Abstract
The linear complexity of a sequence has been used as an important measure of keystream strength, hence designing a sequence with high linear complexity and \(k\)-error linear complexity is a popular research topic in cryptography. In this paper, the concept of stable \(k\)-error linear complexity is proposed to study sequences with stable and large \(k\)-error linear complexity. In order to study linear complexity of binary sequences with period \(2^n\), a new tool called cube theory is developed. By using the cube theory, one can easily construct sequences with the maximum stable \(k\)-error linear complexity. For such purpose, we first prove that a binary sequence with period \(2^n\) can be decomposed into some disjoint cubes. Second, it is proved that the maximum \(k\)-error linear complexity is \(2^n-(2^l-1)\) over all \(2^n\)-periodic binary sequences, where \(2^{l-1}\le k<2^{l}\). Finally, continuing the work of Kurosawa et al., a characterization is presented about the minimum number \(k\) for which the second decrease occurs in the \(k\)-error linear complexity of a \(2^n\)-periodic binary sequence \(s\).
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Ding, C., Xiao, G., Shan, W., et al. (eds.): The Stability Theory of Stream Ciphers. LNCS, vol. 561, pp. 85–88. Springer, Heidelberg (1991)
Etzion, T., Kalouptsidis, N., Kolokotronis, N., Limniotis, K., Paterson, K.G.: Properties of the error linear complexity spectrum. IEEE Trans. Inf. Theory 55(10), 4681–4686 (2009)
Games, R.A., Chan, A.H.: A fast algorithm for determining the complexity of a binary sequence with period \(2^n\). IEEE Trans. Inf. Theory 29(1), 144–146 (1983)
Fu, F.-W., Niederreiter, H., Su, M.: The characterization of 2\(^{n}\)-periodic binary sequences with fixed 1-error linear complexity. In: Gong, G., Helleseth, T., Song, H.-Y., Yang, K. (eds.) SETA 2006. LNCS, vol. 4086, pp. 88–103. Springer, Heidelberg (2006)
Hu, H., Feng, D.: Periodic sequences with very large 1-error linear complexity over \(Fq\). J. Softw. 16(5), 940–945 (2005)
Kurosawa, K., Sato, F., Sakata, T., Kishimoto, W.: A relationship between linear complexity and \(k\)-error linear complexity. IEEE Trans. Inf. Theory 46(2), 694–698 (2000)
Meidl, W.: How many bits have to be changed to decrease the linear complexity? Des. Codes Cryptogr. 33, 109–122 (2004)
Meidl, W.: On the stability of \(2^n\)-periodic binary sequences. IEEE Trans. Inf. Theory 51(3), 1151–1155 (2005)
Niederreiter, H.: Periodic sequences with large \(k\)-error linear complexity. IEEE Trans. Inf. Theory 49, 501–505 (2003)
Stamp, M., Martin, C.F.: An algorithm for the \(k\)-error linear complexity of binary sequences with period \(2^{n}\). IEEE Trans. Inf. Theory 39, 1398–1401 (1993)
Zhou, J.Q.: On the \(k\)-error linear complexity of sequences with period 2\(p^n\) over GF(q). Des. Codes Cryptogr. 58(3), 279–296 (2011)
Zhou, J.Q., Liu, W.Q.: The \(k\)-error linear complexity distribution for \(2^n\)-periodic binary sequences. Des. Codes Cryptogr. (2013). http://link.springer.com/article/10.1007/s10623-013-9805-8
Zhu, F.X., Qi, W.F.: The 2-error linear complexity of \(2^n\)-periodic binary sequences with linear complexity \(2^n-1.\) J. Electron. (China) 24(3), 390–395 (2007). http://www.springerlink.com/content/3200vt810p232769/
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Zhou, J., Liu, W., Zhou, G. (2014). Cube Theory and Stable \(k\)-Error Linear Complexity for Periodic Sequences. In: Lin, D., Xu, S., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2013. Lecture Notes in Computer Science(), vol 8567. Springer, Cham. https://doi.org/10.1007/978-3-319-12087-4_5
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DOI: https://doi.org/10.1007/978-3-319-12087-4_5
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