Abstract
The linear complexity and the \(k\)-error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By using the sieve method of combinatorics, we investigate the \(k\)-error linear complexity distribution of \(2^n\)-periodic binary sequences in this paper based on Games–Chan algorithm. First, for \(k=2,3\), the complete counting functions for the \(k\)-error linear complexity of \(2^n\)-periodic binary sequences (with linear complexity less than \(2^n\)) are characterized. Second, for \(k=3,4\), the complete counting functions for the \(k\)-error linear complexity of \(2^n\)-periodic binary sequences with linear complexity \(2^n\) are presented. Third, as a consequence of these results, the counting functions for the number of \(2^n\)-periodic binary sequences with the \(k\)-error linear complexity for \(k = 2\) and \(3\) are obtained.
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References
Ding C.S., Xiao G.Z., Shan W.J.: The Stability Theory of Stream Ciphers[M]. Lecture Notes in Computer Science, vol. 561, pp. 85–88. Springer, Berlin (1991).
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., 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, Lecture Notes in Computer Science, vol. 4086, pp. 88–103. Springer, Berlin (2006).
Kavuluru R.: \(2^n\)-periodic binary sequences with fixed 2-error or 3-error linear complexity. In: Golomb S., Parker M., Pott A., Winterhof A. (eds.) SETA 2008. Lecture Notes in Computer Science, vol. 5203, pp. 252–265. Springer, Berlin (2008).
Kavuluru R.: Characterization of \(2^n\)-periodic binary sequences with fixed 2-error or 3-error linear complexity. Des. Codes Cryptogr. 53, 75–97 (2009).
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.: On the stablity of \(2^{n}\)-periodic binary sequences. IEEE Trans. Inf. Theory 51(3), 1151–1155 (2005).
Rueppel, R.A.: Analysis and Design of Stream Ciphers (chap. 4). Springer, Berlin (1986).
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.: A counterexample concerning the 3-error linear complexity of \(2^n\)-periodic binary sequences. Des. Codes Cryptogr. 64(3), 285–286 (2012).
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).
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The research was supported by Anhui Natural Science Foundation (No.1208085MF106).
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Communicated by T. Etzion.
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Zhou, J., Liu, W. The \(k\)-error linear complexity distribution for \(2^n\)-periodic binary sequences. Des. Codes Cryptogr. 73, 55–75 (2014). https://doi.org/10.1007/s10623-013-9805-8
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DOI: https://doi.org/10.1007/s10623-013-9805-8
Keywords
- Periodic sequence
- Linear complexity
- \(k\)-error linear complexity
- \(k\)-error linear complexity distribution