Constructing permutation polynomials is a hot topic in finite fields. Recently, huge kinds of permutation polynomials over $ {\bf F}_{q^2} $ have been studied. In this paper, by using AGW criterion and piecewise method, we construct several classes of permutation polynomials over $ {\bf F}_{q^3} $ of the forms similar to $ (x^{q^2}+x^q+x+\delta)^{\frac{q^{3}-1}{d}+1}+L(x) $, for $ d = 2, 3, 4, 6, $ where $ L(x) $ is a linearized polynomial over $ {\bf F}_{q} $.
Citation: Xiaoer Qin, Li Yan. Some specific classes of permutation polynomials over $ {\textbf{F}}_{q^3} $[J]. AIMS Mathematics, 2022, 7(10): 17815-17828. doi: 10.3934/math.2022981
Constructing permutation polynomials is a hot topic in finite fields. Recently, huge kinds of permutation polynomials over $ {\bf F}_{q^2} $ have been studied. In this paper, by using AGW criterion and piecewise method, we construct several classes of permutation polynomials over $ {\bf F}_{q^3} $ of the forms similar to $ (x^{q^2}+x^q+x+\delta)^{\frac{q^{3}-1}{d}+1}+L(x) $, for $ d = 2, 3, 4, 6, $ where $ L(x) $ is a linearized polynomial over $ {\bf F}_{q} $.
| [1] |
A. Akbary, D. Ghioca, Q. Wang, On constructing permutations of finite fields, Finite Fields Appl., 17 (2011), 51–67. https://doi.org/10.1016/j.ffa.2010.10.002 doi: 10.1016/j.ffa.2010.10.002
|
| [2] |
C. S. Ding, Q. Xiang, J. Yuan, P. Z. Yuan, Explicit classes of permutation polynomials of $F_{3^3m}$, Sci. China Ser. A: Math., 52 (2009), 639–647. https://doi.org/10.1007/s11425-008-0142-8 doi: 10.1007/s11425-008-0142-8
|
| [3] |
X. D. Hou, Permutation polynomials over finite fields–A survey of recent advances, Finite Fields Appl., 32 (2015), 82–119. https://doi.org/10.1016/j.ffa.2014.10.001 doi: 10.1016/j.ffa.2014.10.001
|
| [4] |
T. Helleseth, V. Zinoviev, New Kloosterman sums identities over ${{\textbf{F}}}_{2^m}$ for all $m$, Finite Fields Appl., 9 (2003), 187–193. https://doi.org/10.1016/S1071-5797(02)00028-X doi: 10.1016/S1071-5797(02)00028-X
|
| [5] |
K. Q. Li, L. J. Qu, Q. Wang, Compositional inverses of permutation polynomials of the form $x^rh(x^s)$ over finite fields, Cryptogr. Commun., 11 (2019), 279–298. https://doi.org/10.1007/s12095-018-0292-7 doi: 10.1007/s12095-018-0292-7
|
| [6] |
N. Li, T. Helleseth, X. H. Tang, Further results on a class of permutation polynomials over finite fields, Finite Fields Appl., 22 (2013), 16–23. https://doi.org/10.1016/j.ffa.2013.02.004 doi: 10.1016/j.ffa.2013.02.004
|
| [7] | R. Lidl, H. Niederreiter, Finite fields, 2 Eds, Cambridge: Cambridge University Press, 1997. |
| [8] |
Q. Liu, Y. J. Sun, W. G. Zhang, Some classes of permutation polynomials over finite fields with odd characteristic, AAECC, 29 (2018), 409–431. https://doi.org/10.1007/s00200-018-0350-6 doi: 10.1007/s00200-018-0350-6
|
| [9] | G. L. Mullen, D. Panario, Handbook of finite fields, Chapman and Hall/CRC, 2013. https://doi.org/10.1201/b15006 |
| [10] | G. L. Mullen, Q. Wang, Permutation polynomials in one variable, In: Handbook of finite fields, Chapman and Hall/CRC, 2013,215–229. |
| [11] |
X. E. Qin, S. F. Hong, Constructing permutation polynomials over finite fields, Bull. Aust. Math. Soc., 89 (2014), 420–430. https://doi.org/10.1017/S0004972713000646 doi: 10.1017/S0004972713000646
|
| [12] |
X. E. Qin, G. Y. Qian, S. F. Hong, New results on permutation polynomials over finite fields, Int. J. Number Theory, 11 (2015), 437–449. https://doi.org/10.1142/S1793042115500220 doi: 10.1142/S1793042115500220
|
| [13] |
Z. R. Tu, X. Y. Zeng, Y. P. Jiang, Two classes of permutation polynomials having the form $(x^{2^m}+x +\delta)^s+x$, Finite Fields Appl., 31 (2015), 12–24. https://doi.org/10.1016/j.ffa.2014.09.005 doi: 10.1016/j.ffa.2014.09.005
|
| [14] | Q. Wang, Polynomials over finite fields: An index approach, In: Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications, 2019,319–348. https://doi.org/10.1515/9783110642094-015 |
| [15] | Y. P. Wang, W. G. Zhang, D. Bartoli, Q. Wang, Permutation polynomials and complete permutation polynomials over ${{\textbf{F}}}_{q^3}$, 2018, arXiv: 1806.05712v1. |
| [16] |
P. Z. Yuan, C. S. Ding, Permutation polynomials over finite fields from a powerful lemma, Finite Fields Appl., 17 (2011), 560–574. https://doi.org/10.1016/j.ffa.2011.04.001 doi: 10.1016/j.ffa.2011.04.001
|
| [17] |
P. Z. Yuan, C. S. Ding, Further results on permutation polynomials over finite fields, Finite Fields Appl., 27 (2014), 88–103. https://doi.org/10.1016/j.ffa.2014.01.006 doi: 10.1016/j.ffa.2014.01.006
|
| [18] |
P. Z. Yuan, C. S. Ding, Permutation polynomials of the form $L(x) +S_a^2k+S_b^2k$ over ${{\textbf{F}}}_{q^3k}$, Finite Fields Appl., 29 (2014), 106–117. https://doi.org/10.1016/j.ffa.2014.04.004 doi: 10.1016/j.ffa.2014.04.004
|
| [19] |
P. Z. Yuan, Y. B. Zheng, Permutation polynomials from piecewise functions, Finite Fields Appl., 35 (2015), 215–230. https://doi.org/10.1016/j.ffa.2015.05.001 doi: 10.1016/j.ffa.2015.05.001
|
| [20] |
Z. B. Zha, L. Hu, Z. Z. Zhang, New results on permutation polynomials of the form $(x^{p^m}-x+\delta)^s +x^{p^m}+x$ over ${{\textbf{F}}}_{p^2m}$, Cryptogr. Commun., 10 (2018), 567–578. https://doi.org/10.1007/s12095-017-0234-9 doi: 10.1007/s12095-017-0234-9
|
| [21] |
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. https://doi.org/10.1007/s00200-016-0305-8 doi: 10.1007/s00200-016-0305-8
|
| [22] |
Y. B. Zheng, P. Z. Yuan, D. Y. Pei, Large classes of permutation polynomials over ${{\textbf{F}}}_{ q^2}$, Des. Codes Cryptogr., 81 (2016), 505–521. https://doi.org/10.1007/s10623-015-0172-5 doi: 10.1007/s10623-015-0172-5
|
Xiaoer Qin, Li Yan. Some specific classes of permutation polynomials over $ {\textbf{F}}_{q^3} $[J]. AIMS Mathematics, 2022, 7(10): 17815-17828. doi: 10.3934/math.2022981
DownLoad: