Compositional inverses of permutation polynomials of the form x r h(x s ) over finite fields

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Abstract

The study of computing compositional inverses of permutation polynomials over finite fields efficiently is motivated by an open problem proposed by G. L. Mullen (1991), as well as the potential applications of these permutation polynomials (Dillon 1974, Khachatrian and Kyureghyan, Discrete Appl. Math. 216, 622–626 2017, Lidl 1985, Lidl and Müller 1984, Rivest et al., ACM Commun. Comput. Algebra. 1978, 120–126 1976, Schwenk and Huber, Electron. Lett. 34, 759–760 1998). It is well known that every permutation polynomial over a finite field \(\mathbb {F}_{q}\) can be reduced to a permutation polynomial of the form x r h(x s ) with s∣(q − 1) and \(h(x) \in \mathbb {F}_{q}[x]\) (Akbary et al., Finite Fields Appl. 15(2), 195–206 2009, Wang, Finite Fields Appl. 22, 57–69 2013). Recently, several explicit classes of permutation polynomials of the form x r h(x s ) over \({\mathbb F}_{q}\) have been constructed. However, all the known methods to compute the compositional inverses of permutation polynomials of this form seem to be inadequately explicit, which could be a hurdle to potential applications. In this paper, for any prime power q, we introduce a new approach to explicitly compute the compositional inverse of a permutation polynomial of the form x r h(x s ) over \({\mathbb F}_{q}\), where s∣(q − 1) and \(\gcd (r,q-1)= 1\). The main idea relies on transforming the problem of computing the compositional inverses of permutation polynomials over \({\mathbb F}_{q}\) into computing the compositional inverses of two restricted permutation mappings, where one of them is a monomial over \(\mathbb {F}_{q}\) and the other is the polynomial x r h(x) s over a particular subgroup of \(\mathbb {F}_{q}^{*}\) with order (q − 1)/s. This is a multiplicative analog of Tuxanidy and Wang (Finite Fields Appl. 28, 244–281 2014), Wu and Liu (Finite Fields Appl. 24, 136–147 2013). We demonstrate that the inverses of these two restricted permutations can be explicitly obtained in many cases. As consequences, many explicit compositional inverses of permutation polynomials given in Zieve (Proc. Am. Math. Soc. 137, 2209–2216 2009), Zieve (arXiv:1310.0776, 2013), Zieve (arXiv:1312.1325v3, 2013) are obtained using this method.

Keywords

Finite fields Permutation polynomials Compositional inverses 

Mathematics Subject Classification (2010)

11T06 

Notes

Acknowledgments

We would like to thank the editor and the anonymous referees whose valuable comments and suggestions improve both the technical quality and the editorial quality of this paper.

References

  1. 1.
    Akbary, A., Ghioca, D., Wang, Q.: On permutation polynomials of prescribed shape. Finite Fields Appl. 15(2), 195–206 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Akbary, A., Ghioca, D., Wang, Q.: On constructing permutations of finite fields. Finite Fields Appl. 17, 51–67 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Akbary, A., Wang, Q.: On polynomials of the form x r h(x (q− 1)/l). Int. J. Math. Math. Sci., Art. ID 23408, 7 (2007)Google Scholar
  4. 4.
    Charpin, P., Mesnager, S., Sarkar, S.: Involutions over the Galois field \({F}_{2^{n}}\). IEEE Trans. Inform. Theory. 62, 2266–2276 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Coulter, R.S., Henderson, M.: The compositional inverse of a class of permutation polynomials over a finite field. Bull. Aust. Math. Soc. 65, 521–526 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dillon, J.F.: Elementary Hadamard difference sets. PhD thesis, University of Maryland (1974)Google Scholar
  7. 7.
    Hou, X.: Permutation polynomials over finite fields — a survey of recent advances. Finite Fields Appl. 32, 82–119 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Khachatrian, G., Kyureghyan, M.: Permutation polynomials and a new public-key encryption. Discrete Appl. Math. 216, 622–626 (2017)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Laigle-Chapuy, Y.: A note on a class of quadratic permutation polynomials over \({F}_{2^{n}}\). In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, LNCS, vol. 4851, pp 130–137. Springer (2007)Google Scholar
  10. 10.
    Lidl, R., Niederreiter, H.: Finite Fields, 2nd ed. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  11. 11.
    Lidl, R., Mullen, G.L., Turnwald, G.: Dickson polynomials, Longman Scientific and Technical (1993)Google Scholar
  12. 12.
    Lidl, R.: On Cryptosystems Based on Polynomials and Finite Fields. In: Advances in Cryptology—Proceedings of CRYPTO’83, EUROCRYPT ’84 LNCS, vol. 209, pp 10–15. Springer, Berlin (1985)Google Scholar
  13. 13.
    Lidl, R., Müller, W.B.: Permutation Polynomials in RSA-Cryptosystems. In: Advances in Cryptology, pp 293–301. Plenum Press, New York (1984)Google Scholar
  14. 14.
    Li, K., Qu, L., Chen, X.: New classes of permutation binomials and permutation trinomials over finite fields. Finite Fields Appl. 43, 69–85 (2017)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Li, K., Qu, L., Chen, X., Li, C.: Permutation polynomials of the form \(cx+ \text {Tr}_{q^{l}/q}\left (x^{a}\right )\) and permutation trinomials over finite fields with even characteristic. Cryptogr. Commun. 10, 531–554 (2018)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Li, N., Helleseth, T.: Several classes of permutation trinomials from Niho exponents. Cryptogr. Commun. 9, 693–705 (2017)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Li, N., Helleseth, T.: New permutation trinomials from Niho exponents over finite fields with even characteristic. arXiv:1606.03768v1 (2016)
  18. 18.
    Mesnager, S.: Bent Functions: Fundamentals and Results. Springer, Berlin (2016)CrossRefMATHGoogle Scholar
  19. 19.
    Mullen, G.L.: Permutation polynomials over finite fields, Finite Fields, Coding Theory, and Advances in Communication and Computing, Las Vegas, NY, 131–151 (1991)Google Scholar
  20. 20.
    Muratović-Ribić, A.: A note on the coefficients of inverse polynomials. Finite Fields Appl. 13, 977–980 (2007)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Niederreiter, H., Winterhof, A.: Cyclotomic \(\mathscr {R}\)-orthomorprhisms of finite fields. Discrete Math. 295, 161–171 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Park, Y.H., Lee, J.B.: Permutation polynomials and group permutation polynomials. Bull. Aust. Math. Soc. 63, 67–74 (2001)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rivest, R.L., Shamir, A., Adelman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. ACM Commun. Comput. Algebra. 21, 120–126 (1978)MathSciNetMATHGoogle Scholar
  24. 24.
    Schwenk, J., Huber, K.: Public key encryption and digital signatures based on permutation polynomials. Electron. Lett. 34, 759–760 (1998)CrossRefGoogle Scholar
  25. 25.
    Tuxanidy, A., Wang, Q.: On the inverse of some classes of permutations of finite fields. Finite Fields Appl. 28, 244–281 (2014)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Tuxanidy, A., Wang, Q.: Compositional inverses and complete mappings over finite fields. Discrete Appl. Math. 217, part 2, 318–329 (2017)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wang, Q.: Cyclotomic Mapping Permutation Polynomials over Finite Fields. In: Golomb, S.W., Gong, G., Helleseth, T., Song, H.-Y. (eds.) Sequences, Subsequences, and Consequences, In: Lecture Notes in Comput. Sci., vol. 4893, pp 119–128. Springer (2007)Google Scholar
  28. 28.
    Wang, Q.: On inverse permutation polynomials. Finite Fields Appl. 15, 207–213 (2009)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Wang, Q.: Cyclotomy and permutation polynomials of large indices. Finite Fields Appl. 22, 57–69 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Wang, Q.: A note on inverses of cyclotomic mapping permutation polynomials over finite fields. Finite Fields Appl. 45, 422–427 (2017)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Wu, B., Liu, Z.: Linearized polynomials over finite fields revisited. Finite Fields Appl. 22, 79–100 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Wu, B., Liu, Z.: The compositional inverse of a class of bilinear permutation polynomials over finite fields of characteristic 2. Finite Fields Appl. 24, 136–147 (2013)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Wu, B.: Linearized and Linearized Derived Permutation Polynomials over Finite Fields and Their Compositional Inverses. Ph.D thesis, University of Chinese Academy of Sciences. (in Chinese) (2013)Google Scholar
  34. 34.
    Wu, B.: The compositional inverses of linearized permutation binomials over finite fields. arXiv:1311.2154v1 (2013)
  35. 35.
    Wu, B.: The compositional inverse of a class of linearized permutation polynomials over \({F}_{2^{n}}\), n odd. Finite Fields Appl. 29, 34–48 (2014)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Yuan, P., Ding, C.: Permutation polynomials over finite fields from a powerful lemma. Finite Fields Appl. 17(6), 560–574 (2011)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Zheng, Y., Yuan, P., Pei, D.: Piecewise constructions of inverses of some permutation polynomials. Finite Fields Appl. 36, 151–169 (2015)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Zheng, Y., Yu, Y., Pei, D.: Piecewise constructions of inverses of cyclotomic mapping permutations. Finite Fields Appl. 40, 1–9 (2016)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Zieve, M.E.: On some permutation polynomials over \(\mathbb {F}_{q} \) of the form x r h(x (q− 1)/d). Proc. Am. Math. Soc. 137, 2209–2216 (2009)CrossRefMATHGoogle Scholar
  40. 40.
    Zieve, M.E.: 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)
  41. 41.
    Zieve, M.E.: Permutation polynomials induced from permutations of subfields, and some complete sets of mutually orthogonal latin squares. arXiv:1312.1325v3 (2013)

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Authors and Affiliations

  1. 1.College of ScienceNational University of Defense TechnologyChangshaChina
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada

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