A lower bound on the 2-adic complexity of the modified Jacobi sequence

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Abstract

Let p, q be distinct primes satisfying gcd(p −  1, q −  1) = d and let D i , i =  0, 1, · · · ,d −  1, be Whiteman’s generalized cyclotomic classes with \(\mathbb {Z}_{pq}^{\ast }=\cup _{i = 0}^{d-1}D_{i}\). In this paper, we give the values of Gauss periods based on the generalized cyclotomic sets \(D_{0}^{\ast }=\cup _{i = 0}^{\frac {d}{2}-1}D_{2i}\) and \(D_{1}^{\ast }=\cup _{i = 0}^{\frac {d}{2}-1}D_{2i + 1}\). As an application, we determine a lower bound on the 2-adic complexity of the modified Jacobi sequence. Our result shows that the 2-adic complexity of the modified Jacobi sequence is at least pqpq − 1 with period N = pq. This indicates that the 2-adic complexity of the modified Jacobi sequence is large enough to resist the attack of the rational approximation algorithm (RAA) for feedback with carry shift registers (FCSRs).

Keywords

Gauss period Generalized cyclotomic class Modified Jacobi sequence 2-adic complexity 

Mathematics Subject Classification (2010)

11B50 94A55 94A60 

Notes

Acknowledgements

Parts of this work were done during a very pleasant visit of the first author to the School of Mathematics and Statistics at Carleton University. She wishes to thank the hosts for their hospitality. We also thank anonymous referees for their helpful suggestions.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of SciencesChina University of PetroleumQingdaoChina
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
  3. 3.Qilu University of Technology (Shandong Academy of Sciences), Shandong Computer Science Center (National Supercomputer Center in Jinan, Shandong Provincial Key Laboratory of Computer NetworksJinanChina
  4. 4.Key Laboratory of Applied MathematicsFujian Province University (Putian University)PutianChina

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