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On the linear complexity and autocorrelation of generalized cyclotomic binary sequences with period $ 4p^n $
Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China |
In this paper, a new class of generalized cyclotomic binary sequences with period $ 4p^n $ is proposed. These sequences are almost balanced, and the explicit formulas of their linear complexity and autocorrelation are presented.
References:
[1] |
E. Bai, X. Liu and G. Xiao,
Linear complexity of new generalized cyclotomic sequences of order two of length $pq$, IEEE Trans. Inf. Theory, 51 (2005), 1849-1853.
doi: 10.1109/TIT.2005.846450. |
[2] |
D.M. Burton, Elementary Number Theory, McGram-Hill, New York, 1998. |
[3] |
T. W. Cusick, C. Ding and A. Renvall, Stream Ciphers and Number Theory, Elsevier Science B. V., Amsterdam, 2004. |
[4] |
C. Ding,
Linear complexity of generalized cyclotomic binary sequences of order $2$, Finite Fields Appl., 3 (1997), 159-174.
doi: 10.1006/ffta.1997.0181. |
[5] |
C. Ding,
Autocorrelation values of generalized cyclotomic sequences of order two, IEEE Trans. Inf. Theory, 44 (1998), 1699-1702.
doi: 10.1109/18.681354. |
[6] |
C. Ding and T. Helleseth,
New generalized cyclotomy and its applications, Finite Fields Appl., 4 (1998), 140-166.
doi: 10.1006/ffta.1998.0207. |
[7] |
C. Ding and T. Helleseth,
Generalized cyclotomy codes of length $p^{m_{1}}_{1}p^{m_{2}}_{2}\cdots p^{m_{t}}_{t}$, IEEE Trans. Inf. Theory, 45 (1999), 467-474.
doi: 10.1109/18.748996. |
[8] |
X. Dong,
Linear complexity of generalized cyclotomic binary sequences of length $4p^n$, Inf. Sci. Lett., 4 (2015), 67-70.
|
[9] |
V. Edemskiy,
About computation of the linear complexity of generalized cyclotomic sequences with period $p^{n+1}$, Des. Codes Cryptogr., 61 (2011), 251-260.
doi: 10.1007/s10623-010-9474-9. |
[10] |
V. Edemskiy and O. Antonova,
The evaluation of the linear complexity and the autocorrelation of generalized cyclotomic binary sequences of length $2^np^m$, International Journal of Mathematical Models and Methods in Applied Sciences, 9 (2015), 512-517.
|
[11] |
V. Edemskiy, C. Li, X. Zeng and T. Helleseth,
The linear complexity of generalized cyclotomic binary sequences of period $p^n$, Des. Codes Cryptogr., 87 (2018), 1183-1197.
doi: 10.1007/s10623-018-0513-2. |
[12] |
V. Edemskiy and C. Wu,
On the linear complexity of binary sequences derived from generalized cyclotomic classes modulo $2^np^m$, WSEAS Transactions on Mathematics, 18 (2019), 197-202.
|
[13] |
C. Fan and G. Ge,
A unified approach to Whiteman's and Ding-Helleseth's generalized cyclotomy over residue class rings, IEEE Trans. Inf. Theory, 60 (2014), 1326-1336.
doi: 10.1109/TIT.2013.2290694. |
[14] |
S. W. Golomb and G. Gong, Signal Design for Good Correlation, for Wireless Communications, Cryptography and Radar Applications, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511546907.![]() ![]() ![]() |
[15] |
P. Ke, J. Zhang and S. Zhang,
On the linear complexity and the autocorrelation of generalized cyclotomic binary sequences of length $2p^n$, Des. Codes Cryptogr., 67 (2013), 325-339.
doi: 10.1007/s10623-012-9610-9. |
[16] |
Y. J. Kim, S. Y. Jin and H. Y. Song,
Linear complexity and autocorrelation of prime cube sequences, AAECC, LNCS, 4851 (2007), 188-197.
doi: 10.1007/978-3-540-77224-8_23. |
[17] |
J. L. Massey,
Shift-register synthesis and BCH decoding, IEEE Trans. Inf. Theory, 15 (1969), 122-127.
doi: 10.1109/tit.1969.1054260. |
[18] |
M. B. Nathnson, Elementary Methods in Number Theory, Springer, Berlin, GTM 195, 2003. |
[19] |
A. L. Whiteman,
A family of difference sets, Illinois J. Math., 6 (1962), 107-121.
|
[20] |
Z. Xiao, X. Zeng, C. Li and T. Helleseth,
New generalized cyclotomic binary sequences of period $p^2$, Des. Codes Cryptogr., 86 (2018), 1483-1497.
doi: 10.1007/s10623-017-0408-7. |
[21] |
T. Yan, B. Huang and G. Xiao,
Cryptographic properties of some binary generalized cyclotomic sequences with length $p^2$, Inf. Sci., 178 (2008), 1078-1086.
doi: 10.1016/j.ins.2007.02.040. |
[22] |
T. Yan, S. Li and G. Xiao,
On the linear complexity of generalized cyclotomic sequences with period $p^m$, Appl. Math. Lett., 21 (2008), 187-193.
doi: 10.1016/j.aml.2007.03.011. |
[23] |
T. Yan and X. Li,
Some notes on the generalized cyclotomic binary sequences of length $2p^m$ and $p^m$, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, 96 (2013), 2049-2051.
doi: 10.1007/s00200-012-0177-5. |
[24] |
T. Yan, R. Sun and G. Xiao,
Autocorrelation and linear complexity of the new generalized cyclotomic sequences, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, 90 (2007), 857-864.
|
[25] |
X. Zeng, H. Cai, X. Tang and Y. Yang,
Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59 (2013), 3237-3248.
doi: 10.1109/TIT.2013.2237754. |
[26] |
J. Zhang, C. Zhao and X. Ma,
Linear complexity of generalized cyclotomic binary sequences of length $2p^m$, Applicable Algebra in Engineering Communication and Computing, 21 (2010), 93-108.
doi: 10.1007/s00200-009-0116-2. |
show all references
References:
[1] |
E. Bai, X. Liu and G. Xiao,
Linear complexity of new generalized cyclotomic sequences of order two of length $pq$, IEEE Trans. Inf. Theory, 51 (2005), 1849-1853.
doi: 10.1109/TIT.2005.846450. |
[2] |
D.M. Burton, Elementary Number Theory, McGram-Hill, New York, 1998. |
[3] |
T. W. Cusick, C. Ding and A. Renvall, Stream Ciphers and Number Theory, Elsevier Science B. V., Amsterdam, 2004. |
[4] |
C. Ding,
Linear complexity of generalized cyclotomic binary sequences of order $2$, Finite Fields Appl., 3 (1997), 159-174.
doi: 10.1006/ffta.1997.0181. |
[5] |
C. Ding,
Autocorrelation values of generalized cyclotomic sequences of order two, IEEE Trans. Inf. Theory, 44 (1998), 1699-1702.
doi: 10.1109/18.681354. |
[6] |
C. Ding and T. Helleseth,
New generalized cyclotomy and its applications, Finite Fields Appl., 4 (1998), 140-166.
doi: 10.1006/ffta.1998.0207. |
[7] |
C. Ding and T. Helleseth,
Generalized cyclotomy codes of length $p^{m_{1}}_{1}p^{m_{2}}_{2}\cdots p^{m_{t}}_{t}$, IEEE Trans. Inf. Theory, 45 (1999), 467-474.
doi: 10.1109/18.748996. |
[8] |
X. Dong,
Linear complexity of generalized cyclotomic binary sequences of length $4p^n$, Inf. Sci. Lett., 4 (2015), 67-70.
|
[9] |
V. Edemskiy,
About computation of the linear complexity of generalized cyclotomic sequences with period $p^{n+1}$, Des. Codes Cryptogr., 61 (2011), 251-260.
doi: 10.1007/s10623-010-9474-9. |
[10] |
V. Edemskiy and O. Antonova,
The evaluation of the linear complexity and the autocorrelation of generalized cyclotomic binary sequences of length $2^np^m$, International Journal of Mathematical Models and Methods in Applied Sciences, 9 (2015), 512-517.
|
[11] |
V. Edemskiy, C. Li, X. Zeng and T. Helleseth,
The linear complexity of generalized cyclotomic binary sequences of period $p^n$, Des. Codes Cryptogr., 87 (2018), 1183-1197.
doi: 10.1007/s10623-018-0513-2. |
[12] |
V. Edemskiy and C. Wu,
On the linear complexity of binary sequences derived from generalized cyclotomic classes modulo $2^np^m$, WSEAS Transactions on Mathematics, 18 (2019), 197-202.
|
[13] |
C. Fan and G. Ge,
A unified approach to Whiteman's and Ding-Helleseth's generalized cyclotomy over residue class rings, IEEE Trans. Inf. Theory, 60 (2014), 1326-1336.
doi: 10.1109/TIT.2013.2290694. |
[14] |
S. W. Golomb and G. Gong, Signal Design for Good Correlation, for Wireless Communications, Cryptography and Radar Applications, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511546907.![]() ![]() ![]() |
[15] |
P. Ke, J. Zhang and S. Zhang,
On the linear complexity and the autocorrelation of generalized cyclotomic binary sequences of length $2p^n$, Des. Codes Cryptogr., 67 (2013), 325-339.
doi: 10.1007/s10623-012-9610-9. |
[16] |
Y. J. Kim, S. Y. Jin and H. Y. Song,
Linear complexity and autocorrelation of prime cube sequences, AAECC, LNCS, 4851 (2007), 188-197.
doi: 10.1007/978-3-540-77224-8_23. |
[17] |
J. L. Massey,
Shift-register synthesis and BCH decoding, IEEE Trans. Inf. Theory, 15 (1969), 122-127.
doi: 10.1109/tit.1969.1054260. |
[18] |
M. B. Nathnson, Elementary Methods in Number Theory, Springer, Berlin, GTM 195, 2003. |
[19] |
A. L. Whiteman,
A family of difference sets, Illinois J. Math., 6 (1962), 107-121.
|
[20] |
Z. Xiao, X. Zeng, C. Li and T. Helleseth,
New generalized cyclotomic binary sequences of period $p^2$, Des. Codes Cryptogr., 86 (2018), 1483-1497.
doi: 10.1007/s10623-017-0408-7. |
[21] |
T. Yan, B. Huang and G. Xiao,
Cryptographic properties of some binary generalized cyclotomic sequences with length $p^2$, Inf. Sci., 178 (2008), 1078-1086.
doi: 10.1016/j.ins.2007.02.040. |
[22] |
T. Yan, S. Li and G. Xiao,
On the linear complexity of generalized cyclotomic sequences with period $p^m$, Appl. Math. Lett., 21 (2008), 187-193.
doi: 10.1016/j.aml.2007.03.011. |
[23] |
T. Yan and X. Li,
Some notes on the generalized cyclotomic binary sequences of length $2p^m$ and $p^m$, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, 96 (2013), 2049-2051.
doi: 10.1007/s00200-012-0177-5. |
[24] |
T. Yan, R. Sun and G. Xiao,
Autocorrelation and linear complexity of the new generalized cyclotomic sequences, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, 90 (2007), 857-864.
|
[25] |
X. Zeng, H. Cai, X. Tang and Y. Yang,
Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59 (2013), 3237-3248.
doi: 10.1109/TIT.2013.2237754. |
[26] |
J. Zhang, C. Zhao and X. Ma,
Linear complexity of generalized cyclotomic binary sequences of length $2p^m$, Applicable Algebra in Engineering Communication and Computing, 21 (2010), 93-108.
doi: 10.1007/s00200-009-0116-2. |
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