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Published in: Applicable Algebra in Engineering, Communication and Computing 1/2020

12-07-2019 | Original Paper

Determination of a type of permutation binomials and trinomials

Authors: R. K. Sharma, Rohit Gupta

Published in: Applicable Algebra in Engineering, Communication and Computing | Issue 1/2020

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Abstract

Let \({\mathbb {F}}_q\) denote the finite field of order q. In this paper, we determine certain permutation binomials and permutation trinomials of the form \(x^{r}h(x^{q+1})\) over \(\mathbb {F}_{q^2}\). Some of them are generalizations of known ones.

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Appendix
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Literature
1.
go back to reference Bassalygo, L.A., Zinoviev, V.A.: Permutation and complete permutation polynomials. Finite Fields Appl. 33, 198–211 (2015)MathSciNetCrossRef Bassalygo, L.A., Zinoviev, V.A.: Permutation and complete permutation polynomials. Finite Fields Appl. 33, 198–211 (2015)MathSciNetCrossRef
2.
go back to reference Dickson, L.E.: The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, part I. Ann. Math. 11, 65–120 (1896–1897)MathSciNetCrossRef Dickson, L.E.: The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, part I. Ann. Math. 11, 65–120 (1896–1897)MathSciNetCrossRef
3.
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
4.
5.
go back to reference Fernando, N., Hou, X., Lappano, S.D.: Permutation polynomials over finite fields involving \(x+x^q+\dots +x^{q^{a-1}}\). Discrete Math. 315–316, 173–184 (2014)CrossRef Fernando, N., Hou, X., Lappano, S.D.: Permutation polynomials over finite fields involving \(x+x^q+\dots +x^{q^{a-1}}\). Discrete Math. 315–316, 173–184 (2014)CrossRef
6.
go back to reference Gupta, R., Sharma, R.K.: Some new classes of permutation trinomials over finite fields with even characteristic. Finite Fields Appl. 41, 89–96 (2016)MathSciNetCrossRef Gupta, R., Sharma, R.K.: Some new classes of permutation trinomials over finite fields with even characteristic. Finite Fields Appl. 41, 89–96 (2016)MathSciNetCrossRef
7.
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
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, Contemp. Math., Magdeburg, Germany, July 2013, vol. 632, pp. 177–191. AMS (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, Contemp. Math., Magdeburg, Germany, July 2013, vol. 632, pp. 177–191. AMS (2015)
9.
go back to reference Hou, X.: Permutation polynomials over finite fields—a survey of recent advances. Finite Fields Appl. 32, 82–119 (2015)MathSciNetCrossRef Hou, X.: Permutation polynomials over finite fields—a survey of recent advances. Finite Fields Appl. 32, 82–119 (2015)MathSciNetCrossRef
10.
11.
go back to reference Hou, X.: Determination of a type of permutation trinomials over finite fields II. Finite Fields Appl. 35, 16–35 (2015)MathSciNetCrossRef Hou, X.: Determination of a type of permutation trinomials over finite fields II. Finite Fields Appl. 35, 16–35 (2015)MathSciNetCrossRef
12.
go back to reference Hou, X., Lappano, S.D.: Determination of a type of permutation binomials over finite fields. J. Number Theory 147, 14–23 (2015)MathSciNetCrossRef Hou, X., Lappano, S.D.: Determination of a type of permutation binomials over finite fields. J. Number Theory 147, 14–23 (2015)MathSciNetCrossRef
13.
go back to reference Laigle-Chapuy, Y.: Permutation polynomials and applications to coding theory. Finite Fields Appl. 13, 58–70 (2007)MathSciNetCrossRef Laigle-Chapuy, Y.: Permutation polynomials and applications to coding theory. Finite Fields Appl. 13, 58–70 (2007)MathSciNetCrossRef
14.
go back to reference Li, J., Chandler, D.B., Xiang, Q.: Permutation polynomials of degree \(6\) or \(7\) over finite fields of characterstic \(2\). Finite Fields Appl. 16, 406–419 (2010)MathSciNetCrossRef Li, J., Chandler, D.B., Xiang, Q.: Permutation polynomials of degree \(6\) or \(7\) over finite fields of characterstic \(2\). Finite Fields Appl. 16, 406–419 (2010)MathSciNetCrossRef
15.
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
16.
go back to reference Lidl, R., Mullen, W.B.: Permutation Polynomials in RSA-Cryptosystems. Advances in Cryptology, pp. 293–301. Plenum, New York (1984) Lidl, R., Mullen, W.B.: Permutation Polynomials in RSA-Cryptosystems. Advances in Cryptology, pp. 293–301. Plenum, New York (1984)
17.
go back to reference Lidl, R., Niederreiter, H.: Finite Fields, 2nd edn. Cambridge Univ. Press, Cambridge (1997)MATH Lidl, R., Niederreiter, H.: Finite Fields, 2nd edn. Cambridge Univ. Press, Cambridge (1997)MATH
18.
go back to reference Marcos, J.E.: Some permutation polynomials over finite fields. Appl. Algebra Eng. Commun. Comput. 26, 465–474 (2015)MathSciNetCrossRef Marcos, J.E.: Some permutation polynomials over finite fields. Appl. Algebra Eng. Commun. Comput. 26, 465–474 (2015)MathSciNetCrossRef
19.
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
20.
go back to reference Shallue, C.J., Wanless, I.M.: Permutation polynomials and orthomorphism polynomials of degree six. Finite Fields Appl. 20, 84–92 (2013)MathSciNetCrossRef Shallue, C.J., Wanless, I.M.: Permutation polynomials and orthomorphism polynomials of degree six. Finite Fields Appl. 20, 84–92 (2013)MathSciNetCrossRef
21.
22.
go back to reference Yuan, P., Ding, C.: Further results on permutation polynomials over finite fields. Finite Fields Appl. 27, 88–103 (2014)MathSciNetCrossRef Yuan, P., Ding, C.: Further results on permutation polynomials over finite fields. Finite Fields Appl. 27, 88–103 (2014)MathSciNetCrossRef
23.
go back to reference Yuan, P., Ding, C.: Permutation polynomials of the form \(L(x)+S_{2k}^a+S_{2k}^b\) over \(\mathbb{F}_{q^{3k}}\). Finite Fields Appl. 29, 106–117 (2014)MathSciNetCrossRef Yuan, P., Ding, C.: Permutation polynomials of the form \(L(x)+S_{2k}^a+S_{2k}^b\) over \(\mathbb{F}_{q^{3k}}\). Finite Fields Appl. 29, 106–117 (2014)MathSciNetCrossRef
24.
go back to reference Zieve, M.E.: Some families of permutation polynomials over finite fields. Int. J. Number Theory 4, 851–857 (2008)MathSciNetCrossRef Zieve, M.E.: Some families of permutation polynomials over finite fields. Int. J. Number Theory 4, 851–857 (2008)MathSciNetCrossRef
25.
go back to reference Zieve, M.E.: Permutation polynomials induced from permutations of subfields, and some complete sets of mutually orthogonal Latin squares. arXiv:1312.1325 Zieve, M.E.: Permutation polynomials induced from permutations of subfields, and some complete sets of mutually orthogonal Latin squares. arXiv:​1312.​1325
Metadata
Title
Determination of a type of permutation binomials and trinomials
Authors
R. K. Sharma
Rohit Gupta
Publication date
12-07-2019
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 1/2020
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-019-00394-y

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