## Abstract

The linear complexity of sequences is an important measure of the cryptographic strength of key streams used in stream ciphers. The instability of linear complexity caused by changing a few symbols of sequences can be measured using k-error linear complexity. In their SETA 2006 paper, Fu et al. (SETA, pp. 88-103, 2006) studied the linear complexity and the 1-error linear complexity of 2 ^{n} -periodic binary sequences to characterize such sequences with fixed 1-error linear complexity. In this paper we study the linear complexity and the k-error linear complexity of 2 ^{n} -periodic binary sequences in a more general setting using a combination of algebraic, combinatorial, and algorithmic methods. This approach allows us to characterize 2 ^{n} -periodic binary sequences with fixed 2- or 3-error linear complexity. Using this characterization we obtain the counting function for the number of 2 ^{n} -periodic binary sequences with fixed k-error linear complexity for k = 2 and 3.

Original language | English |
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Pages (from-to) | 75-97 |

Number of pages | 23 |

Journal | Designs, Codes, and Cryptography |

Volume | 53 |

Issue number | 2 |

DOIs | |

State | Published - Nov 2009 |

### Bibliographical note

Funding Information:A portion of this paper has appeared in the proceedings of the 5th international conference on Sequences and their Applications (SETA 2008). This material is based upon work supported by the National Science Foundation under Grant No. CCF-0514660. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

## Keywords

- K-error linear complexity
- Linear complexity
- Periodic sequence

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Applied Mathematics

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