Abstract
In this paper, we construct two generalized cyclotomic binary sequences of period \(2p^{m}\) based on the generalized cyclotomy and compute their linear complexity, showing that they are of high linear complexity when \(m\ge 2\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bai E., Liu X., Xiao G.: Linear complexity of new generalized cyclotomic sequences of order two of length $pq$. IEEE Trans. Inf. Theory 51(5), 1849–1853 (2005).
Chang Z., Li D.: On the linear complexity of generalized cyclotomic binary sequence of length $2pq$. Concurr. Comput. Pract. Exp. 26(8), 1520–1530 (2014).
Cusick T., Ding C., Renvall A.: Stream Ciphers and Number Theory. North-Holland Mathematical Library, vol. 55, pp. 198–212. Elsevier, Amsterdam (1998).
Ding C.: Binary cyclotomic generators. Fast Software Encryption. LNCS, vol. 1008, pp. 29–60. Springer, Leuven (1995).
Ding C.: Linear complexity of generalized cyclotomic binary sequences of order $2$. Finite Fields Appl. 3, 159–174 (1997).
Ding C., Helleseth T.: New generalized cyclotomy and its applications. Finite Fields Appl. 4, 140–166 (1998).
Ding C., Helleseth T.: Generalized cyclotomy codes of length $p_{1}^{m_{1}}p_{2}^{m_{2}}\cdots p_{t}^{m_{t}}$. IEEE Trans. Inf. Theory 45(2), 467–474 (1999).
Edemskiy V.: About computation of the linear complexity of generalized cyclotomic sequences with period $p^{n+1}$. Des. Codes Cryptogr. 61, 251–260 (2011).
Edemskiy V., Li C., Zeng X., Helleseth T.: The linear complexity of generalized cyclotomic binary cyclotomic sequences of period $p^{n}$. Des. Codes Cryptogr. (2018). https://doi.org/10.1007/s10623-018-0513-2.
Golomb S.W.: Shift register sequence. Holden Day, San Francisco (1967).
Hu H.G., Gong G.: New sets of zeros or low correlation zone sequences via interleaving technique. IEEE Trans. Inf. Theory 56, 1702–1713 (2010).
Hu L., Yue Q., Wang M.: The linear complexity of Whiteman’s generalized cyclotomic sequences of period $p^{m+1}q^{n+1}$. IEEE Trans. Inf. Theory 58, 5534–5543 (2012).
Ke P.H., Zhang J., Zhang S.Y.: On the of linear complexity and the autocorrelation of generalized cyclotomic binary sequences with the period $2p^{m}$. Des. Codes Cryptogr. 67(3), 325–339 (2013).
Kim Y. J., Song H. Y.: Linear complexity of prime $n$-square sequences. In: IEEE International Symposium on Information Theory, Toronto, Canada, pp. 2405–2408 (2008).
Kim Y.J., Jin S.Y., Song H.Y.: Linear complexity and autocorrelation of prime cube sequences. LNCS, vol. 4851, pp. 188–197. Springer, Berlin (2007).
Li D.D., Wen Q.Y.: Linear complexity of generalized cyclotomic binary sequences with period $2p^{m+1}q^{n+1}$. IEICE Trans. Fund. Electron. E 98A, 1244–1254 (2015).
Liu F., Peng D.Y., Tang X.H.: On the autocorrelation and the linear complexity of $q$-ary prime $n$-square sequence. Sequences and Their Applications. LNCS, vol. 6338, pp. 139–150. Springer, Berlin (2010).
Park Y.H., Hong D., Chun E.: On the linear complexity of some generalized cyclotomic sequences. Int. J. Algebra Comput. 14(4), 431–439 (2004).
Whiteman A.L.: A family of difference sets. Illionis J. Math. 6(1), 107–121 (1962).
Xiao Z.B., Zeng X.Y., Li C.L., Helleseth T.: New generalized cyclotomic binary sequences of period $p^{2}$. Des. Codes Cryptogr. 86, 1483–1497 (2018).
Yan T.: Some notes on the generalized cyclotomic binary sequences of length $2p^{m}$ and $p^{m}$. IEICE Trans. Fund. E96–A 10, 2049–2051 (2013).
Yan T., Sun R., Xiao G.: Autocorrelation and linear complexity of the new generalized cyclotomic sequences. IEICE Trans. Fund. Electron. E90–A, 857–864 (2007).
Ye Z.F., Ke P.H., Wu C.H.: A further study on the linear complexity of new binary cyclotomic sequence of length $p^{r}$. arXiv:1712.08886V2 [cs.CR] 15 Mar 2018.
Zeng X., Cai H., Tang X., Yang Y.: Optimal frequency hopping sequences of odd length. IEEE Trans. Inf. Theory 59(5), 3237–3248 (2013).
Zhang J.W., Zhao C.A., Ma X.: Linear complexity of generalized cyclotomic binary sequences of length $2p^{m}$. Applicable Algebra in Engineering, Communication and Computing, vol. 21, pp. 93–108. Springer, Berlin (2010).
Acknowledgements
Y. O. would like to thank the Morningside Center of Mathematics for hospitality where part of this paper was written. We thank the referees for many helpful comments, especially for providing a counterexample for a previous conjecture about the linear complexity of sequences in the second class.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by T. Helleseth.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Partially supported by Anhui Initiative in Quantum Information Technologies (Grant No. AHY150200) and NSFC (Grant No. 11571328).
Rights and permissions
About this article
Cite this article
Ouyang, Y., Xie, X. Linear complexity of generalized cyclotomic sequences of period \(2p^{m}\). Des. Codes Cryptogr. 87, 2585–2596 (2019). https://doi.org/10.1007/s10623-019-00638-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-019-00638-5