Abstract

We propose a scheme whereby a time domain fractional Fourier transform (FRFT) is used to post process the optical chaotic carrier generated by an electro-optic oscillator. The time delay signature of the delay dynamics is successfully masked by the FRFT when some conditions are satisfied. Meanwhile the dimension space of the physical parameters is increased. Pseudo random binary sequence (PRBS) with low bit rate (hundreds of Mbps) is introduced to control the parameters of the FRFT. The chaotic optical carrier, FRFT parameters and the PRBS are covered by each other so that the eavesdropper has to search the whole key space to crack the system. The scheme allows enhancing the security of communication systems based on delay dynamics without modifying the chaotic source. In this way, the design of chaos based communication systems can be implemented in a modular manner.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  33. B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(5), 056205 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2012 (3)

2011 (3)

R. M. Nguimdo, M. C. Soriano, P. Colet, “Role of the phase in the identification of delay time in semiconductor lasers with optical feedback,” Opt. Lett. 36(22), 4332–4334 (2011).
[CrossRef] [PubMed]

Q. Han, W. Li, M. Yang, “An optical waveform pre-distortion method based on time domain fractional Fourier transformation,” Opt. Commun. 284(2), 660–664 (2011).
[CrossRef]

R. M. Nguimdo, P. Colet, L. Larger, L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. 107(3), 034103 (2011).
[CrossRef] [PubMed]

2010 (6)

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(4), 046212 (2010).
[CrossRef] [PubMed]

P. H. Bryant, “Optimized synchronization of chaotic and hyperchaotic systems,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(1), 015201 (2010).
[CrossRef] [PubMed]

A. Argyris, E. Grivas, M. Hamacher, A. Bogris, D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18(5), 5188–5198 (2010).
[CrossRef] [PubMed]

H. Ma, B. Xu, W. Lin, J. Feng, “Adaptive identification of time delays in nonlinear dynamical models,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(6), 066210 (2010).
[CrossRef] [PubMed]

J. Hizanidis, S. Deligiannidis, A. Bogris, D. Syvridis, “Enhancement of chaos encryption potential by combining all-optical and electrooptical chaos generators,” IEEE J. Quantum Electron. 46(11), 1642–1649 (2010).
[CrossRef]

R. M. Nguimdo, R. Lavrov, P. Colet, M. Jacquot, Y. K. Chembo, L. Larger, “Effect of fiber dispersion on broadband chaos communications implemented by electro-optic nonlinear delay phase dynamics,” J. Lightwave Technol. 28(18), 2688–2696 (2010).
[CrossRef]

2009 (7)

F. Sorrentino, E. Ott, “Using synchronization of chaos to identify the dynamics of unknown systems,” Chaos 19(3), 033108 (2009).
[CrossRef] [PubMed]

D. Rontani, M. Sciamanna, A. Locquet, D. S. Citrin, “Multiplexed encryption using chaotic systems with multiple stochastic-delayed feedbacks,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(6), 066209 (2009).
[CrossRef] [PubMed]

B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(5), 056205 (2009).
[CrossRef] [PubMed]

Q. Ran, H. Zhang, J. Zhang, L. Tan, J. Ma, “Deficiencies of the cryptography based on multiple-parameter fractional Fourier transform,” Opt. Lett. 34(11), 1729–1731 (2009).
[CrossRef] [PubMed]

D. Yang, S. Kumar, “Realization of optical OFDM using time lenses and its comparison with optical OFDM using FFT,” Opt. Express 17(20), 17214–17226 (2009).
[CrossRef] [PubMed]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–891 (2009).
[CrossRef]

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 79(2), 026208 (2009).
[CrossRef] [PubMed]

2008 (1)

A. B. Cohen, B. Ravoori, T. E. Murphy, R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101(15), 154102 (2008).
[CrossRef] [PubMed]

2007 (1)

M. Joshi, Chandrashakher, K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279(1), 35–42 (2007).
[CrossRef]

2006 (1)

X. Li, W. Pan, B. Luo, D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
[CrossRef]

2005 (3)

Y. C. Kouomou, P. Colet, L. Larger, N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95(20), 203903 (2005).
[CrossRef] [PubMed]

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

M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3–4), 209–223 (2005).
[CrossRef]

2004 (2)

L. Larger, J. Goedgebuer, V. Udaltsov, “Ikeda-based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C. R. Phys. 5(6), 669–681 (2004).
[CrossRef]

Y. C. Kouomou, P. Colet, N. Gastaud, L. Larger, “Effect of parameter mismatch on the synchronization of chaotic semiconductor lasers with electro-optical feedback,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 69(5), 056226 (2004).
[CrossRef] [PubMed]

2003 (2)

B. Hennelly, J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003).
[CrossRef] [PubMed]

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[CrossRef]

2002 (1)

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

2001 (1)

1998 (1)

R. Hegger, M. J. Bünner, H. Kantz, A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[CrossRef]

1996 (1)

U. Parlitz, “Estimating model parameters from time series by autosynchronization,” Phys. Rev. Lett. 76(8), 1232–1235 (1996).
[CrossRef] [PubMed]

1995 (1)

S. C. Phatak, S. S. Rao, “Logistic map: a possible random-number generator,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(4), 3670–3678 (1995).
[CrossRef] [PubMed]

1994 (2)

P. Colet, R. Roy, “Digital communication with synchronized chaotic lasers,” Opt. Lett. 19(24), 2056–2058 (1994).
[CrossRef] [PubMed]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
[CrossRef]

1978 (1)

R. L. Rivest, A. Shamir, L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978).
[CrossRef]

Adleman, L.

R. L. Rivest, A. Shamir, L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978).
[CrossRef]

Annovazzi-Lodi, V.

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

Argyris, A.

A. Argyris, E. Grivas, M. Hamacher, A. Bogris, D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18(5), 5188–5198 (2010).
[CrossRef] [PubMed]

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

Bezruchko, B. P.

M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3–4), 209–223 (2005).
[CrossRef]

Bogris, A.

J. Hizanidis, S. Deligiannidis, A. Bogris, D. Syvridis, “Enhancement of chaos encryption potential by combining all-optical and electrooptical chaos generators,” IEEE J. Quantum Electron. 46(11), 1642–1649 (2010).
[CrossRef]

A. Argyris, E. Grivas, M. Hamacher, A. Bogris, D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18(5), 5188–5198 (2010).
[CrossRef] [PubMed]

Bryant, P. H.

P. H. Bryant, “Optimized synchronization of chaotic and hyperchaotic systems,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(1), 015201 (2010).
[CrossRef] [PubMed]

Bünner, M. J.

R. Hegger, M. J. Bünner, H. Kantz, A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[CrossRef]

Chandrashakher,

M. Joshi, Chandrashakher, K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279(1), 35–42 (2007).
[CrossRef]

Chembo, Y. K.

R. M. Nguimdo, R. Lavrov, P. Colet, M. Jacquot, Y. K. Chembo, L. Larger, “Effect of fiber dispersion on broadband chaos communications implemented by electro-optic nonlinear delay phase dynamics,” J. Lightwave Technol. 28(18), 2688–2696 (2010).
[CrossRef]

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 79(2), 026208 (2009).
[CrossRef] [PubMed]

Citrin, D. S.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–891 (2009).
[CrossRef]

D. Rontani, M. Sciamanna, A. Locquet, D. S. Citrin, “Multiplexed encryption using chaotic systems with multiple stochastic-delayed feedbacks,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(6), 066209 (2009).
[CrossRef] [PubMed]

Cohen, A. B.

B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(5), 056205 (2009).
[CrossRef] [PubMed]

A. B. Cohen, B. Ravoori, T. E. Murphy, R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101(15), 154102 (2008).
[CrossRef] [PubMed]

Colet, P.

R. M. Nguimdo, P. Colet, “Electro-optic phase chaos systems with an internal variable and a digital key,” Opt. Express 20(23), 25333–25344 (2012).
[CrossRef] [PubMed]

R. M. Nguimdo, P. Colet, L. Larger, L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. 107(3), 034103 (2011).
[CrossRef] [PubMed]

R. M. Nguimdo, M. C. Soriano, P. Colet, “Role of the phase in the identification of delay time in semiconductor lasers with optical feedback,” Opt. Lett. 36(22), 4332–4334 (2011).
[CrossRef] [PubMed]

R. M. Nguimdo, R. Lavrov, P. Colet, M. Jacquot, Y. K. Chembo, L. Larger, “Effect of fiber dispersion on broadband chaos communications implemented by electro-optic nonlinear delay phase dynamics,” J. Lightwave Technol. 28(18), 2688–2696 (2010).
[CrossRef]

Y. C. Kouomou, P. Colet, L. Larger, N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95(20), 203903 (2005).
[CrossRef] [PubMed]

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

Y. C. Kouomou, P. Colet, N. Gastaud, L. Larger, “Effect of parameter mismatch on the synchronization of chaotic semiconductor lasers with electro-optical feedback,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 69(5), 056226 (2004).
[CrossRef] [PubMed]

P. Colet, R. Roy, “Digital communication with synchronized chaotic lasers,” Opt. Lett. 19(24), 2056–2058 (1994).
[CrossRef] [PubMed]

Cuenot, J. B.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[CrossRef]

Danckaert, J.

Deligiannidis, S.

J. Hizanidis, S. Deligiannidis, A. Bogris, D. Syvridis, “Enhancement of chaos encryption potential by combining all-optical and electrooptical chaos generators,” IEEE J. Quantum Electron. 46(11), 1642–1649 (2010).
[CrossRef]

Erneux, T.

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 79(2), 026208 (2009).
[CrossRef] [PubMed]

Feng, J.

H. Ma, B. Xu, W. Lin, J. Feng, “Adaptive identification of time delays in nonlinear dynamical models,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(6), 066210 (2010).
[CrossRef] [PubMed]

Fischer, I.

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(4), 046212 (2010).
[CrossRef] [PubMed]

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

García-Ojalvo, J.

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

Gastaud, N.

Y. C. Kouomou, P. Colet, L. Larger, N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95(20), 203903 (2005).
[CrossRef] [PubMed]

Y. C. Kouomou, P. Colet, N. Gastaud, L. Larger, “Effect of parameter mismatch on the synchronization of chaotic semiconductor lasers with electro-optical feedback,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 69(5), 056226 (2004).
[CrossRef] [PubMed]

Giaquinta, A.

R. Hegger, M. J. Bünner, H. Kantz, A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[CrossRef]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

Goedgebuer, J.

L. Larger, J. Goedgebuer, V. Udaltsov, “Ikeda-based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C. R. Phys. 5(6), 669–681 (2004).
[CrossRef]

Goedgebuer, J. P.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[CrossRef]

Gonzalez, J. A.

J. J. Suárez-Vargas, B. A. Marquez, J. A. Gonzalez, “Highly complex optical signal generation using electro-optical systems with non-linear, non-invertible transmission functions,” Appl. Phys. Lett. 101(7), 071115 (2012).
[CrossRef]

Grivas, E.

Hamacher, M.

Han, Q.

Q. Han, W. Li, M. Yang, “An optical waveform pre-distortion method based on time domain fractional Fourier transformation,” Opt. Commun. 284(2), 660–664 (2011).
[CrossRef]

Hegger, R.

R. Hegger, M. J. Bünner, H. Kantz, A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[CrossRef]

Hennelly, B.

Hizanidis, J.

J. Hizanidis, S. Deligiannidis, A. Bogris, D. Syvridis, “Enhancement of chaos encryption potential by combining all-optical and electrooptical chaos generators,” IEEE J. Quantum Electron. 46(11), 1642–1649 (2010).
[CrossRef]

Jacquot, M.

R. M. Nguimdo, R. Lavrov, P. Colet, M. Jacquot, Y. K. Chembo, L. Larger, “Effect of fiber dispersion on broadband chaos communications implemented by electro-optic nonlinear delay phase dynamics,” J. Lightwave Technol. 28(18), 2688–2696 (2010).
[CrossRef]

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 79(2), 026208 (2009).
[CrossRef] [PubMed]

Joshi, M.

M. Joshi, Chandrashakher, K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279(1), 35–42 (2007).
[CrossRef]

Kantz, H.

R. Hegger, M. J. Bünner, H. Kantz, A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[CrossRef]

Karavaev, A. S.

M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3–4), 209–223 (2005).
[CrossRef]

Kolner, B. H.

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
[CrossRef]

Kouomou, Y. C.

Y. C. Kouomou, P. Colet, L. Larger, N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95(20), 203903 (2005).
[CrossRef] [PubMed]

Y. C. Kouomou, P. Colet, N. Gastaud, L. Larger, “Effect of parameter mismatch on the synchronization of chaotic semiconductor lasers with electro-optical feedback,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 69(5), 056226 (2004).
[CrossRef] [PubMed]

Kumar, S.

Larger, L.

R. M. Nguimdo, P. Colet, L. Larger, L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. 107(3), 034103 (2011).
[CrossRef] [PubMed]

R. M. Nguimdo, R. Lavrov, P. Colet, M. Jacquot, Y. K. Chembo, L. Larger, “Effect of fiber dispersion on broadband chaos communications implemented by electro-optic nonlinear delay phase dynamics,” J. Lightwave Technol. 28(18), 2688–2696 (2010).
[CrossRef]

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 79(2), 026208 (2009).
[CrossRef] [PubMed]

Y. C. Kouomou, P. Colet, L. Larger, N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95(20), 203903 (2005).
[CrossRef] [PubMed]

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

L. Larger, J. Goedgebuer, V. Udaltsov, “Ikeda-based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C. R. Phys. 5(6), 669–681 (2004).
[CrossRef]

Y. C. Kouomou, P. Colet, N. Gastaud, L. Larger, “Effect of parameter mismatch on the synchronization of chaotic semiconductor lasers with electro-optical feedback,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 69(5), 056226 (2004).
[CrossRef] [PubMed]

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[CrossRef]

Lavrov, R.

Levy, P.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[CrossRef]

Li, W.

Q. Han, W. Li, M. Yang, “An optical waveform pre-distortion method based on time domain fractional Fourier transformation,” Opt. Commun. 284(2), 660–664 (2011).
[CrossRef]

Li, X.

X. Li, W. Pan, B. Luo, D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
[CrossRef]

Lin, W.

H. Ma, B. Xu, W. Lin, J. Feng, “Adaptive identification of time delays in nonlinear dynamical models,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(6), 066210 (2010).
[CrossRef] [PubMed]

Liu, J. M.

Locquet, A.

D. Rontani, M. Sciamanna, A. Locquet, D. S. Citrin, “Multiplexed encryption using chaotic systems with multiple stochastic-delayed feedbacks,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(6), 066209 (2009).
[CrossRef] [PubMed]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–891 (2009).
[CrossRef]

Luo, B.

X. Li, W. Pan, B. Luo, D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
[CrossRef]

Ma, D.

X. Li, W. Pan, B. Luo, D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
[CrossRef]

Ma, H.

H. Ma, B. Xu, W. Lin, J. Feng, “Adaptive identification of time delays in nonlinear dynamical models,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(6), 066210 (2010).
[CrossRef] [PubMed]

Ma, J.

Marquez, B. A.

J. J. Suárez-Vargas, B. A. Marquez, J. A. Gonzalez, “Highly complex optical signal generation using electro-optical systems with non-linear, non-invertible transmission functions,” Appl. Phys. Lett. 101(7), 071115 (2012).
[CrossRef]

Mirasso, C. R.

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(4), 046212 (2010).
[CrossRef] [PubMed]

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

Murphy, T. E.

B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(5), 056205 (2009).
[CrossRef] [PubMed]

A. B. Cohen, B. Ravoori, T. E. Murphy, R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101(15), 154102 (2008).
[CrossRef] [PubMed]

Nguimdo, R. M.

Ortin, S.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–891 (2009).
[CrossRef]

Ott, E.

F. Sorrentino, E. Ott, “Using synchronization of chaos to identify the dynamics of unknown systems,” Chaos 19(3), 033108 (2009).
[CrossRef] [PubMed]

B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(5), 056205 (2009).
[CrossRef] [PubMed]

Pan, W.

X. Li, W. Pan, B. Luo, D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
[CrossRef]

Parlitz, U.

U. Parlitz, “Estimating model parameters from time series by autosynchronization,” Phys. Rev. Lett. 76(8), 1232–1235 (1996).
[CrossRef] [PubMed]

Peil, M.

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 79(2), 026208 (2009).
[CrossRef] [PubMed]

Pesquera, L.

R. M. Nguimdo, P. Colet, L. Larger, L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. 107(3), 034103 (2011).
[CrossRef] [PubMed]

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

Phatak, S. C.

S. C. Phatak, S. S. Rao, “Logistic map: a possible random-number generator,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(4), 3670–3678 (1995).
[CrossRef] [PubMed]

Ponomarenko, V. I.

M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3–4), 209–223 (2005).
[CrossRef]

Prokhorov, M. D.

M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3–4), 209–223 (2005).
[CrossRef]

Ran, Q.

Rao, S. S.

S. C. Phatak, S. S. Rao, “Logistic map: a possible random-number generator,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(4), 3670–3678 (1995).
[CrossRef] [PubMed]

Ravoori, B.

B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(5), 056205 (2009).
[CrossRef] [PubMed]

A. B. Cohen, B. Ravoori, T. E. Murphy, R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101(15), 154102 (2008).
[CrossRef] [PubMed]

Rhodes, W. T.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[CrossRef]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

Rivest, R. L.

R. L. Rivest, A. Shamir, L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978).
[CrossRef]

Rontani, D.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–891 (2009).
[CrossRef]

D. Rontani, M. Sciamanna, A. Locquet, D. S. Citrin, “Multiplexed encryption using chaotic systems with multiple stochastic-delayed feedbacks,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(6), 066209 (2009).
[CrossRef] [PubMed]

Rosso, O. A.

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(4), 046212 (2010).
[CrossRef] [PubMed]

Roy, R.

B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(5), 056205 (2009).
[CrossRef] [PubMed]

A. B. Cohen, B. Ravoori, T. E. Murphy, R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101(15), 154102 (2008).
[CrossRef] [PubMed]

P. Colet, R. Roy, “Digital communication with synchronized chaotic lasers,” Opt. Lett. 19(24), 2056–2058 (1994).
[CrossRef] [PubMed]

Sciamanna, M.

D. Rontani, M. Sciamanna, A. Locquet, D. S. Citrin, “Multiplexed encryption using chaotic systems with multiple stochastic-delayed feedbacks,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(6), 066209 (2009).
[CrossRef] [PubMed]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–891 (2009).
[CrossRef]

Setty, A. V.

B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(5), 056205 (2009).
[CrossRef] [PubMed]

Shamir, A.

R. L. Rivest, A. Shamir, L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978).
[CrossRef]

Sheridan, J. T.

Shore, K. A.

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

Singh, K.

M. Joshi, Chandrashakher, K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279(1), 35–42 (2007).
[CrossRef]

Soriano, M. C.

R. M. Nguimdo, M. C. Soriano, P. Colet, “Role of the phase in the identification of delay time in semiconductor lasers with optical feedback,” Opt. Lett. 36(22), 4332–4334 (2011).
[CrossRef] [PubMed]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(4), 046212 (2010).
[CrossRef] [PubMed]

Sorrentino, F.

F. Sorrentino, E. Ott, “Using synchronization of chaos to identify the dynamics of unknown systems,” Chaos 19(3), 033108 (2009).
[CrossRef] [PubMed]

B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(5), 056205 (2009).
[CrossRef] [PubMed]

Suárez-Vargas, J. J.

J. J. Suárez-Vargas, B. A. Marquez, J. A. Gonzalez, “Highly complex optical signal generation using electro-optical systems with non-linear, non-invertible transmission functions,” Appl. Phys. Lett. 101(7), 071115 (2012).
[CrossRef]

Syvridis, D.

J. Hizanidis, S. Deligiannidis, A. Bogris, D. Syvridis, “Enhancement of chaos encryption potential by combining all-optical and electrooptical chaos generators,” IEEE J. Quantum Electron. 46(11), 1642–1649 (2010).
[CrossRef]

A. Argyris, E. Grivas, M. Hamacher, A. Bogris, D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18(5), 5188–5198 (2010).
[CrossRef] [PubMed]

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

Tan, L.

Tang, S.

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

Udaltsov, V.

L. Larger, J. Goedgebuer, V. Udaltsov, “Ikeda-based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C. R. Phys. 5(6), 669–681 (2004).
[CrossRef]

Udaltsov, V. S.

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[CrossRef]

Van der Sande, G.

Verschaffelt, G.

Xu, B.

H. Ma, B. Xu, W. Lin, J. Feng, “Adaptive identification of time delays in nonlinear dynamical models,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(6), 066210 (2010).
[CrossRef] [PubMed]

Yang, D.

Yang, M.

Q. Han, W. Li, M. Yang, “An optical waveform pre-distortion method based on time domain fractional Fourier transformation,” Opt. Commun. 284(2), 660–664 (2011).
[CrossRef]

Zbinden, H.

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

Zhang, H.

Zhang, J.

Zunino, L.

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(4), 046212 (2010).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

J. J. Suárez-Vargas, B. A. Marquez, J. A. Gonzalez, “Highly complex optical signal generation using electro-optical systems with non-linear, non-invertible transmission functions,” Appl. Phys. Lett. 101(7), 071115 (2012).
[CrossRef]

C. R. Phys. (1)

L. Larger, J. Goedgebuer, V. Udaltsov, “Ikeda-based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C. R. Phys. 5(6), 669–681 (2004).
[CrossRef]

Chaos (1)

F. Sorrentino, E. Ott, “Using synchronization of chaos to identify the dynamics of unknown systems,” Chaos 19(3), 033108 (2009).
[CrossRef] [PubMed]

Commun. ACM (1)

R. L. Rivest, A. Shamir, L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978).
[CrossRef]

IEEE J. Quantum Electron. (4)

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
[CrossRef]

X. Li, W. Pan, B. Luo, D. Ma, “Mismatch robustness and security of chaotic optical communications based on injection-locking chaos synchronization,” IEEE J. Quantum Electron. 42(9), 953–960 (2006).
[CrossRef]

J. Hizanidis, S. Deligiannidis, A. Bogris, D. Syvridis, “Enhancement of chaos encryption potential by combining all-optical and electrooptical chaos generators,” IEEE J. Quantum Electron. 46(11), 1642–1649 (2010).
[CrossRef]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–891 (2009).
[CrossRef]

J. Lightwave Technol. (1)

Nature (1)

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

Opt. Commun. (2)

M. Joshi, Chandrashakher, K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279(1), 35–42 (2007).
[CrossRef]

Q. Han, W. Li, M. Yang, “An optical waveform pre-distortion method based on time domain fractional Fourier transformation,” Opt. Commun. 284(2), 660–664 (2011).
[CrossRef]

Opt. Express (3)

Opt. Lett. (6)

Phys. Lett. A (1)

V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003).
[CrossRef]

Phys. Rev. E Stat. Nonlinear Soft Matter Phys. (7)

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(4), 046212 (2010).
[CrossRef] [PubMed]

D. Rontani, M. Sciamanna, A. Locquet, D. S. Citrin, “Multiplexed encryption using chaotic systems with multiple stochastic-delayed feedbacks,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(6), 066209 (2009).
[CrossRef] [PubMed]

H. Ma, B. Xu, W. Lin, J. Feng, “Adaptive identification of time delays in nonlinear dynamical models,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(6), 066210 (2010).
[CrossRef] [PubMed]

M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 79(2), 026208 (2009).
[CrossRef] [PubMed]

P. H. Bryant, “Optimized synchronization of chaotic and hyperchaotic systems,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(1), 015201 (2010).
[CrossRef] [PubMed]

B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(5), 056205 (2009).
[CrossRef] [PubMed]

Y. C. Kouomou, P. Colet, N. Gastaud, L. Larger, “Effect of parameter mismatch on the synchronization of chaotic semiconductor lasers with electro-optical feedback,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 69(5), 056226 (2004).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

S. C. Phatak, S. S. Rao, “Logistic map: a possible random-number generator,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 51(4), 3670–3678 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

Y. C. Kouomou, P. Colet, L. Larger, N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95(20), 203903 (2005).
[CrossRef] [PubMed]

R. M. Nguimdo, P. Colet, L. Larger, L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. 107(3), 034103 (2011).
[CrossRef] [PubMed]

R. Hegger, M. J. Bünner, H. Kantz, A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[CrossRef]

A. B. Cohen, B. Ravoori, T. E. Murphy, R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101(15), 154102 (2008).
[CrossRef] [PubMed]

U. Parlitz, “Estimating model parameters from time series by autosynchronization,” Phys. Rev. Lett. 76(8), 1232–1235 (1996).
[CrossRef] [PubMed]

Physica D (1)

M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3–4), 209–223 (2005).
[CrossRef]

Rev. Mod. Phys. (1)

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Setup of communication system based on electro-optic delay dynamics and time domain FRFT: LD: laser diode, MZ: Mach-Zehnder modulator, OC: optical coupler, DL: delay line, PD: photodiode, IPD: sign-inverting photo diode, RF: radio frequency driver, m(t): message.

Fig. 2
Fig. 2

Setup of time domain FRFT module, PM: phase modulator, V(t): drive voltage.

Fig. 3
Fig. 3

Autocorrelation function C(s) (a) and delayed mutual information D(s) (b) of chaotic carrier before transform. C(s) (c) and D(s) (d) after transform with p = 1 when Tw = 0.9ns (TDMTw). C(s) (e) and D(s) (f) after transform with p = 1 when Tw = 1ns (TD = MTw).

Fig. 4
Fig. 4

Peak sizes in C(s) (a) and D(s) (b) at s = TD and s = Tw as a function of |p|, Tw = 0.9ns.

Fig. 5
Fig. 5

Influence of the fractional order mismatch (a), dispersion (b) and time window (c) on synchronization. In (a), p = 1 in the FRFT at the transmitter side and p’ is varying from 1.02 to 1.2 in the IFRFT at the receiver side while other parameters remain the same, Tw = Tw’ = 0.9ns, S = S’ = 3000ps2. In (b), S = 3000ps2 in the FRFT and S’ varying from 2550 ps2 to 3000 ps2 in the IFRFT, Tw = Tw’ = 0.9ns, p = p’ = 1. In (c), the time window mismatch is 1ps, p = p’ = 1, S = S= 3000ps2.

Fig. 6
Fig. 6

Autocorrelation function C(s) (a) and delayed mutual information D(s) (b) when time window and fractional order controlled by PRBS. Tw {0.25ns,0.35ns,0.45ns,0.55ns}, p {0.8, 1.0, 1.2, 1.4}, S = 3000ps2.

Fig. 7
Fig. 7

Autocorrelation function (a) and delayed mutual information (b) after IFRFT with p’ = 0.8 Tw’ = 0.25ns S’ = 3000ps2. The parameters in the FRFT are S = 3000ps2 Tw {0.25ns,0.35ns,0.45ns,0.55ns}, p {0.8, 1.0, 1.2, 1.4}. The vertical dashed lines indicate s = TD

Tables (1)

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Table 1 Values of Parameters Used in the Simulation

Equations (17)

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x 1 + τ d x 1 d t + 1 θ t 0 t x 1 ( ε ) d ε = β { cos 2 [ x 1 ( t T D ) + Φ ] + m ( t T D ) } ,
x 2 + τ d x 2 d t + 1 θ t 0 t x 1 ( ε ) d ε = β { cos 2 [ x 1 ( t T C ) + Φ ] + m ( t T C ) } .
x ˜ ( u ) = F p [ x ( t ) ] = | sin ( p π 2 ) | 1 2 exp { j π 4 [ p s i g n [ sin ( p π 2 ) ] ] } x ( t ) exp { j π [ ( t 2 + u 2 ) cot ( p π 2 ) 2 t u csc ( p π 2 ) ] } d t ,
ψ ( t ) = K 1 exp ( i c 1 2 t 2 ) ,
H ( ω ) = K 2 exp ( i S 2 ω 2 ) ,
c 1 = 2 π λ f 0 sin ( p 2 π ) ,
S = λ f 0 2 π tan ( p 4 π ) ,
cos ( ω m t ) = 1 ( ω m t ) 2 2 ! + ( ω m t ) 4 4 ! + + ( 1 ) n ( ω m t ) 2 n ( 2 n ) ! .
cos ( ω m t ) = 1 ( ω m t ) 2 2 !
h P M ( t ) = exp [ i π V ( t ) V π ] ,
V ( t ) = c 1 V π 2 π t 2 .
V ( t ) = c 1 V π [ 1 cos ( ω m t ) ] π ω m 2 .
C ( s ) = [ v ( t ) v ( t ) ] [ v ( t s ) v ( t ) ] [ v ( t ) v ( t ) ] 2 ,
D ( s ) = v ( t ) , v ( t s ) P ( v ( t ) , v ( t s ) ) ln P ( v ( t ) , v ( t s ) ) P ( v ( t ) ) P ( v ( t s ) ) ,
P A C F ( γ ) = | C ( γ ) min { C ( γ ) | γ ( γ η , γ + η ) } | , P D M I ( γ ) = | D ( γ ) min { D ( γ ) | γ ( γ η , γ + η ) } | ,
σ = δ ( t ) 2 / x 1 ( t ) 2 ,
p ( t ) = 0.8 + 0.2 P B 1 i + 0.4 P B 2 i , T w ( t ) = 0.55 0.2 P B 1 i 0.1 P B 2 i , P B 1 i , P B 2 i { 0 , 1 } i T U t ( i + 1 ) T U i = 0 , 1 , 2 ,

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