Skip to main content
SpringerLink
Log in

A Back-Iteration Method for Reconstructing Chaotic Sequences in Finite-Precision Machines

  • Nonlinear Circuits and Systems
  • Published:
Circuits, Systems & Signal Processing Aims and scope Submit manuscript

Abstract

Discrete-valued chaotic sequences, when used as long-period spreading codes or stream ciphers, are expected to offer a low probability of interception and thus an enhancement in security. In this paper, the weakness of digital chaotic sequences is reported. We demonstrate that it is possible to reconstruct a long-period discrete-valued sequence generated by a chaotic system based on very few past values when the sequence generator is implemented in finite-wordlength digital hardware. The proposed back-iteration reconstruction method is based on the connection between the symbolic sequence and the initial condition of a chaotic system.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Bergamo, P. D’Arco, A. De Santis, L. Kocarev, Security of public key cryptosystems based on Chebyshev polynomials. IEEE Trans. Circuits Syst. I, Reg. Pap. 52(7), 1382–1393 (2005)

    Article  Google Scholar 

  2. J. Cernak, Digital generators of chaos. Phys. Lett. A 214, 151–160 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. K.Y. Cheong, T. Koshiba, More on security of public-key cryptosystems based on Chebyshev polynomials. IEEE Trans. Circuits Syst. II, Exp. Briefs 54(9), 795–799 (2007)

    Article  Google Scholar 

  4. B.V. Chirikov, F. Vivaldi, An algorithmic view of pesudochaos. Physica D, 129, 223–235 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. L. Cong, S. Songgeng, Chaotic frequency hopping sequences. IEEE Trans. Commun. 46, 1433–1437 (1998)

    Article  Google Scholar 

  6. L. Cong, W. Xiaofu, S. Songgeng, A general efficient method for chaotic signal estimation. IEEE Trans. Signal Process. 47, 1424–1427 (1999)

    Article  Google Scholar 

  7. F. Dachselt, W. Schwarz, Chaos and cryptography. IEEE Trans. Circuits Syst. I 48, 1498–1509 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. D.R. Frey, Chaotic digital encoding: An approach to secure communications. IEEE Trans. Circuits Syst. I 40, 660–666 (1993)

    Article  MathSciNet  Google Scholar 

  9. M. Jessa, Designing security for number sequences generated by means of the Sawtooth chaotic map. IEEE Trans. Circuits Syst. I, Reg. Pap. 53(5), 1140–1150 (2006)

    Article  MathSciNet  Google Scholar 

  10. S. Kay, V. Nagesha, Methods for chaotic signal estimation. IEEE Trans. Signal Process. 43, 2013–2016 (1995)

    Article  Google Scholar 

  11. L. Kocarev, J. Szczepanski, J. Amigo, I. Tomovski, Discrete chaos, I: theory. IEEE Trans. Circuits Syst. I, Reg. Pap. 53, 1300–1309 (2006)

    Article  MathSciNet  Google Scholar 

  12. T. Kohda, A. Tsuneda, Statistics of chaotic binary sequences. IEEE Trans. Inform. Theory 43, 104–112 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. D.J. Kuck, The Structure of Computers and Computations, vol. 1 (Wiley, New York, 1978)

    Google Scholar 

  14. S. Li, G. Chen, X. Mou, On the security of the Yi–Tan–Siew chaotic cipher. IEEE Trans. Circuits Syst. II, Exp. Briefs 51(12), 665–669 (2004)

    Article  Google Scholar 

  15. N. Masuda, K. Aihara, Cryptosystems with discretized chaotic maps. IEEE Trans. Circuits Syst. I 49, 28–40 (2002)

    Article  MathSciNet  Google Scholar 

  16. N. Masuda, G. Jakimoski, K. Aihara, L. Kocarev, Chaotic block ciphers: from theory to practical algorithms. IEEE Trans. Circuits Syst. I, Reg. Pap. 53(6), 1341–1352 (2006)

    Article  MathSciNet  Google Scholar 

  17. G. Mazzini, G. Setti, R. Rovatti, Chaotic complex spreading sequences for asynchronous DS-CDMA, part I: system modeling and results. IEEE Trans. Circuits Syst. I 44, 937–947 (1997)

    Article  MathSciNet  Google Scholar 

  18. J. Mullins, Chaotic communication. IEEE Spectr. 43, 11–12 (2006)

    Article  Google Scholar 

  19. J. Schweizer, T. Schimming, Symbolic dynamics for processing chaotic signals, part II: communication and coding. IEEE Trans. Circuits Syst. I 48, 1283–1295 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. T. Stojanovski, L. Kocarev, R. Harris, Applications of symbolic dynamics in chaos synchronization. IEEE Trans. Circuits Syst. I 44, 1014–1018 (1997)

    Article  Google Scholar 

  21. S. Wang, P. Yip, H. Leung, Estimating initial conditions of noisy chaotic signals generated by piece-wise linear Markov maps using itineraries. IEEE Trans. Signal Process. 47, 3289–3302 (1999)

    Article  MATH  Google Scholar 

  22. X. Yi, Hash function based on chaotic tent maps. IEEE Trans. Circuits Syst. II, Exp. Briefs 52(6), 354–357 (2005)

    Article  Google Scholar 

  23. H. Zhou, X. Ling, Realizing finite precision chaotic systems via perturbation of m-sequences. Chin. J. Electron. 25(7), 95–97 (1997) (in Chinese)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, Imperial College London, London, SW7 2AZ, UK

    Cong Ling

  2. National Mobile Communications Research Laboratory, Southeast University, Nanjing, 210096, China

    Xiaofu Wu

  3. Nanjing Institute of Communications Engineering, Nanjing, 210007, China

    Xiaofu Wu

Authors
  1. Cong Ling
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaofu Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cong Ling.

Additional information

This work was performed when the authors were with the Nanjing Institute of Communications Engineering, Nanjing, Jiangsu Province, 210007, China, supported by the National Science Foundation of China under Grants 69931040 and 60672081, and by the Science Foundation of Jiangsu Province under Grant BK2006502.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ling, C., Wu, X. A Back-Iteration Method for Reconstructing Chaotic Sequences in Finite-Precision Machines. Circuits Syst Signal Process 27, 883–891 (2008). https://doi.org/10.1007/s00034-008-9065-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-008-9065-4

Keywords

Search

Navigation