# Counting Functions for the *k*-Error Linear Complexity of 2^{n}-Periodic Binary Sequences

## Abstract

Linear complexity is an important measure of the cryptographic strength of key streams used in stream ciphers. The linear complexity of a sequence can decrease drastically when a few symbols are changed. Hence there has been considerable interest in the *k*-error linear complexity of sequences which measures this instability in linear complexity. For 2^{ n }-periodic sequences it is known that minimum number of changes needed per period to lower the linear complexity is the same for sequences with fixed linear complexity. In this paper we derive an expression to enumerate all possible values for the *k*-error linear complexity of 2^{ n }-periodic binary sequences with fixed linear complexity *L*, when *k* equals the minimum number of changes needed to lower the linear complexity below *L*. For some of these values we derive the expression for the corresponding number of 2^{ n }-periodic binary sequences with fixed linear complexity and *k*-error linear complexity when *k* equals the minimum number of changes needed to lower the linear complexity. These results are of importance to compute some statistical properties concerning the stability of linear complexity of 2^{ n }-periodic binary sequences.

## Keywords

Periodic sequence linear complexity*k*-error linear complexity

## References

- 1.Fengxiang, Z., Wenfeng, Q.: The 2-error linear complexity of 2
^{n}-periodic binary sequences with linear complexity 2^{n}− 1. Journal of Electronics (China) 24(3), 390–395 (2007)CrossRefGoogle Scholar - 2.Games, R.A., Chan, A.H.: A fast algorithm for determining the complexity of a pseudo-random sequence with period 2
^{n}. IEEE Trans. Inform. Theory 29(1), 144–146 (1983)MathSciNetCrossRefMATHGoogle Scholar - 3.Kurosawa, K., Sato, F., Sakata, T., Kishimoto, W.: A relationship between linear complexity and
*k*-error linear complexity. IEEE Trans. Inform. Theory 46(2), 694–698 (2000)MathSciNetCrossRefMATHGoogle Scholar - 4.Meidl, W.: On the stability of 2
^{n}-periodic binary sequences. IEEE Trans. Inform. Theory 51(3), 1151–1155 (2005)MathSciNetCrossRefMATHGoogle Scholar - 5.Meidl, W., Venkateswarlu, A.: Remarks on the
*k*-error linear complexity of*p*^{n}-periodic sequences. Design, Codes and Cryptography 42(2), 181–193 (2007)MathSciNetCrossRefMATHGoogle Scholar - 6.Rueppel, R.A.: Analysis and Design of Stream Ciphers. Springer, Heidelberg (1986)CrossRefMATHGoogle Scholar
- 7.Stamp, M., Martin, C.F.: An algorithm for the
*k*-error linear complexity of binary sequences with period 2^{n}. IEEE Trans. Inform. Theory 39(4), 1398–1401 (1993)MathSciNetCrossRefMATHGoogle Scholar