In this paper, by employing the AGW criterion and determining the number of solutions to some equations over finite fields, we further investigate nine classes of permutation polynomials over $ \mathbb{F}_{p^n} $ with the form $ (x^{p^m}-x+\delta)^{s_1}+(x^{p^m}-x+\delta)^{s_2}+x $ and propose five classes of complete permutation polynomials over $ \mathbb{F}_{p^{2m}} $ with the form $ ax^{p^m}+bx+h(x^{p^m}-x) $.
Citation: Qian Liu, Jianrui Xie, Ximeng Liu, Jian Zou. Further results on permutation polynomials and complete permutation polynomials over finite fields[J]. AIMS Mathematics, 2021, 6(12): 13503-13514. doi: 10.3934/math.2021783
In this paper, by employing the AGW criterion and determining the number of solutions to some equations over finite fields, we further investigate nine classes of permutation polynomials over $ \mathbb{F}_{p^n} $ with the form $ (x^{p^m}-x+\delta)^{s_1}+(x^{p^m}-x+\delta)^{s_2}+x $ and propose five classes of complete permutation polynomials over $ \mathbb{F}_{p^{2m}} $ with the form $ ax^{p^m}+bx+h(x^{p^m}-x) $.
| [1] |
A. Akbary, D. Ghioca, Q. Wang, On constructing permutations of finite fields, Finite Fields Appl., 17 (2011), 51–67. doi: 10.1016/j.ffa.2010.10.002
|
| [2] |
C. S. Ding, J. Yuan, A family of skew Hadamard difference sets, J. Comb. Theory A, 113 (2006), 1526–1535. doi: 10.1016/j.jcta.2005.10.006
|
| [3] | C. S. Ding, Z. C. Zhou, Binary cyclic codes from explicit polynomials over GF($2^m$), Discrete Math., 324 (2014), 76–89. |
| [4] |
X. D. Hou, Permutation polynomials over finite fields $-$ A survey of recent advances, Finite Fields Appl., 32 (2015), 82–119. doi: 10.1016/j.ffa.2014.10.001
|
| [5] |
Y. Laigle-Chapuy, Permutation polynomials and applications to coding theory, Finite Fields Appl., 13 (2007), 58–70. doi: 10.1016/j.ffa.2005.08.003
|
| [6] |
L. S. Li, C. Y. Li, C. L. Li, X. Y. Zeng, New classes of complete permutation polynomials, Finite Fields Appl., 55 (2019), 177–201. doi: 10.1016/j.ffa.2018.10.001
|
| [7] |
K. Q. Li, L. J. Qu, Y. Zhou, A link between two classes of permutation polynomials, Finite Fields Appl., 63 (2020), 101641. doi: 10.1016/j.ffa.2020.101641
|
| [8] |
L. S. Li, S. Wang, C. Y. Li, X. Y. Zeng, Permutation polynomials $(x^{p^m}-x+\delta)^s_1+(x^{p^m}-x+\delta)^s_2+x$ over $ \mathbb{F}_{p^n}$, Finite Fields Appl., 51 (2018), 31–61. doi: 10.1016/j.ffa.2018.01.003
|
| [9] | R. Lidl, H. Niederreiter, Finite fields, Cambridge University Press, Cambridge, 1997. |
| [10] |
Q. Liu, Y. J. Sun, W. G. Zhang, Some classes of permutation polynomials over finite fields with odd characteristic, AAECC, 29 (2018), 409–431. doi: 10.1007/s00200-018-0350-6
|
| [11] | G. L. Mullen, Permutation polynomials over finite fields, In: Finite fields, coding theory, and advances in communications and computing, Marcel Dekker, 141 (1993), 131–151. |
| [12] | G. L. Mullen, D. Pannario, Handbook of Finite Fields, Boca Raton: Taylor & Francis, 2013. |
| [13] |
H. Niederreiter, K. H. Robinson, Complete mappings of finite fields, J. Aust. Math. Soc., 33 (1982), 197–212. doi: 10.1017/S1446788700018346
|
| [14] |
J. Schwenk, K. Huber, Public key encryption and digital signatures based on permutation polynomials, Electron. Lett., 34 (1998), 759–760. doi: 10.1049/el:19980569
|
| [15] |
Z. R. Tu, X. Y. Zeng, C. L. Li, T. Helleseth, Permutation polynomials of the form $(x^{p^m}-x+\delta)^s+L(x)$, Finite Fields Appl., 34 (2015), 20–35. doi: 10.1016/j.ffa.2015.01.002
|
| [16] |
L. B. Wang, B. F. Wu, Z. J. Liu, Further results on permutation polynomials of the form $(x^{p^m}-x+\delta)^s+L(x)$ over $ \mathbb{F}_{p^2m}$, Finite Fields Appl., 44 (2017), 92–112. doi: 10.1016/j.ffa.2016.11.007
|
| [17] | G. F. Wu, N. L, T. Helleseth, Y. Q. Zhang, Some classes of complete permutation polynomials over $ \mathbb{F}_q$, Sci. China Math., 58 (2015), 1–14. |
| [18] |
G. K. Xu, X. W. Cao, Complete permutation polynomials over finite fields of odd characteristic, Finite Fields Appl., 31 (2015), 228–240. doi: 10.1016/j.ffa.2014.08.002
|
| [19] |
X. F. Xu, C. L. Li, X. Y. Zeng, T. Helleseth, Constructions of complete permutation polynomials, Des. Codes Cryptogr., 86 (2018), 2869–2892. doi: 10.1007/s10623-018-0480-7
|
| [20] |
X. Y. Zeng, X. S. Zhu, N. Li, X. P. Liu, Permutation polynomials over $ \mathbb{F}_{2^n}$ of the form $(x^{2^i}+x+\delta)^s_1+(x^{2^i}+x+\delta)^s_2+x$, Finite Fields Appl., 47 (2017), 256–268. doi: 10.1016/j.ffa.2017.06.012
|
| [21] |
Z. B. Zha, L. Hu, Some classes of permutation polynomials of the form $(x^{p^m}-x+\delta)^s+x$ over $ \mathbb{F}_{p^2m}$, Finite Fields Appl., 40 (2016), 150–162. doi: 10.1016/j.ffa.2016.04.003
|
| [22] |
D. B. Zheng, Z. Chen, More classes of permutation polynomial of the form $(x^{p^m}-x+\delta)^s+L(x)$, AAECC, 28 (2017), 215–223. doi: 10.1007/s00200-016-0305-8
|
| [23] |
D. B. Zheng, M. Yuan, L. Yu, Two types of permutation polynomials with sepcial forms, Finite Fields Appl., 56 (2019), 1–16. doi: 10.1016/j.ffa.2018.10.008
|
Qian Liu, Jianrui Xie, Ximeng Liu, Jian Zou. Further results on permutation polynomials and complete permutation polynomials over finite fields[J]. AIMS Mathematics, 2021, 6(12): 13503-13514. doi: 10.3934/math.2021783
DownLoad: