Designs, Codes and Cryptography

, Volume 81, Issue 3, pp 505–521

# Large classes of permutation polynomials over $$\mathbb {F}_{q^2}$$

• Yanbin Zheng
• Pingzhi Yuan
• Dingyi Pei
Article

## Abstract

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.

## Keywords

Permutation Finite field Commutative diagram AGW criterion

11T06 11T71

## Notes

### Acknowledgments

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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