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Linear complexity and trace representation of quaternary sequences over \(\mathbb {Z}_{4}\) based on generalized cyclotomic classes modulo pq

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Abstract

We define a family of quaternary sequences over the residue class ring modulo 4 of length pq, a product of two distinct odd primes, using the generalized cyclotomic classes modulo pq and calculate the discrete Fourier transform (DFT) of the sequences. The DFT helps us to determine the exact values of linear complexity and the trace representation of the sequences.

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Notes

  1. In [10, Theorem 6], the condition in (3) must be p≡5(mod 8) and q≡1(mod 8), while the condition in (4) must be p≡1(mod 8) and q≡5(mod 8).

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Acknowledgments

The author wishes to thank the editor and the reviewers for the valuable comments, which make our work greatly improved. Thanks also go to one reviewer who pointed out a similar quaternary sequence in [10] and some helpful technique in [16].

Parts of this work were written during a very pleasant visit of the author to the University of Kentucky in Lexington, USA. He wishes to thank Prof. Andrew Klapper for his hospitality.

The work was partially supported by the National Natural Science Foundation of China under grant No. 61373140 and by the Fujian Provincial Natural Science Foundation of China under grant No.2015J01662.

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  1. School of Mathematics, Putian University, Putian, Fujian, 351100, People’s Republic of China

    Zhixiong Chen

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  1. Zhixiong Chen
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Chen, Z. Linear complexity and trace representation of quaternary sequences over \(\mathbb {Z}_{4}\) based on generalized cyclotomic classes modulo pq . Cryptogr. Commun. 9, 445–458 (2017). https://doi.org/10.1007/s12095-016-0185-6

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