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

12.07.2019 | Original Paper

Determination of a type of permutation binomials and trinomials

verfasst von: R. K. Sharma, Rohit Gupta

Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 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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Literatur
1.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat Ding, C., Yuan, J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113, 1526–1535 (2006) MathSciNetCrossRef Ding, C., Yuan, J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113, 1526–1535 (2006) MathSciNetCrossRef
5.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat Hou, X.: Determination of a type of permutation trinomials over finite fields. Acta Arith. 166, 253–278 (2014) MathSciNetCrossRef Hou, X.: Determination of a type of permutation trinomials over finite fields. Acta Arith. 166, 253–278 (2014) MathSciNetCrossRef
11.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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.
Zurück zum Zitat 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
Metadaten
Titel
Determination of a type of permutation binomials and trinomials
verfasst von
R. K. Sharma
Rohit Gupta
Publikationsdatum
12.07.2019
Verlag
Springer Berlin Heidelberg
Erschienen in
Applicable Algebra in Engineering, Communication and Computing / Ausgabe 1/2020
Print ISSN: 0938-1279
Elektronische ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-019-00394-y

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