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Permutation polynomials of the form \(cx+\text {Tr}_{q^{l}/ q}(x^{a})\) and permutation trinomials over finite fields with even characteristic

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Abstract

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. Particularly, permutation polynomials with few terms are more popular for their simple algebraic form and additional extraordinary properties. Very recently, G. Kyureghyan and M.E. Zieve (2016) studied permutation polynomials over \(\mathbb {F}_{q^{n}}\) of the form \(x+\gamma \text {Tr}_{q^{n}/q}(x^{k})\), where q is odd, and nine classes of permutation polynomials were constructed. In this paper, we present fifteen new classes of permutation polynomials of the form \(cx+\text {Tr}_{q^{l}/ q}(x^{a})\) over finite fields with even characteristic, which explain most of the examples with q = 2k, k > 1, k l < 14 and \(c\in \mathbb {F}_{q^{l}}^{*}\). Furthermore, we also construct four classes of permutation trinomials.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Akbary, A., Wang, Q.: On polynomials of the form x r h(x)(q−1)/l. International Journal of Mathematics and Mathematical Sciences. Article ID 23408, 7 pages (2007)

  3. Ball, S., Zieve, M.: Symplectic spreads and permutation polynomials. Finite Fields and Applications. In: Lect. Notes Comput. Sci, vol. 2948, pp. 79–88. Springer, Berlin (2004)

    Google Scholar 

  4. Berlekamp, E.R., Rumsey, H., Solomon, G.: On the solution of algebraic equations over finite fields. Information And Control 10(67), 553–564 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  5. Charpin, P., Kyureghyan, G.: On a class of permutation polynomials over \(\mathbb {F}_{2^{n}}\). Sequences and their Applications-SETA 2008, Lecture Notes in Comput Sci. 5203. Springer 3, 368–376 (2008)

    Google Scholar 

  6. Charpin, P., Kyureghyan, G.: Monmial functions with linear structure and permutation polynomials. Finite Fields: Theory and Applications, Contemp. Math. 518, Amer. Math. Soc.. 3(16), 99–111 (2010)

    MATH  Google Scholar 

  7. Charpin, P., Kyureghyan, G.: When does G(x) + γ Tr(H(x)) permutes \(\mathbb {F}_{p^{n}}\). Finite Fields Appl. 15, 615–632 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ding, C., Yuan, J.: A family of skew Hadamard difference sets. J. Combin. Theory Ser. A 113, 1526–1535 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ding, C., Qu, L., Wang, Q., et al.: Permutation trinomials over finite fields with even characteristic. SLAM J. Dis. Math. 29, 79–92 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dobbertin, H.: Almost perfect nonlinear power functions on G F(2n): The Welch case. IEEE Trans. Inform. Theory 45, 1271–1275 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dobbertin, H.: Uniformly representable permutation polynomials. Sequence andtheir Applications-SETA 2001. Springer 2(9), 1–22 (2002)

    MathSciNet  MATH  Google Scholar 

  12. 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 

  13. Hou, X.: A survey of permutation binomials and trinomials over finite fields. In: Kyureghyan, G., Mullen, G.L., Pott, A. (eds.) Topics in Finite Fields, Proceedings of the 11th International Conference on Finite Fields and Their Applications, Contemp. Math, vol. 632, pp. 177–191. Magdeburg, Germany (2015). AMS

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Hou, X.: A class of permutation trinomials over finite fields. Acta Arith 162, 51–64 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hou, X.: Determination of a type of permutaiton trinomials over finite fields. Acta Arith. 162, 253–278 (2014)

    Article  MATH  Google Scholar 

  17. Hou, X.: Determination of a type of permutaiton trinomials over finite fields, I I. Finite Fields Appl. 35, 16–35 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kyureghyan, G., Zieve, M.E.: Permutation polynomials of the form X + γ Tr(X k). Contemporary Developments in Finite Fields and Appl. doi:10.1142/9789814719261-0011 (2016)

  19. Lidl, R., Niederreiter, H.: Finite Fields, 2nd ed. Cambridge Univ. Press, Cambridge (1997)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Leonard, P.A., Williams, K.S.: Quartics over \(\mathbb {GF}\left (2^{n}\right )\). Proc. Am. Math. Soc. 36, 347–350 (1972)

    Google Scholar 

  22. Muller, W.B., Nobauer, R.: Cryptanalysis of the Dickson-scheme. In: Proceedings of EUROCRYPT 85, pp. 215–230. Springer-Verlag, New York (1986)

    Google Scholar 

  23. Park, H., Lee, J.B.: Permutation polynomials and group permutation polynomials. Bull. Aust. Math. Soc. 63, 67–74 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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 

  25. Li, K., Qu, L., Li, C., et al: New permutation trinomials constructed from fractional polynomials. arXiv:1605.06216 (2016)

  26. Qu, L., Tan, Y., Tan, C.H., et al.: Constructing differentially 4-uniform permutations over \(\mathbb {F}_{2^{2k}}\) via the switching method. IEEE Trans. Inform. Theory 59, 4675–4686 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rivest, R.L., Shamir, A., Aselman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Comm. ACM 21, 120–126 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ma, J., Zhang, T., Feng, T., et al.: Some new results on permutation polynomials over finite fields. Des. Codes Cryptog.r 83(2), 425–443 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tu, Z., Zeng, X., Hu, L., et al.: A class of binomial permutation polynomials. arXiv:1310.0337 (2013)

  30. Sun, J., Takeshita, O.Y.: Interleavers for turdo codes using permutation polynomials over integer rings. IEEE Trans. Inform. Theory 51, 101–119 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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, Lect. Notes Comput. Sci., vol. 4893, pp. 119–128. Springer, Berlin (2007)

    Google Scholar 

  32. Williams, K.S.: Note on Cubics over G F(2n) and G F(3n)∗. J. Number Theory 7, 361–365 (1975)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. 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)

    Article  MATH  Google Scholar 

  35. 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)

  36. Zieve, M.E.: Permutation polynomials over \(\mathbb {F}_{q^{2}}\) induced from novel permutations of the (q + 1)-th roots of unity. preprint

  37. Zeng, X., Tian, S., Tu, Z.: Permutation polynomials from trace functions over finite fields. Finite Fields Appl. 35, 36–51 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

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

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Authors and Affiliations

  1. College of Science, National University of Defense Technology, Changsha, 410073, China

    Kangquan Li, Longjiang Qu, Xi Chen & Chao Li

  2. State Key Laboratory of Cryptology, Beijing, 100878, China

    Longjiang Qu

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  1. Kangquan Li
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  2. Longjiang Qu
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  3. Xi Chen
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  4. Chao Li
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Corresponding author

Correspondence to Longjiang Qu.

Additional information

This work is supported by the National Basic Research Program of China (Grant No. 2013CB338002), the Nature Science Foundation of China (NSFC) under Grant 61272484, 11531002, 61572026, the Program for New Century Excellent Talents in University (NCET), the Basic Research Fund of National University of Defense Technology (No. CJ 13-02-01), and the Open Foundation of State Key Laboratory of Cryptology.

This article is part of the Topical Collection on Special Issue on Sequences and Their Applications

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Li, K., Qu, L., Chen, X. et al. Permutation polynomials of the form \(cx+\text {Tr}_{q^{l}/ q}(x^{a})\) and permutation trinomials over finite fields with even characteristic. Cryptogr. Commun. 10, 531–554 (2018). https://doi.org/10.1007/s12095-017-0236-7

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