Linear complexity of Ding-Helleseth generalized cyclotomic sequences of order eight

  • Yana Liang
  • Jiali Cao
  • Xingfa Chen
  • Shiping Cai
  • Xiang FanEmail author


During the last two decades, many kinds of periodic sequences with good pseudorandom properties have been constructed from classical and generalized cyclotomic classes, and used as keystreams for stream ciphers and secure communications. Among them are a family DH-GCSd of generalized cyclotomic sequences on the basis of Ding and Helleseth’s generalized cyclotomy, of length pq and order \(d={\gcd }(p-1,q-1)\) for distinct odd primes p and q. The linear complexity (or linear span), as a valuable measure of unpredictability, is precisely determined for DH-GCS8 in this paper. Our approach is based on Edemskiy and Antonova’s computation method with the help of explicit expressions of Gaussian classical cyclotomic numbers of order 8. Our result for d = 8 is compatible with Yan’s low bound (pq − 1)/2 on the linear complexity for any order d, which is high enough to resist attacks of the Berlekamp–Massey algorithm. Finally, we include SageMath codes to illustrate the validity of our result by examples.


Linear complexity Cyclotomic sequence Cyclotomic number SageMath 

Mathematics Subject Classification (2010)

11B50 94A55 94A60 



We would like to thank the editor and two reviewers for their helpful suggestions to improve this paper. The main part of this work was done during the first author Liang’s one-year visit to Sun Yat-sen University from 2017 to 2018. We are very grateful to Professor Zheng-An Yao for his generous support to our seminar. We would also like to express our deep gratitude to Associate Professor Chang-An Zhao for introducing this subject to us.


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Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhaoqing UniversityZhaoqingChina
  2. 2.School of Applied MathematicsGuangdong University of TechnologyGuangzhouChina
  3. 3.Department of MathematicsGuangdong University of EducationGuangzhouChina
  4. 4.School of MathematicsSun Yat-sen UniversityGuangzhouChina

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