Abstract
A class of binary sequences with period 2p is constructed using generalized cyclotomic classes, and their linear complexity, minimal polynomial over \({\mathbb {F}_{{q}}}\) as well as 2-adic complexity are determined using Gauss period and group ring theory. The results show that the linear complexity of these sequences attains the maximum when \(p\equiv \pm 1\pmod {8}\) and is equal to p+1 when \(p\equiv \pm 3(\bmod 8)\) over extension field. Moreover, the 2-adic complexity of these sequences is maximum. According to Berlekamp–Massey(B–M) algorithm and the rational approximation algorithm (RAA), these sequences have quite good cryptographic properties in the aspect of linear complexity and 2-adic complexity.
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References
Ding C.S., Helleseth T.: New generalized cyclotomy and its applications. Finite Fields Their Appl. 4(2), 140–166 (1998).
Ding C.S., Helleseth T., Shan W.J.: On the linear complexity of Legendre sequences. IEEE Trans. Inform. Theory 44(3), 1276–1278 (1998).
Du X.N., Chen Z.X.: Linear complexity of quaternary sequences generated using generalized cyclotomic classes modulo \(2p\). IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 94(5), 1214–1217 (2011).
Edemskiy V., Li C.L., Zeng X.Y.: The linear complexity of generalized cyclotomic binary sequences of period \({p}^{n}\). Des. Codes Cryptogr. 87(5), 1183–1197 (2019).
Holfer R., Winterholf A.: On the 2-adic complexity of the two-prime generator. IEEE Trans. Inform. Theory 64(8), 5957–5960 (2018).
Hu H.: Comments on “a new method to compute the 2-adic complexity of binary sequences’’. IEEE Trans. Inform. Theory 60(9), 5803–5804 (2014).
Jing, X.Y., Qiang, S.Y., Yang, M.H., Feng, K.Q.:Determined of the autocorrelation distribution and 2-adic complexity of generalized cyclotomic binary sequences of order \(2\) with period \(pq\). arXiv.2105.10947vl [cs.IT](30 May 2021)
Ke P.H., Zhang J., Zhang S.Y.: On the linear complexity and the autocorrelation of generalized cyclotomic binary sequences of length \(2{{p}^{m}}\). Des. Codes Cryptogr. 67(3), 325–339 (2013).
Klapper A., Goresky M.: Feedback shift registers, 2-adic span, and combiners with memory. J. Cryptol. 10(2), 111–147 (1997).
Li D.D., Wen Q.Y., Zhang J.: Linear complexity of generalized cyclotomic quaternary sequences with period \(pq\). IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 97(5), 1153–1158 (2014).
Massey J.L.: Shift-rigister synthesis and BCH decoding. IEEE Trans. Inform. Theroy 15, 122–127 (1969).
Ou Y.Y., Xie X.Y.: Linear complexity of generalized cyclotomic sequences of period \(2{{p}^{m}}\). Des. Codes Cryptogr. 87(11), 2585–2596 (2019).
Qiang, S.Y., Jing, X.Y., Yang, M.H.: The 2-adic complexity of two classes of binary sequences with interleaved structure. arXiv.2011.12080vl [cs.IT](24 Nov 2020)
Sun Y.H., Wang Q.Y.: A lowwer bound on the 2-adic complexity of the modified Jacobi sequences. Ctyptogr. Commun. 11(2), 337–349 (2018).
Tian T., Wang Q.Y., Qi M.L.: 2-Adic complexity of binary m-sequences. IEEE Trans. Inform. Theory 56(1), 450–454 (2009).
Wang, Q.Y., Kong, W.G., Yan, Y.:Autocorrelation of a class of quaternary sequences of period \(2{{p}^{m}}\). arXiv.2002.00375vl [cs.IT](2 Feb 2020)
Wang, Y., Yan, L.T., Tian, Q.Ding, L.P.: Autocorrelation and linear complexity of binary generalized cyclotomic sequences of order \(pq\). J. Math. (2021).
Wang Q.Y., Lin D.D., Guang X.: On the linear complexity of Legendre sequences over \({F_q}\). IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 97(7), 1627–1630 (2014).
Wang Y., Xue G.N., Li S.B., Hui F.F.: The linear complexity of a new class of generalized cyclotomic sequence of order \(q\) with period \(2{{p}^{m}}\). J. Electron. Inform. Technol. 41(9), 2151–2155 (2019).
Xiao Zb., Zeng X.Y., Li C.L.: New generalized cyclotomic binary sequences of period \({p}^{2}\). Des. Codes Cryptogr. 86(7), 1483–1497 (2018).
Xiong H., Qu L., Li C.: A new method to compute the 2-adic complexity of binary sequences. IEEE Trans. Inform. Theory 60(4), 2399–2406 (2014).
Yang, M.H., Feng, K.Q.: Determination of 2-adic complexity of generalized binary sequences of order \(2\). arXiv.2007.15327vl [cs.IT](30 July 2020)
Zhang J.W., Zhao C.A., Ma X.: Linear complexity of generalized cyclotomic binary sequences of length \(2{{p}^{m}}\). Appl. Algebra Eng. Commun. Comput. 21(2), 93–108 (2010).
Acknowledgements
The authors wish to thank the reviwers for their detailed and very helpful comments that improved this paper as well as the editors for all their works on this paper. The work of Yan Wang was supported by the National Natural Science Foundation of China under Grant 61902304. The work of Ziling Heng was supported by the National Natural Science Foundation of China under Grant 11901049.
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Wang, Y., Han, X., Wang, W. et al. Linear complexity over \({\mathbb {F}_{{q}}}\) and 2-adic complexity of a class of binary generalized cyclotomic sequences with good autocorrelation. Des. Codes Cryptogr. 90, 1695–1712 (2022). https://doi.org/10.1007/s10623-022-01068-6
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DOI: https://doi.org/10.1007/s10623-022-01068-6