Designs, Codes and Cryptography

, Volume 81, Issue 3, pp 505–521 | Cite as

Large classes of permutation polynomials over \(\mathbb {F}_{q^2}\)



Permutation polynomials (PPs) of the form \((x^{q} -x + c)^{\frac{q^2 -1}{3}+1} +x\) over \(\mathbb {F}_{q^2}\) were presented by Li et al. (Finite Fields Appl 22:16–23, 2013). More recently, we have constructed PPs of the form \((x^{q} + bx + c)^{\frac{q^2 -1}{d}+1} -bx\) over \(\mathbb {F}_{q^2}\), where \(d=2, 3, 4, 6\) (Yuan and Zheng in Finite Fields Appl 35:215–230, 2015). In this paper we concentrate our efforts on the PPs of more general form
$$\begin{aligned} f(x)=(ax^{q} +bx +c)^r \phi \big ((ax^{q} +bx +c)^{(q^2 -1)/d}\big ) +ux^{q} +vx ~{\text {over}}\; \mathbb {F}_{q^2}, \end{aligned}$$
where \(a,b,c,u,v \in \mathbb {F}_{q^2}\), \(r \in \mathbb {Z}^{+}\), \(\phi (x)\in \mathbb {F}_{q^2}[x]\) 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 \(\mathbb {F}_{q^2}\) to that of verifying whether two more polynomials permute two subsets of \(\mathbb {F}_{q^2}\). As a consequence, we find a series of simple conditions for f(x) to be a PP of \(\mathbb {F}_{q^2}\). These results unify and generalize some known classes of PPs.


Permutation Finite field Commutative diagram AGW criterion 

Mathematics Subject Classification

11T06 11T71 



We are grateful to the two anonymous referees for useful comments and suggestions. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11371106, 11271142, 61363069) and the Guangdong Provincial Natural Science Foundation (Grant No. S2012010009942).


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Guangxi Key Laboratory of Trusted SoftwareGuilin University of Electronic TechnologyGuilinChina
  2. 2.School of Computer Science and EngineeringGuilin University of Electronic TechnologyGuilinChina
  3. 3.School of MathematicsSouth China Normal UniversityGuangzhouChina
  4. 4.School of Mathematics and Information ScienceGuangzhou UniversityGuangzhouChina
  5. 5.Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education InstitutesGuangzhou UniversityGuangzhouChina

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