Skip to main content
SpringerLink
Log in

On k–error Linear Complexity of Zeng-Cai-Tang-Yang Generalized Cyclotomic Binary Sequences of Period \(p^2\)

  • Conference paper
  • First Online:
Frontiers in Cyber Security (FCS 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 879))

Included in the following conference series:

  • 469 Accesses

Abstract

Due to good pseudorandom properties, generalized cyclotomic sequences have been widely used in simulation, radar systems, cryptography, and so on. In this paper, we consider the k-error linear complexity of Zeng-Cai-Tang-Yang generalized cyclotomic binary sequences of period \(p^2\), proposed in the recent paper “New generalized cyclotomic binary sequences of period \(p^2\)”, by Z. Xiao et al., who calculated the linear complexity of the sequence (Designs, Codes and Cryptography, 2018, 86(7): 1483–1497). More exactly, we determine the values of k-error linear complexity over \(\mathbb {F}_2\) for \(f=2\) and almost \(k>0\) in terms of the theory of Fermat quotients. Results indicate that such sequences have good stability.

An extended version of this work will be submitted to Elsevier. This work supported by the National Natural Science Foundation of China under grant No. 61772292, 61872060, by the National Key R&D Program of China under grant No. 2017YFB0802000, by the Provincial Natural Science Foundation of Fujian under grant No. 2018J01425 and by the Program for Innovative Research Team in Science and Technology in Fujian Province University.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For our purpose, we will choose g such that the fermat quotient \(q_p(g)=1\), see the notion in Sect. 2.

  2. 2.

    Such g always exists.

References

  1. Chen, Z.: Trace representation and linear complexity of binary sequences derived from Fermat quotients. Sci. China Inf. Sci. 57(11), 1–10 (2014)

    MathSciNet  Google Scholar 

  2. Chen, Z., Du, X.: On the linear complexity of binary threshold sequences derived from Fermat quotients. Des. Codes Cryptogr. 67(3), 317–323 (2013)

    Article  MathSciNet  Google Scholar 

  3. Chen, Z., Gómez-Pérez, D.: Linear complexity of binary sequences derived from polynomial quotients. In: Helleseth, T., Jedwab, J. (eds.) SETA 2012. LNCS, vol. 7280, pp. 181–189. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-30615-0_17

    Chapter  Google Scholar 

  4. Chen Z., Niu Z., Wu C.: On the \(k\)-error linear complexity of binary sequences derived from polynomial quotients. Sci. China Inf. Sci. 58(9), 092107:1–092107:15 (2015)

    Article  MathSciNet  Google Scholar 

  5. Chen, Z., Ostafe, A., Winterhof, A.: Structure of pseudorandom numbers derived from Fermat quotients. In: Hasan, M.A., Helleseth, T. (eds.) WAIFI 2010. LNCS, vol. 6087, pp. 73–85. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13797-6_6

    Chapter  MATH  Google Scholar 

  6. Chung, J., Yang, K.: k-fold cyclotomy and its application to frequency-hopping sequences. IEEE Trans. Inf. Theory. 57(4), 2306–2317 (2011)

    Article  MathSciNet  Google Scholar 

  7. Ding, C.: Binary cyclotomic generators. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 29–60. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-60590-8_4

    Chapter  Google Scholar 

  8. Cusick, T., Ding, C., Renvall, A.: Stream Ciphers and Number Theory. Gulf Professional Publishing, Houston (2004)

    Google Scholar 

  9. Ding, C., Helleseth, T.: New generalized cyclotomy and its application. Finite Fields Appl. 4(2), 140–166 (1998)

    Article  MathSciNet  Google Scholar 

  10. Ding, C., Helleseth, T., Lam, K.: Several classes of binary sequences with three-level autocorrelation. IEEE Trans. Inf. Theory 45(7), 2606–2612 (1999)

    Article  MathSciNet  Google Scholar 

  11. Ding, C., Hesseseth, T., Shan, W.: On the linear complexity of Legendre sequences. IEEE Trans. Inf. Theory 44(3), 1276–1278 (1998)

    Article  MathSciNet  Google Scholar 

  12. Ding, C., Xiao, G., Shan, W.: The Stability Theory of Stream Ciphers. Springer, Berlin (1991). https://doi.org/10.1007/3-540-54973-0

    Book  MATH  Google Scholar 

  13. Gauss, C.: Disquisitiones Arithmeticae, 1801; English translation: Yale University Press, New Haven (1966). Reprinted by Springer, Berlin (1986)

    Google Scholar 

  14. Golomb, S., Gong, G.: Signal Design for Good Correlation: for Wireless Communication, Cryptography, and Radar. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  15. Gómez-Pérez, D., Winterhof, A.: Multiplicative character sums of Fermat quotients and pseudorandom sequences. Period. Math. Hungar. 64(2), 161–168 (2012)

    Article  MathSciNet  Google Scholar 

  16. Kim, J., Song, H.: On the linear complexity of Hall’s sextic residue sequences. IEEE Trans. Inf. Theory 47(5), 2094–2096 (2001)

    Article  MathSciNet  Google Scholar 

  17. Ostafe, A., Shparlinski, I.: Pseudorandomness and dynamics of Fermat quotients. SIAM J. Discr. Math. 25(1), 50–71 (2011)

    Article  MathSciNet  Google Scholar 

  18. Stamp, M., Martin, C.: An algorithm for the \(k\)-error linear complexity of binary sequences with period \(2^n\). IEEE Trans. Inform. Theory 39(4), 1398–1401 (1993)

    Article  MathSciNet  Google Scholar 

  19. Sun, Y., Ren, W.: Linear complexity of binary sequences based on two-fold cyclotomy. J. Sichuan Univ. (Nat. Sci. Edn.) 52(4), 748–754 (2015). (in chinese)

    MATH  Google Scholar 

  20. Whiteman, A.: A family of difference sets. Illinois J. Math. 1, 107–121 (1962)

    MathSciNet  MATH  Google Scholar 

  21. Xiao, Z., Zeng, X., Li, C., Helleseth, T.: New generalized cyclotomic binary sequences of period \(p^2\). Des. Codes Crypt. 86(7), 1483–1497 (2018)

    Article  MathSciNet  Google Scholar 

  22. Zeng, X., Cai, H., Tang, X., Yang, Y.: Optimal frequency hopping sequences of odd length. IEEE Trans. Inf. Theory 59(5), 3237–3248 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Cyber Security, School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, People’s Republic of China

    Chenhuang Wu & Chunxiang Xu

  2. Provincial Key Laboratory of Applied Mathematics, Putian University, Putian, 351100, Fujian, People’s Republic of China

    Chenhuang Wu

Authors
  1. Chenhuang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunxiang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chunxiang Xu .

Editor information

Editors and Affiliations

  1. University of Electronic Science and Technology of China, Chengdu, China

    Fagen Li

  2. University of Tokyo, Tokyo, Japan

    Tsuyoshi Takagi

  3. University of Electronic Science and Technology of China, Chengdu, China

    Chunxiang Xu

  4. University of Electronic Science and Technology of China, Chengdu, China

    Xiaosong Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Wu, C., Xu, C. (2018). On k–error Linear Complexity of Zeng-Cai-Tang-Yang Generalized Cyclotomic Binary Sequences of Period \(p^2\). In: Li, F., Takagi, T., Xu, C., Zhang, X. (eds) Frontiers in Cyber Security. FCS 2018. Communications in Computer and Information Science, vol 879. Springer, Singapore. https://doi.org/10.1007/978-981-13-3095-7_2

Download citation

  • DOI: https://doi.org/10.1007/978-981-13-3095-7_2

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-13-3094-0

  • Online ISBN: 978-981-13-3095-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation