Ma, Jingxue; Zhang, Tao; Feng, Tao; Ge, Gennian Some new results on permutation polynomials over finite fields. (English) Zbl 1369.11091 Des. Codes Cryptography 83, No. 2, 425-443 (2017). Summary: 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 G. F. Wu et al. [Sci. China, Math. 58, No. 10, 2081–2094 (2015; Zbl 1325.05013)]. 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 C. Blondeau et al. [Int. J. Inf. Coding Theory 1, No. 2, 149–170 (2010; Zbl 1204.94061)]. Cited in 1 ReviewCited in 25 Documents MSC: 11T06 Polynomials over finite fields Keywords:permutation polynomials; complete permutation polynomials; trace function; differential uniformity Citations:Zbl 1325.05013; Zbl 1204.94061 PDFBibTeX XMLCite \textit{J. Ma} et al., Des. Codes Cryptography 83, No. 2, 425--443 (2017; Zbl 1369.11091) Full Text: DOI arXiv References: [1] Berlekamp E.R., Rumsey H., Solomon G.: On the solution of algebraic equations over finite fields. Inform. Control 10, 553-564 (1967). · Zbl 0166.04803 [2] Blondeau C., Canteaut A., Charpin P.: Differential properties of power functions. Int. J. Inf. Coding Theory 1, 149-170 (2010). · Zbl 1204.94061 [3] Charpin P., Kyureghyan G.M.: Cubic monomial bent functions: a subclass of \[{\cal M}\] M. SIAM J. Discret Math. 22, 650-665 (2008). · Zbl 1171.11062 [4] Ding C., Yuan J.: A family of skew Hadamard difference sets. J. Comb. Theory Ser. A 113, 1526-1535 (2006). · Zbl 1106.05016 [5] 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). · Zbl 1352.11102 [6] Ding C., Xiang Q., Yuan J., Yuan P.: Explicit classes of permutation polynomials of \[\mathbb{F}_{3^{3m}}\] F33m. Sci. China Ser. A 52, 639-647 (2009). · Zbl 1215.11113 [7] Akbary A., Ghioca D., Wang Q.: On constructing permutations of finite fields. Finite Fields Appl. 17, 51-67 (2011). · Zbl 1281.11102 [8] Dobbertin H.: Almost perfect nonlinear power functions on \[{\rm GF}(2^n)\] GF(2n): the Welch case. IEEE Trans. Inf. Theory 45, 1271-1275 (1999). · Zbl 0957.94021 [9] 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). · Zbl 0946.05010 [10] Fernando N., Hou X.: A piecewise construction of permutation polynomials over finite fields. Finite Fields Appl. 18, 1184-1194 (2012). · Zbl 1254.05008 [11] Helleseth T.: Some results about the cross-correlation function between two maximal linear sequences. Discret Math. 16, 209-232 (1976). · Zbl 0348.94017 [12] Hollmann H.D.L., Xiang Q.: A class of permutation polynomials of \[{\bf F}_{2^m}\] F2m related to Dickson polynomials. Finite Fields Appl. 11, 111-122 (2005). · Zbl 1073.11074 [13] Hou X.: Two classes of permutation polynomials over finite fields. J. Comb. Theory Ser. A 118, 448-454 (2011). · Zbl 1230.11146 [14] Hou X.: A new approach to permutation polynomials over finite fields. Finite Fields Appl. 18, 492-521 (2012). · Zbl 1273.11169 [15] Hou X.: Permutation polynomials over finite fields—a survey of recent advances. Finite Fields Appl. 32, 82-119 (2015). · Zbl 1325.11128 [16] Laigle-Chapuy Y.: Permutation polynomials and applications to coding theory. Finite Fields Appl. 13, 58-70 (2007). · Zbl 1107.11048 [17] Li N., Helleseth T., Tang X.: Further results on a class of permutation polynomials over finite fields. Finite Fields Appl. 22, 16-23 (2013). · Zbl 1285.05004 [18] Lidl R., Niederreiter H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20. Addison-Wesley Publishing Company Advanced Book Program, Reading, MA (1983). · Zbl 0554.12010 [19] Niederreiter H., Robinson K.H.: Complete mappings of finite fields. J. Aust. Math. Soc. Ser. A 33, 197-212 (1982). · Zbl 0495.12018 [20] Qu L., Tan Y., Tan C.H., Li C.: Constructing differentially 4-uniform permutations over \[{\mathbb{F}}_{2^{2k}}\] F22k via the switching method. IEEE Trans. Inf. Theory 59, 4675-4686 (2013). · Zbl 1364.94565 [21] Sun J., Takeshita O.Y.: Interleavers for turbo codes using permutation polynomials over integer rings. IEEE Trans. Inf. Theory 51, 101-119 (2005). · Zbl 1280.94121 [22] Tu Z., Zeng X., Hu L.: Several classes of complete permutation polynomials. Finite Fields Appl. 25, 182-193 (2014). · Zbl 1284.05012 [23] Tu Z., Zeng X., Hu L., Li C.: A class of binomial permutation polynomials. arXiv:1310.0337 (2013). [24] 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). · Zbl 1314.11073 [25] Wu G., Li N., Helleseth T., Zhang Y.: Some classes of complete permutation polynomials over \[{{\mathbb{F}}_q}\] Fq. Sci. China Math. 58, 2081-2094 (2015). · Zbl 1325.05013 [26] Yuan J., Ding C.: Four classes of permutation polynomials of \[{\mathbb{F}}_{2^m}\] F2m. Finite Fields Appl. 13, 869-876 (2007). · Zbl 1167.11045 [27] Yuan J., Ding C., Wang H., Pieprzyk J.: Permutation polynomials of the form \[(x^p-x+\delta )^s+L(x)\](xp-x+δ)s+L(x). Finite Fields Appl. 14, 482-493 (2008). · Zbl 1211.11136 [28] Zha Z., Hu L.: Two classes of permutation polynomials over finite fields. Finite Fields Appl. 18, 781-790 (2012). · Zbl 1288.11111 [29] Zieve M.E.: Permutation polynomials induced from permutations of subfields, and some complete sets of mutually orthogonal latin squares. arXiv:1312.1325 (2013). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.