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Linear complexity and trace representation of balanced quaternary cyclotomic sequences of prime period p

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Abstract

Let \(p=ef+1\) be an odd prime, where \({e}\equiv 0\,(\bmod \,4)\). A family of balanced quaternary sequences is defined by using the classical cyclotomic classes of order e with respect to p in this paper. We derive the formulas for their linear complexity and trace representation over \(\mathbb {Z}_4\) by computing the discrete Fourier transform of these sequences. As an application, the linear complexity and trace representation over \(\mathbb {Z}_4\) are given for two types of specific sequences with low autocorrelation derived from the cyclotomic classes of order 4 and 8, respectively. Furthermore, we also determine the exact linear complexity and minimal polynomial for each sequence of the second type over the finite field \(\mathbb {F}_4\).

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Acknowledgements

The authors would like to thank the anonymous referees for their very useful comments.

Funding

The work of X. Zeng was supported by the National Nature Science Foundation of China (NSFC) under Grant 62072161.The work of Z. Yang and Z. Xiao was supported by the National Natural Science Foundation of China (NSFC) under Grant 12061027.

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Authors and Affiliations

  1. College of Science, Wuhan University of Science and Technology, Wuhan, 430081, Hubei, China

    Zhiye Yang & Zibi Xiao

  2. Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan, 430062, China

    Xiangyong Zeng

Authors
  1. Zhiye Yang
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  2. Zibi Xiao
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  3. Xiangyong Zeng
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Contributions

Z. Xiao and Z. Yang discussed and developed the basic idea. Z. Yang performed calculations and analyses, and drafted the manuscript. Z. Xiao provided guidance and advice for the research, and critically revised the manuscript for important content. X. Zeng gave some constructive commentaries and supplements, and helped to polish the article. All authors read and approved the final manuscript.

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Correspondence to Zibi Xiao.

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Yang, Z., Xiao, Z. & Zeng, X. Linear complexity and trace representation of balanced quaternary cyclotomic sequences of prime period p. Cryptogr. Commun. 15, 921–940 (2023). https://doi.org/10.1007/s12095-023-00649-z

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