Permutations polynomials of the form G(X)kL(X) and curves over finite fields


For a positive integer k and a linearized polynomial L(X), polynomials of the form \(P(X)=G(X)^{k}-L(X) \in {\mathbb F}_{q^{n}}[X]\) are investigated. It is shown that when L has a non-trivial kernel and G is a permutation of \(\mathbb {F}_{q^{n}}\), then P(X) cannot be a permutation if \(\gcd (k,q^{n}-1)>1\). Further, necessary conditions for P(X) to be a permutation of \(\mathbb {F}_{q^{n}}\) are given for the case that G(X) is an arbitrary linearized polynomial. The method uses plane curves, which are obtained via the multiplicative and the additive structure of \(\mathbb {F}_{q^{n}}\), and their number of rational affine points.

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  1. 1.

    Anbar, N.: Curves over finite fields and permutations of the form xkγ Tr(x). Turkish J. Math. 43(1), 533–538 (2019)

    MathSciNet  Google Scholar 

  2. 2.

    Anbar, N., Odz̆ak, A., Patel, V., Quoos, L., Somoza, A., Topuzoğlu, A.: On the difference between permutation polynomials over finite fields. Finite Fields Appl. 49, 132–142 (2018)

    MathSciNet  Google Scholar 

  3. 3.

    Gerike, D., Kyureghyan, G.: Results on permutation polynomials of shape \(x^{t}+{\gamma } \text {Tr}_{q^{n}/q}(x^{d}) \). Combinatorics and Finite Fields, Radon Ser. Comput. Appl. Math., De Gruyter. Berlin 23, 67–78 (2019)

    Google Scholar 

  4. 4.

    Helleseth, T., Zinoviev, V.: New Kloosterman sums identities over \(\mathbb {F}_{2^{m}}\) for all m. Finite Fields Appl. 9(2), 187–193 (2003)

    MathSciNet  Google Scholar 

  5. 5.

    Hirschfeld, J.W.P., Korchmáros, G., Torres, F.: Algebraic Curves over a Finite Field. Princeton University Press (2013)

  6. 6.

    Kyureghyan, G., Zieve, M.: Permutation Polynomials of the Form X + γ Tr(Xk). Contemporary Developments in Finite Fields and Applications, pp 178–194. World Sci. Publ., Hackensack (2016)

    Google Scholar 

  7. 7.

    Lidl, R., Niederreiter, H.: Finite Fields. With a foreword by P. M. Cohn. Second edition Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  8. 8.

    Liu, Q., Sun, Y., Zhang, W.G.: Some classes of permutation polynomials over finite fields with odd characteristic. Appl. Algebra Engrg. Comm. Comput. 29(5), 409–431 (2018)

    MathSciNet  Google Scholar 

  9. 9.

    Ma, J., Ge, G.: A note on permutation polynomials over finite fields. Finite Fields Appl. 48, 261–270 (2017)

    MathSciNet  Google Scholar 

  10. 10.

    Mullen, G.L., Panario, D.: Handbook of Finite Fields. Chapman and Hall (2013)

  11. 11.

    Niederreiter, H., Xing, C.P.: Algebraic Geometry in Coding Theory and Cryptography. Princeton University Press, Princeton (2009)

    Google Scholar 

  12. 12.

    Stichtenoth, H.: Algebraic Function Fields and Codes. Second edition Graduate Texts in Mathematics, vol. 254. Springer, Berlin (2009)

    Google Scholar 

  13. 13.

    Tu, Z., Zeng, X., Li, C., Helleseth, T.: Permutation polynomials of the form \((x^{p^{m}}-x+{\delta })^{s}+L(x) \) over the finite field \(\mathbb {F}_{p^{2m}}\) of odd characteristic. Finite Fields Appl. 34, 20–35 (2015)

    MathSciNet  Google Scholar 

  14. 14.

    Wang, L., Wu, B., Liu, Z.: 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, 92–112 (2017)

    MathSciNet  Google Scholar 

  15. 15.

    Xu, G., Cao, X., Xu, S.: Further results on permutation polynomials of the form \((x^{p^{m}}-x+{\delta })^{s}+L(x) \) over \(\mathbb {F}_{p^{2m}}\). J. Algebra Appl. 15(5), 1650098 (2016). 13 pp

    MathSciNet  Google Scholar 

  16. 16.

    Yuan, J., Ding, C.: Four classes of permutation polynomials of \(\mathbb {F}_{2^{m}}\). Finite Fields Appl. 13(4), 869–876 (2007)

    MathSciNet  Google Scholar 

  17. 17.

    Yuan, J., Ding, C., Wang, H., Pieprzyk, J.: Permutation polynomials of the form (xpx + δ)s + L(X). Finite Fields Appl. 14(2), 482–493 (2008)

    MathSciNet  Google Scholar 

  18. 18.

    Zheng, D., Chen, Z.: More classes of permutation polynomials of the form \((x^{p^{m}}-x+{\delta })^{s}+L(x) \). Appl. Algebra Engrg. Comm. Comput. 28(3), 215–223 (2017)

    MathSciNet  Google Scholar 

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N.A. is supported by B.A.CF-19-01967.

We would like to thank Wilfried Meidl for his useful comments, which helped to improve the presentation of the manuscript considerably.

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Correspondence to Nurdagül Anbar.

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Anbar, N., Kaşıkcı, C. Permutations polynomials of the form G(X)kL(X) and curves over finite fields. Cryptogr. Commun. (2021).

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  • Curves/function fields
  • Permutation polynomials
  • Rational points/places

Mathematics Subject Classification (2010)

  • 11T06
  • 14H05