form (x + bx + c) q2−1 d +1 − bx over Fq2 , where d = 2, 3, 4, 6 [Finite Fields Appl. 35 (2015) 215–230]. In this paper we concentrate our efforts on the PPs of more general form f(x) = (ax + bx+ c)φ((ax + bx+ c) 2 ) + ux + vx over Fq2 , where a, b, c, u, v ∈ Fq2 , r ∈ Z , φ(x) ∈ Fq2 [x] and d is an arbitrary positive divisor of q 2 − 1. The key step is the construction of a commutative diagram with specific properties, which is the basis of the Akbary–Ghioca–Wang (AGW) criterion. By employing the AGW criterion two times, we reduce the problem of determining whether f(x) permutes Fq2 to that of verifying whether two more polynomials permute two subsets of Fq2 . As a consequence, we find a series of simple conditions for f(x) to be a PP of Fq2 . These results unify and generalize some known classes of PPs.