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Construction of de Bruijn sequences from product of two irreducible polynomials

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Abstract

We study a class of Linear Feedback Shift Registers (LFSRs) with characteristic polynomial f(x) = p(x)q(x) where p(x) and q(x) are distinct irreducible polynomials in 𝔽2[x]. Important properties of the LFSRs, such as the cycle structure and the adjacency graph, are derived. A method to determine a state belonging to each cycle and a generic algorithm to find all conjugate pairs shared by any pair of cycles are given. The process explicitly determines the edges and their labels in the adjacency graph. The results are then combined with the cycle joining method to efficiently construct a new class of de Bruijn sequences. An estimate of the number of resulting sequences is given. In some cases, using cyclotomic numbers, we can determine the number exactly.

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Acknowledgements

The work of Z. Chang is supported by the Joint Fund of the National Natural Science Foundation of China under Grant U1304604. Research Grants TL-9014101684-01 and MOE2013-T2-1-041 support the research carried out by M. F. Ezerman, S. Ling, and H. Wang.

The collaboration leading to this paper was performed while Z. Chang was a visiting scholar at the Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University.

The authors thank the referees and the editor for their kind suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Zuling Chang

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore

    Martianus Frederic Ezerman, San Ling & Huaxiong Wang

Authors
  1. Zuling Chang
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  2. Martianus Frederic Ezerman
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  3. San Ling
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  4. Huaxiong Wang
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Correspondence to Martianus Frederic Ezerman.

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Chang, Z., Ezerman, M.F., Ling, S. et al. Construction of de Bruijn sequences from product of two irreducible polynomials. Cryptogr. Commun. 10, 251–275 (2018). https://doi.org/10.1007/s12095-017-0219-8

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  • DOI: https://doi.org/10.1007/s12095-017-0219-8

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