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Published in: Applicable Algebra in Engineering, Communication and Computing 2/2022

11-06-2020 | Original Paper

Some classes of permutation polynomials of the form \(b(x^q+ax+\delta )^{\frac{i(q^2-1)}{d}+1}+c(x^q+ax+\delta )^{\frac{j(q^2-1)}{d}+1}+L(x)\) over \( {{{\mathbb {F}}}}_{q^2}\)

Authors: Danyao Wu, Pingzhi Yuan

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 2/2022

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Abstract

Let \(q\) be a prime power and \( {{{\mathbb {F}}}}_q\) be a finite field with \(q\) elements. In this paper, we employ the AGW criterion to investigate the permutation behavior of some polynomials of the form
$$\begin{aligned} b(x^q+ax+\delta )^{1+\frac{i(q^2-1)}{d}}+c(x^q+ax+\delta )^{1+\frac{j(q^2-1)}{d}}+L(x) \end{aligned}$$
over \( {{{\mathbb {F}}}}_{q^2}\) with \(a^{1+q}=1, q\equiv \pm 1\pmod {d}\) and \(L(x)=-ax\) or \(x^q-ax.\) Accordingly, we also present the permutation polynomials of the form \(b(x^q+ax+\delta )^s-ax\) by letting \(c=0\) and choosing some special exponent s, which generalize some known results on permutation polynomials of this form.
Literature
1.
go back to reference Akbary, A., Ghioca, D., Wang, Q.: On constructing permutations of finite fields. Finite Fields Appl. 17, 51–67 (2011) MathSciNetCrossRef Akbary, A., Ghioca, D., Wang, Q.: On constructing permutations of finite fields. Finite Fields Appl. 17, 51–67 (2011) MathSciNetCrossRef
2.
go back to reference Berlekamp, E.R., Rumsey, H., Solomon, G.: On the solution of algebraic equations over finite fields. Inf. Control 10(6), 553–564 (1967) MathSciNetCrossRef Berlekamp, E.R., Rumsey, H., Solomon, G.: On the solution of algebraic equations over finite fields. Inf. Control 10(6), 553–564 (1967) MathSciNetCrossRef
3.
4.
go back to reference Ding, C., Helleseth, T.: Optimal ternary cyclic codes from monomials. IEEE Trans. Inf. Theory 59, 5898–5904 (2013) MathSciNetCrossRef Ding, C., Helleseth, T.: Optimal ternary cyclic codes from monomials. IEEE Trans. Inf. Theory 59, 5898–5904 (2013) MathSciNetCrossRef
5.
6.
go back to reference Gupta, R., Sharma, R.K.: Further results on permutation polynomials of the form \((x^{p^m}-x+\delta )^s+x \) over \(\mathbb{F}_{p^{2m}}\). Finite Fields Appl. 50, 196–208 (2018) MathSciNetCrossRef Gupta, R., Sharma, R.K.: Further results on permutation polynomials of the form \((x^{p^m}-x+\delta )^s+x \) over \(\mathbb{F}_{p^{2m}}\). Finite Fields Appl. 50, 196–208 (2018) MathSciNetCrossRef
7.
go back to reference Helleset, T., Zinoviev, V.: New Kloostermans sums identities over \({\mathbb{F}}_{2^m}\) for all \(m\). Finite Fields Appl. 9(2), 187–193 (2003) MathSciNetCrossRef Helleset, T., Zinoviev, V.: New Kloostermans sums identities over \({\mathbb{F}}_{2^m}\) for all \(m\). Finite Fields Appl. 9(2), 187–193 (2003) MathSciNetCrossRef
8.
go back to reference Hou, X.: A survey of permutation binomials and trinomials over finite fields. In: Proceedings of the 11th International Conference on Finite Fields and Their Applications, Contemporary Mathematics, Magdeburg, Germany, July 2013, 632 AMS 177–191 (2015) Hou, X.: A survey of permutation binomials and trinomials over finite fields. In: Proceedings of the 11th International Conference on Finite Fields and Their Applications, Contemporary Mathematics, Magdeburg, Germany, July 2013, 632 AMS 177–191 (2015)
9.
go back to reference Laigle-Chapuy, Y.: Permutation polynomial and applications to coding theory. Finite Fields Appl. 13, 58–70 (2007) MathSciNetCrossRef Laigle-Chapuy, Y.: Permutation polynomial and applications to coding theory. Finite Fields Appl. 13, 58–70 (2007) MathSciNetCrossRef
10.
go back to reference Li, N., Helleseth, T., Tang, X.: Further results on a class of permutation polynomials over finite fields. Finite Fields Appl. 22, 16–23 (2013) MathSciNetCrossRef Li, N., Helleseth, T., Tang, X.: Further results on a class of permutation polynomials over finite fields. Finite Fields Appl. 22, 16–23 (2013) MathSciNetCrossRef
11.
go back to reference Li, K., Qu, L., Chen, X.: New classes of permutation binomials and permutation trinomials over finite fields. Finite Fields Appl. 43, 69–85 (2017) MathSciNetCrossRef Li, K., Qu, L., Chen, X.: New classes of permutation binomials and permutation trinomials over finite fields. Finite Fields Appl. 43, 69–85 (2017) MathSciNetCrossRef
12.
go back to reference Li, L., Wang, S., Li, C., Zeng, X.: 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, 31–61 (2018) MathSciNetCrossRef Li, L., Wang, S., Li, C., Zeng, X.: 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, 31–61 (2018) MathSciNetCrossRef
13.
go back to reference Li, Z., Wang, M., Wu, J., Zhu, X.: Some new forms of permutation polynomials based on the AGW criterion. Finite Fields Appl. 61, 101584 (2020) MathSciNetCrossRef Li, Z., Wang, M., Wu, J., Zhu, X.: Some new forms of permutation polynomials based on the AGW criterion. Finite Fields Appl. 61, 101584 (2020) MathSciNetCrossRef
14.
go back to reference Lidl, R., Niederreiter, H.: Finite fields. In: Encyclopedia of Mathematics and its Applications, 2 edn, vol. 20. Cambridge University Press, Cambridge (1997) Lidl, R., Niederreiter, H.: Finite fields. In: Encyclopedia of Mathematics and its Applications, 2 edn, vol. 20. Cambridge University Press, Cambridge (1997)
15.
go back to reference Mullen, G.L.: Permutation polynomials over finite fields. In: Proceedings of Conference on Finite Fields and Their Applications, Lecture Notes in Pure and Applied Mathematics, vol. 141. Marcel Dekker, pp. 131–151 (1993) Mullen, G.L.: Permutation polynomials over finite fields. In: Proceedings of Conference on Finite Fields and Their Applications, Lecture Notes in Pure and Applied Mathematics, vol. 141. Marcel Dekker, pp. 131–151 (1993)
16.
go back to reference Rivest, R.L., Shamir, A., Adelman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21, 120–126 (1978) MathSciNetCrossRef Rivest, R.L., Shamir, A., Adelman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21, 120–126 (1978) MathSciNetCrossRef
17.
go back to reference Schwenk, J., Huber, K.: Public key encryption and digital signatures based on permutation polynomials. Electron. Lett. 34, 759–760 (1998) CrossRef Schwenk, J., Huber, K.: Public key encryption and digital signatures based on permutation polynomials. Electron. Lett. 34, 759–760 (1998) CrossRef
18.
go back to reference Tu, Z., Zeng, X., Jiang, X.: Two classes of permutation polynomials having the form \((x^{2^m}+x+\delta )^s+x\). Finite Fields Appl. 31, 12–24 (2015) MathSciNetCrossRef Tu, Z., Zeng, X., Jiang, X.: Two classes of permutation polynomials having the form \((x^{2^m}+x+\delta )^s+x\). Finite Fields Appl. 31, 12–24 (2015) MathSciNetCrossRef
19.
go back to reference Tu, Z., Zeng, X., Li, C., Helleseth, T.: Permutation polynomials of the from \((x^{p^m}-x+\delta )^s+L(x) \) over the finite field \({\mathbb{F}}_{p^{2m}}\) of odd characteristic. Finite Fields Appl. 34, 20–35 (2015) MathSciNetCrossRef Tu, Z., Zeng, X., Li, C., Helleseth, T.: Permutation polynomials of the from \((x^{p^m}-x+\delta )^s+L(x) \) over the finite field \({\mathbb{F}}_{p^{2m}}\) of odd characteristic. Finite Fields Appl. 34, 20–35 (2015) MathSciNetCrossRef
20.
go back to reference Wang, L., Wu, B., Liu, Z.: Further results on permutation polynomials of the form \((x^{p}-x+\delta )^s+x \) over \({\mathbb{F}}_{p^{2m}}\). Finite Fields Appl. 44, 92–112 (2017) MathSciNetCrossRef Wang, L., Wu, B., Liu, Z.: Further results on permutation polynomials of the form \((x^{p}-x+\delta )^s+x \) over \({\mathbb{F}}_{p^{2m}}\). Finite Fields Appl. 44, 92–112 (2017) MathSciNetCrossRef
21.
go back to reference Wang, L., Wu, B.: General constructions of permutation polynomials of the form \((x^{2^m}+x+\delta )^{i(2^m-1)+1}+x \) over \({\mathbb{F}}_{2^{2m}}\). Finite Fields Appl. 52, 137–155 (2018) MathSciNetCrossRef Wang, L., Wu, B.: General constructions of permutation polynomials of the form \((x^{2^m}+x+\delta )^{i(2^m-1)+1}+x \) over \({\mathbb{F}}_{2^{2m}}\). Finite Fields Appl. 52, 137–155 (2018) MathSciNetCrossRef
22.
go back to reference Yuan, J., Ding, C.: Four classes of permutation polynomials over \({\mathbb{F}}_{2^m}\). Finite Fields Appl. 13(4), 869–876 (2007) MathSciNetCrossRef Yuan, J., Ding, C.: Four classes of permutation polynomials over \({\mathbb{F}}_{2^m}\). Finite Fields Appl. 13(4), 869–876 (2007) MathSciNetCrossRef
23.
go back to reference Yuan, J., Ding, C., Wang, H., Pieprzyk, J.: Permutation polynomials of the form \((x^p-x+\delta )^s+L(x)\). Finite Fields Appl. 14(2), 482–493 (2008) MathSciNetCrossRef Yuan, J., Ding, C., Wang, H., Pieprzyk, J.: Permutation polynomials of the form \((x^p-x+\delta )^s+L(x)\). Finite Fields Appl. 14(2), 482–493 (2008) MathSciNetCrossRef
24.
go back to reference Yuan, P., Ding, C.: Permutation polynomials over finite fields from a powerful lemma. Finite Fields Appl. 17, 560–574 (2011) MathSciNetCrossRef Yuan, P., Ding, C.: Permutation polynomials over finite fields from a powerful lemma. Finite Fields Appl. 17, 560–574 (2011) MathSciNetCrossRef
25.
go back to reference Yuan, J., Ding, C.: Four classes of permutation polynomials of \({\mathbb{F}}_{2^m}\). Finite Fields Appl. 13(4), 869–876 (2007) MathSciNetCrossRef Yuan, J., Ding, C.: Four classes of permutation polynomials of \({\mathbb{F}}_{2^m}\). Finite Fields Appl. 13(4), 869–876 (2007) MathSciNetCrossRef
26.
27.
go back to reference Zeng, X., Zhu, X., Hu, L.: Two new permutation polynomials with the form \((x^{2^k}+x+\delta )^s+x\) over \({\mathbb{F}}_{2^n}\). Appl. Algebra Eng. Commun. Comput. 21(2), 145–150 (2010) CrossRef Zeng, X., Zhu, X., Hu, L.: Two new permutation polynomials with the form \((x^{2^k}+x+\delta )^s+x\) over \({\mathbb{F}}_{2^n}\). Appl. Algebra Eng. Commun. Comput. 21(2), 145–150 (2010) CrossRef
28.
go back to reference Zeng, X., Zhu, X., Li, N., Liu, X.: 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, 256–268 (2017) MathSciNetCrossRef Zeng, X., Zhu, X., Li, N., Liu, X.: 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, 256–268 (2017) MathSciNetCrossRef
29.
go back to reference Zha, Z., Hu, L.: Two classes of permutation polynomials over finite fields. Finite Fields Appl. 18, 781–790 (2012) MathSciNetCrossRef Zha, Z., Hu, L.: Two classes of permutation polynomials over finite fields. Finite Fields Appl. 18, 781–790 (2012) MathSciNetCrossRef
30.
go back to reference Zha, Z., Hu, L.: Some classes of permuation polynomials of the form \((x^{p^m}-x+\delta )^s+L(x)\) over \({\mathbb{F}}_{p^{2m}}\). Finite Fields Appl. 40, 150–162 (2016) MathSciNetCrossRef Zha, Z., Hu, L.: Some classes of permuation polynomials of the form \((x^{p^m}-x+\delta )^s+L(x)\) over \({\mathbb{F}}_{p^{2m}}\). Finite Fields Appl. 40, 150–162 (2016) MathSciNetCrossRef
31.
go back to reference Zha, Z., Hu, L., Zhang, Z.: Permutation polynomials of the form \(x+\gamma {\rm Tr}_q^{q^n}(h(x))\). Finite Fields Appl. 60, 101573 (2019) MathSciNetCrossRef Zha, Z., Hu, L., Zhang, Z.: Permutation polynomials of the form \(x+\gamma {\rm Tr}_q^{q^n}(h(x))\). Finite Fields Appl. 60, 101573 (2019) MathSciNetCrossRef
32.
go back to reference Zheng, D., Chen, Z.: More classes of permutation polynomials of the form \((x^{2^m}+x+\delta )^s+L(x)\). Appl. Algebra Eng. Commun. Comput. 28(3), 215–223 (2017) CrossRef Zheng, D., Chen, Z.: More classes of permutation polynomials of the form \((x^{2^m}+x+\delta )^s+L(x)\). Appl. Algebra Eng. Commun. Comput. 28(3), 215–223 (2017) CrossRef
33.
go back to reference Zheng, D., Yuan, M., Yuan, L.: Two types of permutation polynomials with special forms. Finite Fields Appl. 56, 1–16 (2019) MathSciNetCrossRef Zheng, D., Yuan, M., Yuan, L.: Two types of permutation polynomials with special forms. Finite Fields Appl. 56, 1–16 (2019) MathSciNetCrossRef
34.
go back to reference Zheng, Y., Wang, Q., Wei, W.: On inverses of permutation polynomials of small degree over finite fields. IEEE Trans. Inf. Theory 66(2), 914–922 (2020) MathSciNetCrossRef Zheng, Y., Wang, Q., Wei, W.: On inverses of permutation polynomials of small degree over finite fields. IEEE Trans. Inf. Theory 66(2), 914–922 (2020) MathSciNetCrossRef
35.
go back to reference Zheng, Y., Yuan, P., Pei, D.: Large classes of permutation polynomials over \({\mathbb{F}}_{q^2}\). Des. Codes Cryptogr. 81, 505–521 (2016) MathSciNetCrossRef Zheng, Y., Yuan, P., Pei, D.: Large classes of permutation polynomials over \({\mathbb{F}}_{q^2}\). Des. Codes Cryptogr. 81, 505–521 (2016) MathSciNetCrossRef
Metadata
Title
Some classes of permutation polynomials of the form over
Authors
Danyao Wu
Pingzhi Yuan
Publication date
11-06-2020
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 2/2022
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00441-z

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