Abstract
The construction of permutation trinomials over finite fields attracts people’s interest recently due to their simple form and some additional properties. Motivated by some results on the construction of permutation trinomials with Niho exponents, in this paper, by constructing some new fractional polynomials that permute the set of the (q + 1)-th roots of unity in \(\mathbb {F}_{q^{2}}\), we present several classes of permutation trinomials with Niho exponents over \(\mathbb {F}_{q^{2}}\), where q = 5k.
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Ding, C., Qu, L., Wang, Q., Yuan, J., Yuan, P.: Permutation trinomials over finite fields with even characteristic. SIAM J. Dis. Math. 29, 79–92 (2015)
Ding, C., Yuan, J.: A family of skew Hadamard difference sets. J. Comb. Theory, Ser. A 113, 1526–1535 (2006)
Ding, C.: A construction of binary linear codes from Boolean functions. Discret. Math. 339(9), 2288–2303 (2016)
Dobbertin, H.: Almost perfect nonlinear power functions on G F(2n): The Welch case. IEEE Trans. Inf. Theory 45(4), 1271–1275 (1999)
Gupta, R., Sharma, R.K.: Some new classes of permutation trinomials over finite fields with even characteristic. Finite Fields Appl. 41, 89–96 (2016)
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(1), 58–70 (2007)
Li, K., Qu, L., Chen, X.: New classes of permutation binomials and permutation trinomials over finite fields. Finite Fields Appl. 43, 69–85 (2017)
Li, K., Qu, L., Li, C., Fu, S.: New permutation trinomials constructed from fractional polynomials, available online: https://arxiv.org/pdf/1605.06216v1.pdf
Li, N., Helleseth, T.: Several classes of permutation trinomials from Niho exponents. Cryptogr. Commun. 9(6), 693–705 (2017)
Li, N., Helleseth, T.: New permutation trinomials from Niho exponents over finite fields with even characteristic, available online: http://arxiv.org/pdf/1606.03768v1.pdf
Li, N.: On two conjectures about permutation trinomials over \(\mathbb {F}_{3^{2k}}\). Finite Fields Appl. 47, 1–10 (2017)
Lidl, R., Niederreiter, H.: Finite Fields, 2nd edn. Cambridge Univ. Press, Cambridge (1997)
Niho, Y.: Multivalued Cross-Correlation Functions between Two Maximal Linear Recursive Sequences. PhD dissertation, Univ. Southern Calif. Los Angeles (1972)
Park, Y.H., Lee, J.B.: Permutation polynomials and group permutation polynomials. Bull. Aus- tral. Math. Soc. 63, 67–74 (2001)
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(7), 4675–4686 (2013)
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)
Zieve, M.E.: On some permutation polynomials over F q of the form x r h(x (q− 1)/d). Proc. Amer. Math. Soc. 137, 2209–2216 (2009)
Acknowledgements
The authors would like to thank the referees for their comments that improved the presentation and quality of this paper. This work of G. Wu was supported by the Fundamental Research Funds for the Central Universities (No. JB161504) and the National Natural Science Foundation of China (Grants Nos. 61602361, 61671013). This work of N. Li was supported in part by the National Natural Science Foundation of China under Grant 61702166 and the Natural Science Foundation of Hubei Province of China under Grant 2017CFB143.
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Wu, G., Li, N. Several classes of permutation trinomials over \(\mathbb {F}_{5^{n}}\) from Niho exponents. Cryptogr. Commun. 11, 313–324 (2019). https://doi.org/10.1007/s12095-018-0291-8
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DOI: https://doi.org/10.1007/s12095-018-0291-8