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An algorithm for computing the error sequence of \(p^{n}\)-periodic binary sequences

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Abstract

For binary sequences with period \(p^{n}\), where \(p\) is an odd prime and 2 is a primitive root modulo \(p^{2}\), we present an algorithm which computes the minimum number \(k\) so that the \(k\)-error linear complexity is not greater than a given constant \(c\). An associated error sequence which gives the \(k\)-error linear complexity is also obtained.

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References

  1. Ding, C., Xiao, G., Shan, W.: The Stability Theory of Stream Ciphers. Lecture Notes in Computer Science, 561. Springer, Berlin (1991)

    Book  Google Scholar 

  2. 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)

    Article  MathSciNet  Google Scholar 

  3. Games, R.A., Chan, A.H.: A fast algorithm for determining the linear complexity of a binary sequence with period \(2^{n}\). IEEE Trans. Inf. Theory, IT 29(1), 144–146 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Han, Y.K., Chung, J., Yang, K.: On the \(k\)-error linear complexity of \(p^{m}\)-periodic binary sequences. IEEE Trans. Inf. Theory 53(6), 2297–2304 (2007)

    Article  MathSciNet  Google Scholar 

  5. Kaida, T., Uehara, S., Imamura, K.: An algorithm for the \(k\)-error linear complexity of sequences over GF(\(p^{m}\))with period \(p^{n}\), \(p\) a prime. Inf. Comput. 151, 134–147 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lauder, A., Paterson, K.G.: Computing the error linear complexity spectrum of a binary sequence of period \(2^{n}\). IEEE Trans. Inf. Theory 49(1), 273–280 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Meidl, W.: How many bits have to be changed to decrease the linear complexity? Des. Codes Cryptogr. 33, 109–122 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Meidl, W.: On the stability of \(2^{n}\)-periodic binary sequences. IEEE Trans. Inf. Theory 51(3), 1151–1155 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Meidl, W., Niederreiter, H.: Counting functions and expected values for the \(k\)-error linear complexity. Finite Fields Appl. 8, 142–154 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Meidl, W., Niederreiter, H.: On the expected value of the linear complexity and \(k\)-error linear complexity of periodic sequences. IEEE Trans. Inf. Theory 48(11), 2817–2825 (2002)

    Article  MathSciNet  Google Scholar 

  12. Rueppel, R.A.: Analysis and Design of Stream Ciphers. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  13. Sălăgean, A.: On the computation of the linear complexity and the \(k\)-error linear complexity of binary sequences with period a power of two. IEEE Trans. Inf. Theory 51(3), 1145–1150 (2005)

    Article  MATH  Google Scholar 

  14. Stamp, M., Martin, C.F.: An algorithm for the \(k\)-error linear complexity of binary sequences of period \(2^{n}\). IEEE Trans. Inf. Theory 39(4), 1398–1401 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Wei, S., Chen, Z., Xiao, G.: A fast algorithm for the \(k\)-error linear complexity of a binary sequence. In: 2001 International Conferences on Info-tech and Info-net Proceedings, ICII 2001-Beijing, 5, pp. 152–157 (2001).

  16. Xiao, G., Wei, S., Lam, K.Y., Imamura, K.: A fast algorithm for determining the linear complexity of a sequence with period \(p^{n}\) over GF(q). IEEE Trans. Inf. Theory 46(6), 2203–2206 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The author would like to thank the anonymous referees for their helpful suggestions on the manuscript.

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Correspondence to Miao Tang.

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Tang, M. An algorithm for computing the error sequence of \(p^{n}\)-periodic binary sequences. AAECC 25, 197–212 (2014). https://doi.org/10.1007/s00200-014-0222-7

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  • DOI: https://doi.org/10.1007/s00200-014-0222-7

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