Enhanced secure strategy for electro-optic chaotic systems with delayed dynamics by using fractional Fourier transformation Mengfan Cheng,1 Lei Deng,1,2,* Hao Li,1 and Deming Liu1,2 1
Next generation Internet Access National Engineering lab (NGIA), School of Optoelectronic Science and Engineering, Huazhong University of Sci.&Tech. (HUST), Wuhan, 430074 China 2 Wuhan National lab for Optoelectronics (WNLO), Huazhong University of Sci.&Tech., Wuhan, 430074 China * [email protected]
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 OCIS codes: (250.0250) Optoelectronics; (140.1540) Chaos; (060.4785) Optical security and encryption.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
P. Colet and R. Roy, “Digital communication with synchronized chaotic lasers,” Opt. Lett. 19(24), 2056–2058 (1994). S. Tang and J. M. Liu, “Message encoding-decoding at 2.5 Gbits/s through synchronization of chaotic pulsing semiconductor lasers,” Opt. Lett. 26(23), 1843–1845 (2001). 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). L. Larger, J. Goedgebuer, and V. Udaltsov, “Ikeda-based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C. R. Phys. 5(6), 669–681 (2004). A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101(15), 154102 (2008). M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, and 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). Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95(20), 203903 (2005). R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. 107(3), 034103 (2011). R. M. Nguimdo and P. Colet, “Electro-optic phase chaos systems with an internal variable and a digital key,” Opt. Express 20(23), 25333–25344 (2012). R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998). V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003). M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3–4), 209–223 (2005). L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and 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).
#202795 - $15.00 USD (C) 2014 OSA
Received 10 Dec 2013; revised 29 Jan 2014; accepted 20 Feb 2014; published 27 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005241 | OPTICS EXPRESS 5241
14. U. Parlitz, “Estimating model parameters from time series by autosynchronization,” Phys. Rev. Lett. 76(8), 1232–1235 (1996). 15. F. Sorrentino and E. Ott, “Using synchronization of chaos to identify the dynamics of unknown systems,” Chaos 19(3), 033108 (2009). 16. P. H. Bryant, “Optimized synchronization of chaotic and hyperchaotic systems,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(1), 015201 (2010). 17. D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and 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). 18. R. M. Nguimdo, M. C. Soriano, and 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). 19. D. Rontani, M. Sciamanna, A. Locquet, and 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). 20. H. Ma, B. Xu, W. Lin, and J. Feng, “Adaptive identification of time delays in nonlinear dynamical models,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(6), 066210 (2010). 21. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett. 37(13), 2541–2543 (2012). 22. J. Hizanidis, S. Deligiannidis, A. Bogris, and D. Syvridis, “Enhancement of chaos encryption potential by combining all-optical and electrooptical chaos generators,” IEEE J. Quantum Electron. 46(11), 1642–1649 (2010). 23. J. J. Suárez-Vargas, B. A. Marquez, and 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). 24. B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003). 25. Q. Ran, H. Zhang, J. Zhang, L. Tan, and J. Ma, “Deficiencies of the cryptography based on multiple-parameter fractional Fourier transform,” Opt. Lett. 34(11), 1729–1731 (2009). 26. M. Joshi, Chandrashakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279(1), 35–42 (2007). 27. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951– 1963 (1994). 28. Q. Han, W. Li, and M. Yang, “An optical waveform pre-distortion method based on time domain fractional Fourier transformation,” Opt. Commun. 284(2), 660–664 (2011). 29. D. Yang and S. Kumar, “Realization of optical OFDM using time lenses and its comparison with optical OFDM using FFT,” Opt. Express 17(20), 17214–17226 (2009). 30. X. Li, W. Pan, B. Luo, and 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). 31. Y. C. Kouomou, P. Colet, N. Gastaud, and 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). 32. A. Argyris, E. Grivas, M. Hamacher, A. Bogris, and D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18(5), 5188–5198 (2010). 33. B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, and R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(5), 056205 (2009). 34. R. M. Nguimdo, R. Lavrov, P. Colet, M. Jacquot, Y. K. Chembo, and 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). 35. R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978). 36. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). 37. S. C. Phatak and 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).
1. Introduction Chaos based optical communication systems have attracted considerable attention due to their superiority [1, 2]. It is worth noting that most recent results reported in the literatures focus on electro-optic chaotic systems with delayed dynamics [3–9]. In these systems, the message security relies essentially on the difficulty of identifying the hardware parameters of chaotic transmitters. However, many different techniques have been proposed to estimate the parameters of chaotic system. Of all the parameters in delayed dynamical systems, time delay is of great importance, since it is the key to generate high dimensional chaotic signals which is the foundation of chaos based communications. An effective type of attack is to recover the time delay by using some statistical methods, such as autocorrelation function (ACF), delayed #202795 - $15.00 USD (C) 2014 OSA
Received 10 Dec 2013; revised 29 Jan 2014; accepted 20 Feb 2014; published 27 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005241 | OPTICS EXPRESS 5242
mutual information (DMI), extrema statistics and filling factor [10–13]. On the other hand, since U. Parlitz suggested exploiting the adaptive synchronization of chaotic systems to solve parameter estimation problem, various synchronization-based schemes have been studied by researchers [14–16]. To resist these attacks, several methods attempt to conceal the delay signature by adjusting the delay time itself. In [17], the delay time is chosen to be close to the laser relaxation period. However, chaos complexity is weak in this regime. Moreover, in [18], it has been demonstrated that the delay signature can still be retrieved from the phase time series, even in the presence of noise. Time delay modulation [19] has also been considered as a theoretically feasible way to prevent the time delay extraction, but it is difficult to implement practically. These methods are also threatened by synchronization-based parameter identification methods [20]. Therefore efforts have been made to modify the system structure. A cross feedback configuration is introduced in semiconductor ring lasers [21]. With such a feedback, the time delay signature can be eliminated both in the intensity and the phase dynamics. In [22], the time delay is successfully hidden by combining alloptical and electro-optical schemes. Another currently effective method is proposed by R. M. Nguimdo, et al. [8, 9]. Both serial and parallel configurations of the electro-optic phase chaos system with two feedback loops have been considered in their schemes. An external PRBS was mixed within the chaotic carrier to perform time delay concealment. In this paper, we propose an alternative way to enhance the secure strategy for electrooptic chaotic system by using a post-processing technique, without modifying the chaotic source. Similar concept has been proposed in [23]. A non-linear, non-invertible transmission function (implemented by a Mach-Zehnder (MZ) modulator) is used to improve the security of chaotic signal. Although time delay concealment is not one of the concerns in the literature, it gives us an inspiration. In our scheme, the transform adopted should be invertible, since a non-invertible transform may have some difficulties in synchronization. Another concern is that the transform should be an encryption transform. Otherwise the original chaotic carrier will easily be obtained by an illegal receiver. At last, the transform should be implementable in optical field. Motivated by the above discussion, we found that the FRFT in time domain fits our requirements perfectly. FRFT has been widely used in signal processing and image encryption [24–29]. Thanks to the principle of space-time duality theory [27], FRFT becomes a new way to analyze and process optical time signals [28, 29]. In this work we introduce, time domain FRFT is used as an encryption transform to post process the electro-optical chaotic carrier. As we will show below, when certain conditions are satisfied, the time delay signature can be masked perfectly. Last but not least, PRBS is also introduced in our scheme. It is noteworthy that the PRBS is not necessary for time delay mask, which makes our method conceptually different from the method proposed in [8], but still, it is indispensable for resisting brute force attack. 2. System setup Here we propose a configuration built on electro-optic chaotic system and time domain FRFT. The proposed setup is illustrated in Fig. 1. Both the transmitter and the receiver consist of an electro-optic chaotic system and a FRFT (IFRFT) module, connected in serial. In the electro-optic chaotic system [3], a continuous-wave (CW) laser diode (LD) at the telecom wavelength of 1550 nm is seeded to a MZ modulator whose radio-frequency (RF) and directcurrent (DC) half-wave voltages are Vπ and Vπdc. After an optical coupler (OC) and an optical fiber delay line (DL) of delay time TD, the optical signal is detected by a photodiode (PD). Before fed back into the MZ modulator, the generated electrical signal is amplified by a RF driver. In
#202795 - $15.00 USD (C) 2014 OSA
Received 10 Dec 2013; revised 29 Jan 2014; accepted 20 Feb 2014; published 27 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005241 | OPTICS EXPRESS 5243
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.
transmitter, the message is added inside the loop by the OC. And in receiver, the message is obtained via canceling the chaotic carrier. The dynamical modeling of chaotic system in the transmitter and the receiver can be described in Eqs. (1) and (2), which were proposed and studied in [4–7]. x1 + τ
dx1 1 t x1 (ε )d ε = β {cos 2 [ x1 (t − TD ) + Φ ] + m(t − TD )}, + dt θ t0
(1)
x2 + τ
dx2 1 t x1 (ε )d ε = β {cos 2 [ x1 (t − TC ) + Φ ] + m(t − TC )}. + dt θ t0
(2)
In Eqs. (1) and (2), x1,2(t) = πV1,2(t)/2Vπ, V1,2(t) are the input voltages for the MZ electrode. m(t) is the message signal, β is the feedback strength of the loop, Φ is the offset phase, τ and θ are the characteristic response times of the loop, and TC is the propagation time of light between the transmitter and receiver. FRFT is a generalization of the Fourier transform, and can be seen as the projection of a given signal between time and frequency axis. Next, let us consider the implementation of the FRFT in time domain. The FRFT of a given signal x(t) is defined as
π
x (u ) = F [ x(t )] = sin( p ) 2 p
−
1 2
π π exp j p − sign[sin( p )] 4 2
∞
π π 2 2 −∞ x(t ) exp jπ (t + u ) cot( p 2 ) − 2tu csc( p 2 ) dt ,
(3)
where p is the fractional order and 0
Next generation Internet Access National Engineering lab (NGIA), School of Optoelectronic Science and Engineering, Huazhong University of Sci.&Tech. (HUST), Wuhan, 430074 China 2 Wuhan National lab for Optoelectronics (WNLO), Huazhong University of Sci.&Tech., Wuhan, 430074 China * [email protected]
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 OCIS codes: (250.0250) Optoelectronics; (140.1540) Chaos; (060.4785) Optical security and encryption.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
P. Colet and R. Roy, “Digital communication with synchronized chaotic lasers,” Opt. Lett. 19(24), 2056–2058 (1994). S. Tang and J. M. Liu, “Message encoding-decoding at 2.5 Gbits/s through synchronization of chaotic pulsing semiconductor lasers,” Opt. Lett. 26(23), 1843–1845 (2001). 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). L. Larger, J. Goedgebuer, and V. Udaltsov, “Ikeda-based nonlinear delayed dynamics for application to secure optical transmission systems using chaos,” C. R. Phys. 5(6), 669–681 (2004). A. B. Cohen, B. Ravoori, T. E. Murphy, and R. Roy, “Using synchronization for prediction of high-dimensional chaotic dynamics,” Phys. Rev. Lett. 101(15), 154102 (2008). M. Peil, M. Jacquot, Y. K. Chembo, L. Larger, and 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). Y. C. Kouomou, P. Colet, L. Larger, and N. Gastaud, “Chaotic breathers in delayed electro-optical systems,” Phys. Rev. Lett. 95(20), 203903 (2005). R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. 107(3), 034103 (2011). R. M. Nguimdo and P. Colet, “Electro-optic phase chaos systems with an internal variable and a digital key,” Opt. Express 20(23), 25333–25344 (2012). R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998). V. S. Udaltsov, J. P. Goedgebuer, L. Larger, J. B. Cuenot, P. Levy, and W. T. Rhodes, “Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations,” Phys. Lett. A 308(1), 54–60 (2003). M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, and B. P. Bezruchko, “Reconstruction of time-delayed feedback systems from time series,” Physica D 203(3–4), 209–223 (2005). L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and 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).
#202795 - $15.00 USD (C) 2014 OSA
Received 10 Dec 2013; revised 29 Jan 2014; accepted 20 Feb 2014; published 27 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005241 | OPTICS EXPRESS 5241
14. U. Parlitz, “Estimating model parameters from time series by autosynchronization,” Phys. Rev. Lett. 76(8), 1232–1235 (1996). 15. F. Sorrentino and E. Ott, “Using synchronization of chaos to identify the dynamics of unknown systems,” Chaos 19(3), 033108 (2009). 16. P. H. Bryant, “Optimized synchronization of chaotic and hyperchaotic systems,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(1), 015201 (2010). 17. D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and 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). 18. R. M. Nguimdo, M. C. Soriano, and 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). 19. D. Rontani, M. Sciamanna, A. Locquet, and 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). 20. H. Ma, B. Xu, W. Lin, and J. Feng, “Adaptive identification of time delays in nonlinear dynamical models,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 82(6), 066210 (2010). 21. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett. 37(13), 2541–2543 (2012). 22. J. Hizanidis, S. Deligiannidis, A. Bogris, and D. Syvridis, “Enhancement of chaos encryption potential by combining all-optical and electrooptical chaos generators,” IEEE J. Quantum Electron. 46(11), 1642–1649 (2010). 23. J. J. Suárez-Vargas, B. A. Marquez, and 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). 24. B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003). 25. Q. Ran, H. Zhang, J. Zhang, L. Tan, and J. Ma, “Deficiencies of the cryptography based on multiple-parameter fractional Fourier transform,” Opt. Lett. 34(11), 1729–1731 (2009). 26. M. Joshi, Chandrashakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279(1), 35–42 (2007). 27. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951– 1963 (1994). 28. Q. Han, W. Li, and M. Yang, “An optical waveform pre-distortion method based on time domain fractional Fourier transformation,” Opt. Commun. 284(2), 660–664 (2011). 29. D. Yang and S. Kumar, “Realization of optical OFDM using time lenses and its comparison with optical OFDM using FFT,” Opt. Express 17(20), 17214–17226 (2009). 30. X. Li, W. Pan, B. Luo, and 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). 31. Y. C. Kouomou, P. Colet, N. Gastaud, and 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). 32. A. Argyris, E. Grivas, M. Hamacher, A. Bogris, and D. Syvridis, “Chaos-on-a-chip secures data transmission in optical fiber links,” Opt. Express 18(5), 5188–5198 (2010). 33. B. Ravoori, A. B. Cohen, A. V. Setty, F. Sorrentino, T. E. Murphy, E. Ott, and R. Roy, “Adaptive synchronization of coupled chaotic oscillators,” Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 80(5), 056205 (2009). 34. R. M. Nguimdo, R. Lavrov, P. Colet, M. Jacquot, Y. K. Chembo, and 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). 35. R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978). 36. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). 37. S. C. Phatak and 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).
1. Introduction Chaos based optical communication systems have attracted considerable attention due to their superiority [1, 2]. It is worth noting that most recent results reported in the literatures focus on electro-optic chaotic systems with delayed dynamics [3–9]. In these systems, the message security relies essentially on the difficulty of identifying the hardware parameters of chaotic transmitters. However, many different techniques have been proposed to estimate the parameters of chaotic system. Of all the parameters in delayed dynamical systems, time delay is of great importance, since it is the key to generate high dimensional chaotic signals which is the foundation of chaos based communications. An effective type of attack is to recover the time delay by using some statistical methods, such as autocorrelation function (ACF), delayed #202795 - $15.00 USD (C) 2014 OSA
Received 10 Dec 2013; revised 29 Jan 2014; accepted 20 Feb 2014; published 27 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005241 | OPTICS EXPRESS 5242
mutual information (DMI), extrema statistics and filling factor [10–13]. On the other hand, since U. Parlitz suggested exploiting the adaptive synchronization of chaotic systems to solve parameter estimation problem, various synchronization-based schemes have been studied by researchers [14–16]. To resist these attacks, several methods attempt to conceal the delay signature by adjusting the delay time itself. In [17], the delay time is chosen to be close to the laser relaxation period. However, chaos complexity is weak in this regime. Moreover, in [18], it has been demonstrated that the delay signature can still be retrieved from the phase time series, even in the presence of noise. Time delay modulation [19] has also been considered as a theoretically feasible way to prevent the time delay extraction, but it is difficult to implement practically. These methods are also threatened by synchronization-based parameter identification methods [20]. Therefore efforts have been made to modify the system structure. A cross feedback configuration is introduced in semiconductor ring lasers [21]. With such a feedback, the time delay signature can be eliminated both in the intensity and the phase dynamics. In [22], the time delay is successfully hidden by combining alloptical and electro-optical schemes. Another currently effective method is proposed by R. M. Nguimdo, et al. [8, 9]. Both serial and parallel configurations of the electro-optic phase chaos system with two feedback loops have been considered in their schemes. An external PRBS was mixed within the chaotic carrier to perform time delay concealment. In this paper, we propose an alternative way to enhance the secure strategy for electrooptic chaotic system by using a post-processing technique, without modifying the chaotic source. Similar concept has been proposed in [23]. A non-linear, non-invertible transmission function (implemented by a Mach-Zehnder (MZ) modulator) is used to improve the security of chaotic signal. Although time delay concealment is not one of the concerns in the literature, it gives us an inspiration. In our scheme, the transform adopted should be invertible, since a non-invertible transform may have some difficulties in synchronization. Another concern is that the transform should be an encryption transform. Otherwise the original chaotic carrier will easily be obtained by an illegal receiver. At last, the transform should be implementable in optical field. Motivated by the above discussion, we found that the FRFT in time domain fits our requirements perfectly. FRFT has been widely used in signal processing and image encryption [24–29]. Thanks to the principle of space-time duality theory [27], FRFT becomes a new way to analyze and process optical time signals [28, 29]. In this work we introduce, time domain FRFT is used as an encryption transform to post process the electro-optical chaotic carrier. As we will show below, when certain conditions are satisfied, the time delay signature can be masked perfectly. Last but not least, PRBS is also introduced in our scheme. It is noteworthy that the PRBS is not necessary for time delay mask, which makes our method conceptually different from the method proposed in [8], but still, it is indispensable for resisting brute force attack. 2. System setup Here we propose a configuration built on electro-optic chaotic system and time domain FRFT. The proposed setup is illustrated in Fig. 1. Both the transmitter and the receiver consist of an electro-optic chaotic system and a FRFT (IFRFT) module, connected in serial. In the electro-optic chaotic system [3], a continuous-wave (CW) laser diode (LD) at the telecom wavelength of 1550 nm is seeded to a MZ modulator whose radio-frequency (RF) and directcurrent (DC) half-wave voltages are Vπ and Vπdc. After an optical coupler (OC) and an optical fiber delay line (DL) of delay time TD, the optical signal is detected by a photodiode (PD). Before fed back into the MZ modulator, the generated electrical signal is amplified by a RF driver. In
#202795 - $15.00 USD (C) 2014 OSA
Received 10 Dec 2013; revised 29 Jan 2014; accepted 20 Feb 2014; published 27 Feb 2014 10 March 2014 | Vol. 22, No. 5 | DOI:10.1364/OE.22.005241 | OPTICS EXPRESS 5243
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.
transmitter, the message is added inside the loop by the OC. And in receiver, the message is obtained via canceling the chaotic carrier. The dynamical modeling of chaotic system in the transmitter and the receiver can be described in Eqs. (1) and (2), which were proposed and studied in [4–7]. x1 + τ
dx1 1 t x1 (ε )d ε = β {cos 2 [ x1 (t − TD ) + Φ ] + m(t − TD )}, + dt θ t0
(1)
x2 + τ
dx2 1 t x1 (ε )d ε = β {cos 2 [ x1 (t − TC ) + Φ ] + m(t − TC )}. + dt θ t0
(2)
In Eqs. (1) and (2), x1,2(t) = πV1,2(t)/2Vπ, V1,2(t) are the input voltages for the MZ electrode. m(t) is the message signal, β is the feedback strength of the loop, Φ is the offset phase, τ and θ are the characteristic response times of the loop, and TC is the propagation time of light between the transmitter and receiver. FRFT is a generalization of the Fourier transform, and can be seen as the projection of a given signal between time and frequency axis. Next, let us consider the implementation of the FRFT in time domain. The FRFT of a given signal x(t) is defined as
π
x (u ) = F [ x(t )] = sin( p ) 2 p
−
1 2
π π exp j p − sign[sin( p )] 4 2
∞
π π 2 2 −∞ x(t ) exp jπ (t + u ) cot( p 2 ) − 2tu csc( p 2 ) dt ,
(3)
where p is the fractional order and 0
