## Synchronization of random bit generators based on coupled chaotic lasers and application to cryptography |

Optics Express, Vol. 18, Issue 17, pp. 18292-18302 (2010)

http://dx.doi.org/10.1364/OE.18.018292

Acrobat PDF (1210 KB)

### Abstract

Random bit generators (RBGs) constitute an important tool in cryptography, stochastic simulations and secure communications. The later in particular has some difficult requirements: high generation rate of unpredictable bit strings and secure key-exchange protocols over public channels. Deterministic algorithms generate pseudo-random number sequences at high rates, however, their unpredictability is limited by the very nature of their deterministic origin. Recently, physical RBGs based on chaotic semiconductor lasers were shown to exceed Gbit/s rates. Whether secure synchronization of two high rate physical RBGs is possible remains an open question. Here we propose a method, whereby two fast RBGs based on mutually coupled chaotic lasers, are synchronized. Using information theoretic analysis we demonstrate security against a powerful computational eavesdropper, capable of noiseless amplification, where all parameters are publicly known. The method is also extended to secure synchronization of a small network of three RBGs.

© 2010 OSA

## 1. Introduction

7. N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. **44**(247), 335–341 (1949). [CrossRef] [PubMed]

9. T. E. Murphy and R. Roy, “Chaotic lasers: The world's fastest dice,” Nat. Photonics **2**(12), 714–715 (2008). [CrossRef]

12. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics **4**(1), 58–61 (2010). [CrossRef]

10. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics **2**(12), 728–732 (2008). [CrossRef]

11. I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. **103**(2), 024102 (2009). [CrossRef] [PubMed]

12. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics **4**(1), 58–61 (2010). [CrossRef]

13. K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express **18**(6), 5512–5524 (2010). [CrossRef] [PubMed]

## 2. Results

### Synchronization of two RBGs

16. M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. **85**(6), 60005 (2009). [CrossRef]

_{A}and τ

_{B}, and the mutual coupling time delay, τ, are all equal to 10 ns in the examples below. The strength of self-feedback and the mutual coupling are denoted by κ and σ, respectively. The injection current to the threshold current ratio is selected to be 1.5, so that the lasers operate in the coherence collapse regime [17

17. R. J. Jones, P. S. Spencer, J. Lawrence, and D. M. Kane, “Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback,” IEEE Proc. Optoelectron. **148**(1), 7–12 (2001). [CrossRef]

18. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**(3), 347–355 (1980). [CrossRef]

19. V. Ahlers, U. Parlitz, and W. Lauterborn, “Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(6), 7208–7213 (1998). [CrossRef]

19. V. Ahlers, U. Parlitz, and W. Lauterborn, “Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(6), 7208–7213 (1998). [CrossRef]

20. E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **74**(4), 046201 (2006). [CrossRef] [PubMed]

_{C}=τ

_{A}=τ

_{B}=τ and coupling strengths, σ

_{C}=σ, κ

_{C}=κ, is depicted in Fig. 2(b). A comparison of Fig. 2(a) and 2(b) indicates that ZLS of mutually coupled chaotic lasers is superior to the unidirectional coupling of laser C in a large fraction of the phase space (κ,σ) [14

14. E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **73**(6), 066214 (2006). [CrossRef] [PubMed]

_{C}>σ, while maintaining its total input κ

_{C}+σ

_{C}~κ+σ. In what follows, we first describe the utilization of ZLS as a carrier synchronizing the RBGs of A and B and then we analyze the security of the channel.

7. N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. **44**(247), 335–341 (1949). [CrossRef] [PubMed]

9. T. E. Murphy and R. Roy, “Chaotic lasers: The world's fastest dice,” Nat. Photonics **2**(12), 714–715 (2008). [CrossRef]

^{2}I, where M=1 corresponds to the transmission of “1” while M=M

_{0}corresponds to the transmission of “-1”. In simulations we modulate the intensity by changing the field and in the examples below M

_{0}is set to 0.9 with a bandwidth of 1 Gbit/s, the explicit equations are given in 20. Our simulations indicate that the ZLS between the communicating pair remains robust even in the presence of such independent modulation by each of the parties. B for instance, decodes the massage transmitted from A by dividing the intensity received from A with its own synchronized laser output, prior to his modulation, <I

_{A}

^{R}>/<I

_{B}>, where the average, <…>, is over a predetermined duration of one bit transmission time. If this fraction is larger than (1+M

_{0}

^{2})/2 then the estimated received bit is “1”, otherwise “-1”. The encoding/decoding procedures are implemented simultaneously at both lasers and are known as a mutual chaos pass filter (MCPF) mechanism [20

20. E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **74**(4), 046201 (2006). [CrossRef] [PubMed]

### Protocol

- • To achieve the goal of two identical random bit sequences, an information reconciliation procedure is performed, a form of error correcting code, as for protocols of quantum cryptography [2]. At the end of this procedure the two partners hold
*identical random bit sequences*. - • Inevitably, leakage of information occurs during the information reconciliation procedure and is eliminated by a privacy amplification procedure which is also utilized in quantum cryptography for similar reasons [6,14
6. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys.

**81**(3), 1301–1350 (2009). [CrossRef]].14. E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

**73**(6), 066214 (2006). [CrossRef] [PubMed]

^{−1}and σ=40 ns

^{−1}the cross correlation at zero time lag between the parties is much higher, ~0.94, than correlation between the attacker and the parties ~0.5. An attacker using the same set of parameters as A and B would obtain a very high BER in his CPF mechanism [21

21. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature **438**(7066), 343–346 (2005). [CrossRef] [PubMed]

_{C}while decreasing κ

_{C}so that κ

_{C}+σ

_{C}~κ+σ. (Note that minimizing BER of C differs from maximizing its ZLS). Figure 3(a) indicates that the minimum BER for the attacker, q~0.15, is obtained for (κ=40 ns

^{−1}, σ

_{C}=90 ns

^{−1}) while the parties are operating with (κ=90 ns

^{−1}, σ=40 ns

^{−1}). Though this is a much lower BER then C would obtain without the use of amplification, it remains more than twice as high as the BER of A and B.

### Information theory analysis

_{C}>σ and as a result q>p. A higher BER for C, however, does not necessarily indicate that the fraction of identical bits between S

_{C}and S

_{A}is reduced in comparison to the fraction of identical bits between S

_{A}and S

_{B}. One can show, using symbolic mathematics (Methods section), that the average fraction of identical bits between S

_{C}and S

_{A}(or S

_{B}) is given by

### Extended protocol for non identical delay times

_{A}=τ

_{B}=τ, which might not be realistic in public channel scenarios. However, the ZLS as well as the proposed algorithm can be generalized to the case where the mutual delay differs from the self-coupling delay. For a delay configuration given by τ

_{A}+τ

_{B}=2τ, for instance, and κ=σ it was shown that the two partners are synchronized, where A lags after B by Δ=(τ

_{B}-τ

_{A})/2 assuming τ

_{B}>τ

_{A}[16

16. M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. **85**(6), 60005 (2009). [CrossRef]

_{B}=τ

_{A}, is a particular solution of this more general case. This scenario can be realized in practice where a 50% semi-transparent mirror is located at τ

_{A}/2 and τ

_{B}/2 from the two mutually communicating lasers A and B, respectively (Fig. 6 ) [23

23. V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **79**(6), 065201 (2009). [CrossRef] [PubMed]

16. M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. **85**(6), 60005 (2009). [CrossRef]

- • The two partners start with an identical public random binary sequence of length L, S
_{A}=S_{B}=S and encode random bit streams onto their chaotic intensities as for the ZLS case. - • A calculates the ratio between its received average intensity over one bit length and its own average intensity at time t−τ−Δ. In case of complete Δ lagged synchronization, there are three possible outcomes for the ratio. The ratio is 1 in case both A and B do not modulate their intensities (both send 1), (1+M
_{0}^{2})/2 in case only one of the parties modulated its intensity (one sends −1 and the other sends 1) and M_{0}^{2}in case both parties modulate their intensities (both send −1). To identify the correct ratio, A sets two thresholds between the three ratios, (3+M_{0}^{2})/4 and (1+3M_{0}^{2})/4, as opposed to the one threshold case in ZLS. - • A knows its transmitted bits, thus it can estimate the received bit from B. For instance, if the ratio was determined to be (1+M
_{0}^{2})/2 and A’s sent bit was 1 then the estimation of the bit sent by B is −1. - • Actions of the parties on the public vector, S, remain identical as in ZLS.

_{0}=0.95, τ=20 ns, τ

_{A}=17 ns, τ

_{B}=23 ns, κ=σ=4 ns

*and a bandwidth of 1 Gbit/s, p~0.09 and q~0.23 and the key-exchange protocol is secure, Fig. 4(a). Note that since the message is transmitted in both the mutual and self-feedback, the total modulation is increased. Thus to preserve the noise to signal ratio, a higher M*

^{−1}_{0}is used.

### Secure synchronization of three RBGs

24. J. Kestler, W. Kinzel, and I. Kanter, “Sublattice synchronization of chaotic networks with delayed couplings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **76**(3), 035202 (2007). [CrossRef] [PubMed]

21. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature **438**(7066), 343–346 (2005). [CrossRef] [PubMed]

## 3. Discussion

^{−7}as was observed in a unidirectional configuration and modulation bandwidth of 1Gbit/s [21

21. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature **438**(7066), 343–346 (2005). [CrossRef] [PubMed]

## 4. Methods

### Symbolic mathematics

_{A}, S

_{B}and S

_{C}are identical. All possible probabilities for actions of the parties on the S vectors are calculated and summarized in Table 1 . Using symbolic mathematics, the fraction of identical bits is derived and depicted in Eq. (1). A more complex case, where the joint probability distribution of the attacker and the parties are taken into account using symbolic mathematics, results in Eq. (4). A comparison of Eq. (1) and (3) results in which is the boundary line between blue and white areas in Fig. 4(a). Similar calculation for the case of three communicating parties and an attacker as in Fig. 1(b), results in 2048 different scenarios. The boundary line of the red region in Fig. 4(b) is found to be .

### Correlation of decoded bits

## Acknowledgments

## References and links

1. | M. A. Nielsen, and I. L. Chuang, |

2. | D. R. Stinson, |

3. | R. G. Gallager, |

4. | C. H. Bennett, C. Hand, and G. Brassard, “Quantum Cryptography: Public key distribution and coin tossing,” Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, p. 175 (1984). |

5. | A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. |

6. | V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. |

7. | N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. |

8. | S. Asmussen, and P. W. Glynn, |

9. | T. E. Murphy and R. Roy, “Chaotic lasers: The world's fastest dice,” Nat. Photonics |

10. | A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics |

11. | I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. |

12. | I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics |

13. | K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express |

14. | E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

15. | I. Kanter, E. Kopelowitz, and W. Kinzel, “Public channel cryptography: chaos synchronization and Hilbert’s tenth problem,” Phys. Rev. Lett. |

16. | M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. |

17. | R. J. Jones, P. S. Spencer, J. Lawrence, and D. M. Kane, “Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback,” IEEE Proc. Optoelectron. |

18. | R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. |

19. | V. Ahlers, U. Parlitz, and W. Lauterborn, “Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

20. | E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

21. | A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature |

22. | T. M. Cover, and J. A. Thomas, |

23. | V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

24. | J. Kestler, W. Kinzel, and I. Kanter, “Sublattice synchronization of chaotic networks with delayed couplings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(060.4785) Fiber optics and optical communications : Optical security and encryption

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: July 6, 2010

Revised Manuscript: August 1, 2010

Manuscript Accepted: August 1, 2010

Published: August 10, 2010

**Citation**

Ido Kanter, Maria Butkovski, Yitzhak Peleg, Meital Zigzag, Yaara Aviad, Igor Reidler, Michael Rosenbluh, and Wolfgang Kinzel, "Synchronization of random bit generators based on coupled chaotic lasers and application to cryptography," Opt. Express **18**, 18292-18302 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-18292

Sort: Year | Journal | Reset

### References

- M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information. (Cambridge University Press, Cambridge, 2000).
- D. R. Stinson, Cryptography: Theory and Practice. (CRC Press, Boca Raton, 1995).
- R. G. Gallager, Principles of Digital Communication. (Cambridge University Press, Cambridge, 2008).
- C. H. Bennett, C. Hand, and G. Brassard, “Quantum Cryptography: Public key distribution and coin tossing,” Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, p. 175 (1984).
- A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991). [CrossRef] [PubMed]
- V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009). [CrossRef]
- N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44(247), 335–341 (1949). [CrossRef] [PubMed]
- S. Asmussen, and P. W. Glynn, Stochastic Simulation: Algorithms and Analysis. (Springer-Verlag, New York, 2007).
- T. E. Murphy and R. Roy, “Chaotic lasers: The world's fastest dice,” Nat. Photonics 2(12), 714–715 (2008). [CrossRef]
- A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008). [CrossRef]
- I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009). [CrossRef] [PubMed]
- I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010). [CrossRef]
- K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18(6), 5512–5524 (2010). [CrossRef] [PubMed]
- E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(6), 066214 (2006). [CrossRef] [PubMed]
- I. Kanter, E. Kopelowitz, and W. Kinzel, “Public channel cryptography: chaos synchronization and Hilbert’s tenth problem,” Phys. Rev. Lett. 101(8), 084102 (2008). [CrossRef] [PubMed]
- M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. 85(6), 60005 (2009). [CrossRef]
- R. J. Jones, P. S. Spencer, J. Lawrence, and D. M. Kane, “Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback,” IEEE Proc. Optoelectron. 148(1), 7–12 (2001). [CrossRef]
- R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]
- V. Ahlers, U. Parlitz, and W. Lauterborn, “Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(6), 7208–7213 (1998). [CrossRef]
- E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006). [CrossRef] [PubMed]
- A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005). [CrossRef] [PubMed]
- T. M. Cover, and J. A. Thomas, Elements of Information Theory (John Wiley and Sons, New York, 1991).
- V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009). [CrossRef] [PubMed]
- J. Kestler, W. Kinzel, and I. Kanter, “Sublattice synchronization of chaotic networks with delayed couplings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(3), 035202 (2007). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.