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A recursive construction of permutation polynomials over \({\mathbb {F}}_{q^2}\) with odd characteristic related to Rédei functions

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Abstract

In this paper, we construct two classes of permutation polynomials over \({\mathbb {F}}_{q^2}\) with odd characteristic closely related to rational Rédei functions. Two distinct characterizations of their compositional inverses are also obtained. These permutation polynomials can be generated recursively. As a consequence, we can generate permutation polynomials with an arbitrary number of terms in a very simple way. Moreover, several classes of permutation binomials and trinomials are given. With the help of a computer, we find that the number of permutation polynomials of these types is quite big.

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References

  1. Akbary A., Ghioca D., Wang Q.: On constructing permutations of finite fields. Finite Fields Appl. 17(1), 51–67 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  2. Bartoli D., Masuda A.M., Quoos L.: Permutation polynomials over \({\mathbb{F}}_{q^2}\) from rational functions. arXiv:1802.05260 (2018).

  3. Carlet C., Ding C., Yuan J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inf. Theory 51(6), 2089–2102 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  4. Ding C., Helleseth T.: Optimal ternary cyclic codes from monomials. IEEE Trans. Inf. Theory 59(9), 5898–5904 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  5. Ding C., Yuan J.: A family of skew Hadamard difference sets. J. Comb. Theory A 113(7), 1526–1535 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  6. Ding C., Yin J.: Signal sets from functions with optimum nonlinearity. IEEE Trans. Commun. 55(5), 936–940 (2007).

    Article  Google Scholar 

  7. Ding C., Qu L., Wang Q., Yuan J., Yuan P.: Permutation trinomials over finite fields with even characteristic. SIAM J. Discret. Math. 29(1), 79–92 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  8. Gupta R., Sharma R.K.: Some new classes of permutation trinomials over finite fields with even characteristic. Finite Fields Appl. 41, 89–96 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  9. Hou X.: Permutation polynomials over finite fields—a survey of recent advances. Finite Fields Appl. 32, 82–119 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  10. Hou X., Mullen G.L., Sellers J.A., Yucas J.L.: Reversed Dickson polynomials over finite fields. Finite Fields Appl. 15(6), 748–773 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  11. Laigle-Chapuy Y.: Permutation polynomials and applications to coding theory. Finite Fields Appl. 13(1), 58–70 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  12. Li N., Helleseth T.: New permutation trinomials from Niho exponents over finite fields with even characteristic. arXiv:1606.03768 (2016).

  13. Li N., Helleseth T.: Several classes of permutation trinomials from Niho exponents. Cryptogr. Commun. 9(6), 693–705 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  14. Lidl R., Müller W.B.: Permutation polynomials in RSA-cryptosystems. In: Proceedings of CRYPTO’83 Advances in Cryptology, Santa Barbara, CA, USA, August 21–24, pp. 293–301 (1983).

  15. Lidl R., Mullen G.L., Turnwald G.: Dickson Polynomials. Monographs and Surveys in Pure and Applied Mathematics, vol. 65. Chapman and Hall/CRC, Boca Raton (1993).

    MATH  Google Scholar 

  16. Lidl R., Niederreiter H.: Finite Fields. Encyclopedia of Mathematica and Its Applications, vol. 20, 2nd edn. Cambridge University Press, Cambridge (1997).

    Google Scholar 

  17. Lee J., Park Y.H.: Some permuting trinomials over finite fields. Acta Math. Sci. Ser. B (English Ed.) 17, 250–254, 07 (1997).

  18. Li K., Qu L., Li C., Fu S.: New permutation trinomials constructed from fractional polynomials. arXiv:1605.06216 (2016).

  19. Li K., Qu L., Chen X.: New classes of permutation binomials and permutation trinomials over finite fields. Finite Fields Appl. 43, 69–85 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  20. Li K., Qu L., Chen X., Li C.: Permutation polynomials of the form \(cx+\rm Tr_{q^l/q}(x^a)\) and permutation trinomials over finite fields with even characteristic. Cryptogr. Commun. 10(3), 531–554 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  21. Li K., Qu L., Wang Q.: New constructions of permutation polynomials of the form \(x^rh(x^{q-1})\) over \({\mathbb{F}}_{q^2}\). Des. Codes Cryptogr. (2017). https://doi.org/10.1007/s10623-017-0452-3.

  22. Li K., Qu L., Wang Q.: Compositional inverses of permutation polynomials of the form \(x^rh(x^s)\) over finite fields. Cryptogr. Commun. (2018). https://doi.org/10.1007/s12095-018-0292-7.

  23. Mullen G.L., Panario D.: Handbook of Finite Fields. CRC Press, Boca Raton (2013).

    Book  MATH  Google Scholar 

  24. Niederreiter H., Winterhof A.: Cyclotomic R-orthomorphisms of finite fields. Discret. Math. 295(1–3), 161–171 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  25. Rivest R.L., Shamir A., Adleman L.M.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  26. Schwenk J., Huber K.: Public key encryption and digital signatures based on permutation polynomials. Electron. Lett. 34(8), 759–760 (1998).

    Article  Google Scholar 

  27. Wang Q.: Cyclotomic mapping permutation polynomials over finite fields. In: Sequences, Subsequences, and Consequences, International Workshop, SSC 2007, Los Angeles, CA, USA, May 31–June 2, 2007, Revised Invited Papers, pp. 119–128 (2007).

  28. Wang Q.: A note on inverses of cyclotomic mapping permutation polynomials over finite fields. Finite Fields Appl. 45, 422–427 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  29. Yuan P., Ding C.: Further results on permutation polynomials over finite fields. Finite Fields Appl. 27, 88–103 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  30. Yuan P., Ding C.: Permutation polynomials over finite fields from a powerful lemma. Finite Fields Appl. 17(6), 560–574 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  31. Yuan J., Carlet C., Ding C.: The weight distribution of a class of linear codes from perfect nonlinear functions. IEEE Trans. Inf. Theory 52(2), 712–717 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  32. Zha Z., Hu L., Fan S.: Further results on permutation trinomials over finite fields with even characteristic. Finite Fields Appl. 45, 43–52 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  33. Zieve M.: Permutation polynomials on \(\mathbb{F}_q\) induced from bijective Rédei functions on subgroups of the multiplicative group of \(\mathbb{F}_q^*\). arXiv:1310.0776 (2013).

  34. Zieve M.E.: On some permutation polynomials over \({\mathbb{F}}_{q}\) of the form \(x^rh(x^{(q-1)/d})\). Proc. Am. Math. Soc. 137(7), 2209–2216 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  35. Zheng Y., Yuan P., Pei D.: Large classes of permutation polynomials over \({\mathbb{F}}_{q^2}\). Des. Codes Cryptogr. 81(3), 505–521 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  36. Zheng Y., Yu Y., Zhang Y., Pei D.: Piecewise constructions of inverses of cyclotomic mapping permutation polynomials. Finite Fields Appl. 40, 1–9 (2016).

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank the anonymous referees for their helpful suggestions. This work was supported by the National Key Research and Development Program of China (No. 2016YFB0800401), Hubei Provincial Natural Science Foundation of China (2016CFB454), Science and Technology on Communication Security Laboratory (No. 6142103010701), the National Natural Science Foundation of China (Nos. 61572491 and 11688101) and NSERC of Canada.

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Authors and Affiliations

  1. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Shihui Fu & Xiutao Feng

  2. Science and Technology on Communication Security Laboratory, Chengdu, 610041, China

    Xiutao Feng

  3. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, 100093, China

    Dongdai Lin

  4. School of Mathematics and Statistics, Carleton University, Ottawa, ON, K1S 5B6, Canada

    Qiang Wang

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  1. Shihui Fu
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  2. Xiutao Feng
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  4. Qiang Wang
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Correspondence to Qiang Wang.

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Communicated by C. Carlet.

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Fu, S., Feng, X., Lin, D. et al. A recursive construction of permutation polynomials over \({\mathbb {F}}_{q^2}\) with odd characteristic related to Rédei functions. Des. Codes Cryptogr. 87, 1481–1498 (2019). https://doi.org/10.1007/s10623-018-0548-4

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