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Joint linear complexity of multisequences consisting of linear recurring sequences

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Abstract

The linear complexity of sequences is one of the important security measures for stream cipher systems. Recently, in the study of vectorized stream cipher systems, the joint linear complexity of multisequences has been investigated. In this paper, we study the joint linear complexity of multisequences consisting of linear recurring sequences. The expectation and variance of the joint linear complexity of random multisequences consisting of linear recurring sequences are determined. These results extend the corresponding results on the expectation and variance of the joint linear complexity of random periodic multisequences. Then we enumerate the multisequences consisting of linear recurring sequences with fixed joint linear complexity. A general formula for the appropriate counting function is derived. Some convenient closed-form expressions for the counting function are determined in special cases. Furthermore, we derive tight upper and lower bounds on the counting function in general. Some interesting relationships among the counting functions of certain cases are established. The generating polynomial for the distribution of joint linear complexities is determined. The proofs use new methods that enable us to obtain results of great generality.

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Acknowledgements

A part of this paper was written while Fang-Wei Fu and Ferruh Özbudak were with Temasek Laboratories, National University of Singapore. They would like to express their thanks to Temasek Laboratories and the Department of Mathematics at the National University of Singapore for the hospitality.

This research was supported by the DSTA grant R-394-000-025-422 with Temasek Laboratories in Singapore.

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Authors and Affiliations

  1. Chern Institute of Mathematics and Key Laboratory of Pure Mathematics and Combinatorics, Nankai University, Tianjin, 300071, China

    Fang-Wei Fu

  2. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Republic of Singapore

    Harald Niederreiter

  3. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637616, Republic of Singapore

    Ferruh Özbudak

  4. Department of Mathematics, Middle East Technical University, 06531, Ankara, Turkey

    Ferruh Özbudak

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  1. Fang-Wei Fu
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  2. Harald Niederreiter
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  3. Ferruh Özbudak
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Correspondence to Harald Niederreiter.

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Fu, FW., Niederreiter, H. & Özbudak, F. Joint linear complexity of multisequences consisting of linear recurring sequences. Cryptogr. Commun. 1, 3–29 (2009). https://doi.org/10.1007/s12095-007-0001-4

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