Linear complexity of generalized cyclotomic sequences with period \(p^nq^m\). (English) Zbl 1534.94072
Mesnager, Sihem (ed.) et al., Arithmetic of finite fields. 9th international workshop, WAIFI 2022, Chengdu, China, August 29 – September 2, 2022. Revised selected papers. Cham: Springer. Lect. Notes Comput. Sci. 13638, 320-333 (2023).
Summary: Linear complexity is a very important merit factor for measuring the unpredictability of pseudo-random sequences for applications. The higher the linear complexity, the better the unpredictability of a sequence. In this paper, we continue the investigation of generalized cyclotomic sequences constructed by new generalized cyclotomy presented by X. Zeng et al. [IEEE Trans. Inf. Theory 59, No. 5, 3237–3248 (2013; Zbl 1364.94087)]. In detail, we consider the new generalized cyclotomic sequence with period \(p^nq^m\) where \(p\), \(q\) are odd distinct primes and \(n\), \(m\) are natural numbers. It is shown that these sequences have high linear complexity. Finally, we also give some examples to illustrate the correctness of our results.
For the entire collection see [Zbl 1516.11002].
For the entire collection see [Zbl 1516.11002].
MSC:
| 94A55 | Shift register sequences and sequences over finite alphabets in information and communication theory |
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
| 11K45 | Pseudo-random numbers; Monte Carlo methods |
Citations:
Zbl 1364.94087References:
| [1] | Golomb, SW, Shift Register Sequences (1967), San Francisco: Holden-Day, San Francisco · Zbl 0267.94022 |
| [2] | Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20. Addison-Wesley (1983) · Zbl 0554.12010 |
| [3] | Ding, C.; Helleseth, T., New generalized cyclotomy and its applications, Finite Fields Their Appl., 4, 2, 140-166 (1998) · Zbl 0908.11058 · doi:10.1006/ffta.1998.0207 |
| [4] | Whiteman, A.L.: A family of difference sets. Illinois J. Math. 6, 107-121 (1962) · Zbl 0099.26502 |
| [5] | Ding, C.; Helleseth, T., Generalized cyclotomic codes of length \(p_1^{e_1}\cdots p_t^{e_t}\), IEEE Trans. Inf. Theory, 45, 2, 467-474 (1999) · Zbl 0946.94026 |
| [6] | Fan, C.; Ge, G., A unified approach to Whiteman’s and Ding-Helleseth’s generalized cyclotomy over residue class rings, IEEE Trans. Inf. Theory, 60, 2, 1326-1336 (2014) · Zbl 1364.94675 · doi:10.1109/TIT.2013.2290694 |
| [7] | Du, X.; Chen, Z., A generalization of the Hall’s sextic residue sequences, Inf. Sci., 222, 784-794 (2013) · Zbl 1293.11087 · doi:10.1016/j.ins.2012.07.048 |
| [8] | Yan, T.; Li, S.; Xiao, G., On the linear complexity of generalized cyclotomic sequences with the period \(p^m\), Appl. Math. Lett., 21, 2, 187-193 (2008) · Zbl 1234.94025 · doi:10.1016/j.aml.2007.03.011 |
| [9] | Chen, X.; Chen, Z.; Liu, H., A family of pseudorandom binary sequences derived from generalized cyclotomic classes modulo \(p^{m+1}q^{n+1}\), Int. J. Netw. Secur., 22, 4, 610-620 (2020) |
| [10] | Hu, L.; Yue, Q.; Wang, MH, The linear complexity of Whiteman’s generalized cyclotomic sequences of period \(p^{m+1}q^{n+1}\), IEEE Trans. Inf. Theory, 58, 8, 5533-5543 (2012) · Zbl 1364.11136 · doi:10.1109/TIT.2012.2196254 |
| [11] | Kim, Y.J., Song, H.Y.: Linear complexity of prime \(n\)-square sequences. In: 2008 IEEE International Symposium on Information Theory, pp. 2405-2408 (2008) |
| [12] | Ke, P.; Zhang, J.; Zhang, S., On the linear complexity and the autocorrelation of generalized cyclotomic binary sequences of length \(2p^m\), Des. Codes Cryptogr., 67, 3, 325-339 (2013) · Zbl 1277.11113 · doi:10.1007/s10623-012-9610-9 |
| [13] | Zeng, X.; Cai, H.; Tang, X.; Yang, Y., Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59, 5, 3237-3248 (2013) · Zbl 1364.94087 · doi:10.1109/TIT.2013.2237754 |
| [14] | Xu, S.; Cao, X.; Mi, J.; Tang, C., More cyclotomic constructions of optimal frequency-hopping sequences, Adv. Math. Commun., 13, 3, 373-391 (2019) · Zbl 1443.94040 · doi:10.3934/amc.2019024 |
| [15] | Xiao, Z.; Zeng, X.; Li, C.; Helleseth, T., New generalized cyclotomic binary sequences of period \(p^2\), Des. Codes Cryptogr., 86, 7, 1483-1497 (2018) · Zbl 1391.94718 · doi:10.1007/s10623-017-0408-7 |
| [16] | Edemskiy, V.; Li, C.; Zeng, X.; Helleseth, T., The linear complexity of generalized cyclotomic binary sequences of period \(p^n\), Des. Codes Cryptogr., 87, 5, 1183-1197 (2019) · Zbl 1480.94023 · doi:10.1007/s10623-018-0513-2 |
| [17] | Ye, Z.; Ke, P.; Wu, C., A further study of the linear complexity of new binary cyclotomic sequence of length \(p^n\), Appl. Algebra Eng. Commun. Comput., 30, 3, 217-231 (2019) · Zbl 1416.94037 · doi:10.1007/s00200-018-0368-9 |
| [18] | Ouyang, Y.; Xie, X., Linear complexity of generalized cyclotomic sequences of period \(2p^m\), Des. Codes Cryptogr., 87, 5, 1-12 (2019) |
| [19] | Edemskiy, V.; Wu, C., On the linear complexity of binary sequences derived from generalized cyclotomic classes modulo \(2^np^m\), WSEAS Trans. Math., 18, 197-202 (2019) |
| [20] | Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics. Springer, New York (1990). doi:10.1007/978-1-4757-2103-4 · Zbl 0712.11001 |
| [21] | Cusick, T.; Ding, C.; Renvall, A., Stream Ciphers and Number Theory (2004), North-Holland mathematical library: Elsevier, North-Holland mathematical library · Zbl 1069.11053 |
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