Abstract
Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of trinomial complete permutation polynomials are presented, one of which confirms a conjecture proposed by Wu et al. (Sci China Math 58:2081–2094, 2015). Furthermore, we give two classes of permutation trinomial, and make some progress on a conjecture about the differential uniformity of power permutation polynomials proposed by Blondeau et al. (Int J Inf Coding Theory 1:149–170, 2010).
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References
Berlekamp E.R., Rumsey H., Solomon G.: On the solution of algebraic equations over finite fields. Inform. Control 10, 553–564 (1967).
Blondeau C., Canteaut A., Charpin P.: Differential properties of power functions. Int. J. Inf. Coding Theory 1, 149–170 (2010).
Charpin P., Kyureghyan G.M.: Cubic monomial bent functions: a subclass of \({\cal M}\). SIAM J. Discret Math. 22, 650–665 (2008).
Ding C., Yuan J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113, 1526–1535 (2006).
Ding C., Qu L., Wang Q., Yuan J., Yuan P.: Permutation trinomials over finite fields with even characteristic. SIAM J. Discret Math. 29, 79–92 (2015).
Ding C., Xiang Q., Yuan J., Yuan P.: Explicit classes of permutation polynomials of \(\mathbb{F}_{3^{3m}}\). Sci. China Ser. A 52, 639–647 (2009).
Akbary A., Ghioca D., Wang Q.: On constructing permutations of finite fields. Finite Fields Appl. 17, 51–67 (2011).
Dobbertin H.: Almost perfect nonlinear power functions on \({\rm GF}(2^n)\): the Welch case. IEEE Trans. Inf. Theory 45, 1271–1275 (1999).
Dobbertin H.: Kasami power functions, permutation polynomials and cyclic difference sets. In: Difference Sets, Sequences and Their Correlation Properties (Bad Windsheim, 1998). Nato Advanced Science Institutes Series C Mathematical and Physical Sciences, vol. 542, pp. 133–158. Kluwer, Dordrecht (1999).
Fernando N., Hou X.: A piecewise construction of permutation polynomials over finite fields. Finite Fields Appl. 18, 1184–1194 (2012).
Helleseth T.: Some results about the cross-correlation function between two maximal linear sequences. Discret Math. 16, 209–232 (1976).
Hollmann H.D.L., Xiang Q.: A class of permutation polynomials of \({\bf F}_{2^m}\) related to Dickson polynomials. Finite Fields Appl. 11, 111–122 (2005).
Hou X.: Two classes of permutation polynomials over finite fields. J. Comb. Theory Ser. A 118, 448–454 (2011).
Hou X.: A new approach to permutation polynomials over finite fields. Finite Fields Appl. 18, 492–521 (2012).
Hou X.: Permutation polynomials over finite fields—a survey of recent advances. Finite Fields Appl. 32, 82–119 (2015).
Laigle-Chapuy Y.: Permutation polynomials and applications to coding theory. Finite Fields Appl. 13, 58–70 (2007).
Li N., Helleseth T., Tang X.: Further results on a class of permutation polynomials over finite fields. Finite Fields Appl. 22, 16–23 (2013).
Lidl R., Niederreiter H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20. Addison-Wesley Publishing Company Advanced Book Program, Reading, MA (1983).
Niederreiter H., Robinson K.H.: Complete mappings of finite fields. J. Aust. Math. Soc. Ser. A 33, 197–212 (1982).
Qu L., Tan Y., Tan C.H., Li C.: Constructing differentially 4-uniform permutations over \({\mathbb{F}}_{2^{2k}}\) via the switching method. IEEE Trans. Inf. Theory 59, 4675–4686 (2013).
Sun J., Takeshita O.Y.: Interleavers for turbo codes using permutation polynomials over integer rings. IEEE Trans. Inf. Theory 51, 101–119 (2005).
Tu Z., Zeng X., Hu L.: Several classes of complete permutation polynomials. Finite Fields Appl. 25, 182–193 (2014).
Tu Z., Zeng X., Hu L., Li C.: A class of binomial permutation polynomials. arXiv:1310.0337 (2013).
Wu G., Li N., Helleseth T., Zhang Y.: Some classes of monomial complete permutation polynomials over finite fields of characteristic two. Finite Fields Appl. 28, 148–165 (2014).
Wu G., Li N., Helleseth T., Zhang Y.: Some classes of complete permutation polynomials over \({{\mathbb{F}}_q}\). Sci. China Math. 58, 2081–2094 (2015).
Yuan J., Ding C.: Four classes of permutation polynomials of \({\mathbb{F}}_{2^m}\). Finite Fields Appl. 13, 869–876 (2007).
Yuan J., Ding C., Wang H., Pieprzyk J.: Permutation polynomials of the form \((x^p-x+\delta )^s+L(x)\). Finite Fields Appl. 14, 482–493 (2008).
Zha Z., Hu L.: Two classes of permutation polynomials over finite fields. Finite Fields Appl. 18, 781–790 (2012).
Zieve M.E.: Permutation polynomials induced from permutations of subfields, and some complete sets of mutually orthogonal latin squares. arXiv:1312.1325 (2013).
Acknowledgments
The authors express their gratitude to the anonymous reviewers for their detailed and constructive comments which are very helpful to the improvement of the presentation of this paper. The research of T. Feng was supported by Fundamental Research Fund for the Central Universities of China, the National Natural Science Foundation of China under Grant Nos. 11201418 and 11422112, and the Research Fund for Doctoral Programs from the Ministry of Education of China under Grant 20120101120089. The research of G. Ge was supported by the National Natural Science Foundation of China under Grant Nos. 11431003 and 61571310.
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Ma, J., Zhang, T., Feng, T. et al. Some new results on permutation polynomials over finite fields. Des. Codes Cryptogr. 83, 425–443 (2017). https://doi.org/10.1007/s10623-016-0236-1
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DOI: https://doi.org/10.1007/s10623-016-0236-1