Two-dimensional coupled electro-optic delayed feedback oscillator with complexity improvement and time delay concealment

Results in Physics 7 (2017) 4123–4129

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Two-dimensional coupled electro-optic delayed feedback oscillator with complexity improvement and time delay concealment Lingfeng Liu ⇑, Ming Luo School of Software, Nanchang University, Nanchang 330031, China

a r t i c l e

i n f o

Article history: Received 6 April 2017 Received in revised form 17 October 2017 Accepted 18 October 2017 Available online 22 October 2017 Keywords: Chaos Electro-optic chaotic system Time delay concealment

a b s t r a c t Conceal the time delay of the electro-optic chaotic system is important for its security. In this paper, we propose a new two-dimensional coupled electro-optic chaotic system. Two electro-optic systems are coupled by combining their electronic signals. This system can conceal the time delay naturally without any additional operations. Meanwhile, the new system can also improve the dynamical complexity of the original system, as well as extend the key space by bringing in some new tunable control coefficients. Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Introduction With the requirement of high speed communication, the optical communication network has received much attentions recently. The optic chaotic systems are with wide frequency spectrum, low attenuation and high transmission rate, which have been widely used in the all-optic communication network [1–3]. Presently, several methods are provided to generate optical chaos signal, including optic feedback [4], optic injection [5], optoelectronic feedback [6], and electro-optic delayed feedback system based on nonlinear devices, eg., by Mach-Zehnder modulator (MZ modulator) [7–9]. Comparatively speaking, the electro-optic delayed feedback system based on MZ modulator has high modulation speed, good stability, large Lyapunov exponent, et al., which makes it popular in the research. However, many studies show that the electro-optic chaotic system based on the MZ modulator is not secure enough [10,11], several deficiencies make this kind of chaos system vulnerable to attackers [12]. The output optical chaos signal will always carry the delay time information of original system, and the time delay signal will be recovered by using some appropriate methods, such as auto-correlation function (ACF) [13], delayed mutual information (DMI) [14], permutation information analysis [15], extrema statistics [16], and filling factor [17]. As the time delay is the most important parameter of a delayed chaotic system, the identification of time delay will make the chaotic system be easy to recon-

⇑ Corresponding author.

struct [18–20]. Thus, conceal the time delay is important for the electro-optic chaotic system to be used in secure communications. Till now, some proposals have been proposed by researchers to conceal its time delay. We can classify these methods into two categories. One is parameter modulation [21–23]. Hu et al. use the output signal of an electro-optic oscillator to modulate the gain coefficient of another electro-optic system [21]; Gao et al. proposed a method whereby an intermittent time delay modulation scheme by digital chaotic map [22]. The other is system coupling [24–27]. In [24], a coupled system by all-optical and electro-optic chaotic system is proposed. In [25], a pseudorandom bit sequence is coupled into the electro-optic chaotic system. Obviously, most of these methods need more physical devices, or digital signal processing modules, which leads to a practicality issue. Cheng et al. proposed a three-dimensional coupled electro-optic delayed feedback oscillator in [26]. The time delay can be concealed naturally without any additional operations. However, based on his coupling method, three laser diodes are necessary. The time delay will be easily identified if only two electro-optic chaotic systems are coupled. Beside the concealment of time delay signature, a chaotic system should also has a high complexity level to ensure the security of communications. Therefore, the improvement of complexity of electro-optic delayed feedback oscillator is also important. Motivated by the coupling method and [26], in this paper, we propose a new two-dimensional coupled electro-optic chaotic system, with whose time delay can be concealed naturally without any additional operations. Meanwhile, by using Approximate entropy (ApEn) and permutation entropy (PE) as the complexity measure,

E-mail address: [email protected] (L. Liu). https://doi.org/10.1016/j.rinp.2017.10.037 2211-3797/Ó 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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we have that the new coupled system can also improve the complexity of the original system in [8], as well as extend the key space by bringing in some new tunable control coefficients. Therefore, this new model has greatly improved the security of the electrooptic chaotic system based on MZ modulator, which can be used in secure communications. The rest of this paper is organized as follows. The new coupled electro-optic delayed feedback oscillator with its performances are presented in Section ‘‘The double delayed feedback electro-optic oscillator and its performances”. The analysis of time delay concealment is introduced in Section ‘‘Time-delay concealment analysis”, and finally, Section ‘‘Conclusions” concludes the whole paper. The double delayed feedback electro-optic oscillator and its performances The original electro-optic chaotic system based on the MZ modulator in [8] is described as

xþs

dx 1 þ dt h

Z

t

xðsÞds ¼ b cos2 ½xðt  TÞ þ U

ð1Þ

t0

where b = pgAGP/2Vp, U = pVB/2Vdc, h = 1/2pfL, and s = 1/2pfH. The symbols are defined as, Vp: radio-frequency voltages of MZ modulator; Vdc: dc half-wave voltages of MZ modulator; VB: biased voltage; P: power of semiconductor laser source; T: time delay; g: gain coefficients of photodiode; G: gain coefficients of ratio-frequency amplifier; A: Overall attenuation of feedback loop; fL: low cutoff frequencies; fH: high cutoff frequencies. As presented in many studies, the time delay T can be identified by ACF or DMI method [23,27]. In order to conceal the time delay, we construct the following two-dimensional coupled electro-optic chaotic system, with its mathematical model be written as

8 < x1 þ s1 dx1 þ : x2 þ s

dt

1 h1

dx2 2 dt

þ h12

Rt t0

x1 ðsÞds ¼ b1 cos2 ½x1 ðt  TÞ þ x2 ðt  TÞ þ U1 

t0

x2 ðsÞds ¼ b2 cos2 ½x2 ðt  TÞ  x1 ðt  TÞ þ U2 

Rt

ð2Þ where x1 and x2 are the state variable. Obviously, compared to system (1), system (2) contains more tunable parameters, which means that the key space has been greatly extended. In this paper, we always let s1 = s2 = s, h1 = h2 = h, b1 = b2 = b and U1 = U2 = U for

simplicity. The whole scheme of this system is shown in Fig. 1. For each single chain, the model is similar to system (1). We illustrate the dynamical performances of system (2) by setting s = 25 ps, h = 5 ls, U = p/4, and T = 60 ns. The x1 trajectories of system (2) with different gain coefficient b can be shown in Fig. 2 (The x2 trajectories are similar which is omitted here to avoid redundancy). As Fig. 2 shown, the trajectories are regular with a small gain (b = 1; b = 1.2), and come to be pseudorandom-like with the growth of b (b = 1.5). Fig. 3 plots the attractor of this coupled system when b = 2. The two-dimensional phase points can cover the whole phase space, which indicates that the coupled system has a quite good ergodicity. The bifurcation diagram of system (2) is shown in Fig. 4. The simulation results show that the coupled system is chaotic since b > 1.3, which will enter chaos with a lower gain b than system (1) (The system (1) will be chaotic since b > 2.5 [8]). Next, we map the dynamical complexity by using ApEn and PE, respectively. ApEn measures the probability of the new pattern generated in the sequences with the embedding dimension grows, which was proposed by Pincus [28], and now is regarded as an important complexity measure for time series. We compare the ApEn of our system with the original system (1). The results are shown in Fig. 5. From Fig. 5, we can find that the ApEn of system (2) is obviously much larger, which means that our system has greatly improved the complexity of system (1) in this sense. PE compares the size of some consecutive values in the sequence, and summed up different order types, then use Shannon’s entropy to measure the uncertainty of these ordering, which was introduced by Bandt and Pompe [29]. As suggested in [29], we set L = 6 (ordinal pattern length), D = 2 (embedding delay), and N = 1.2  105 (sequence length). Fig. 6 presents the PE complexity of system (1) and system (2), which indicates that the chaotic trajectories generated by system (2) is rather complex (PE(x1) > 0.99) when b  4, and is always more complex than the system (1) with the same b. Time-delay concealment analysis Among all the time delay identification methods, the ACF and DMI are the most two effective methods due to their robustness to noise. In this section, we will analyze the concealment of time delay against these two methods.

Fig. 1. The schematic of our 2D coupled electro-optic oscillator, where LD is for laser diode; PD is for photodetector; RF driver is for radio frequency amplifier; Delay is for fiber delay lines; ES is for Electrical power splitter.

L. Liu, M. Luo / Results in Physics 7 (2017) 4123–4129

Fig. 2. The trajectories of system (2) with (a1) (a2) b = 1; (b1) (b2) b = 1.2; (c1) (c2) b = 1.5.

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Fig. 3. The attractor of System (2)when b = 2.

Fig. 5. The ApEn of system (1) and system (2). Fig. 4. The bifurcation diagram of system (2).

ACF method Set b = 5, we use ACF to analyze the time delay of systems (1) and (2), see Fig. 7. As Fig. 7 indicated, there is an apparent peak for system (1) when T = 60 ns, which is exactly the delay time of original system. While no apparent peaks can be found in system (2), which means that the time delay cannot be revealed by this method. The time delay signature is eliminated for system (2) may be because that the coupling of systems will reduce the correlation of the output signal. Due to this feature, the coupling method can be used in the time delay concealment [24–27]. As we know, theoretically, the electro-optic delayed feedback oscillator can conceal its time delay with a larger gain b. While in practice, gain coefficient b must be less than the gain limit of amplifier. Therefore, the influence of b in the time delay concealment is important for an identification method. The following definition of background QACF and peak size PACF(b) at the relevant time delay are needed in this analysis.

Fig. 6. The PE of system (1) and system (2).

L. Liu, M. Luo / Results in Physics 7 (2017) 4123–4129

Fig. 7. ACF of (a) system (1); (b) system (2).

Fig. 8. The peak size PACF(b) at 60ns for different b.

Fig. 9. DMI of (a) system (1); (b) system (2).

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Conclusions Improving the security of electro-optic delayed feedback oscillator is necessary for secure communication. In this paper, a new two-dimensional coupled electro-optic delayed feedback chaotic system is proposed. We numerically proves that our coupled system can conceal the time delay against ACF and DMI methods since b = 5. Meanwhile, the attractor, bifurcation diagram, ApEn and PE analysis show that this system can improve the complexity of original system in [8]. Furthermore, the key space can be enlarged by bringing in some new tunable control parameters. In summary, this new coupled system has greatly improved the security of original electro-optic delayed feedback oscillator, and is easier to implement than most other proposed electro-optic delayed feedback systems.

Acknowledgements

(



Fig. 10. The peak size PDMI(b) at 60 ns for different b.

Q ACF ¼ ½P ACF ðbÞ; PACF ðbÞ PACF ðbÞ ¼ ACFðxðbÞ; TÞ

ð3Þ

References

The bounds of background QACF are defined as

PACF ðbÞ ¼ meanfACFðxðbÞÞg  SDðACFðxðbÞÞÞ PACF ðbÞ ¼ meanfACFðxðbÞÞg þ SDðACFðxðbÞÞÞ

ð4Þ

where SD denotes the standard deviation, x(b) is the state variable x1 of system (2) with different b. If the peak size PACF(b) is in the background QACF, the time delay is said to be concealed. The background QACF and peak size PACF(b) of system (2) is shown in Fig. 8. In Fig. 8, the two red lines denote the max and min bounds of background QACF, and the squares denote the peak sizes at 60 ns for different gain b. As Fig. 8 shows, all the peak sizes are located in the background since b = 5, which concludes that the time delay has been concealed since b = 5 of system (2). DMI method The DMI of systems (1) and (2) when b = 5 are presented in Fig. 9(a) and (b), respectively. From Fig. 9(a), we can find an extrema at time delay T = 60 ns, which is actually corresponding to the time delay of system (1). While in Fig. 9(b), no apparent extrema can be found, which implies its time delay has been effectively concealed. Furthermore, we investigate the influence of gain b. The background QDMI and the peak size PDMI are defined as

(

¼ ½P DMI ðbÞ; PDMI ðbÞ

Q DMI

DMIðxðbÞ;TÞ PDMI ðbÞ ¼ meanfDMIðxðbÞÞgSDðDMIðxðbÞÞÞ

ð5Þ

where the bounds of background QDMI are defined as

PDMI ðbÞ ¼ ¼

meanfDMIðxðbÞÞg  SDðDMIðxðbÞÞÞ PDMI ðbÞ meanfDMIðxðbÞÞg  SDðDMIðxðbÞÞÞ meanfDMIðxðbÞÞg þ SDðDMIðxðbÞÞÞ meanfDMIðxðbÞÞg  SDðDMIðxðbÞÞÞ

This work is supported by the National Natural Science Foundation of China (61601215 61662046); Science & technology research project of Education Department of Jiangxi Province (GJJ150104).

ð6Þ

As shown in Eq. (5) and (6), a normalization is used here. The normalization operation has no influence to the final results, which is only used to make the results more clear. The simulation results are shown in Fig. 10, where the red lines denote the max and min bounds of background QDMI, and the squares denote the peak sizes at 60 ns for different b. From Fig. 10 we have that the time delay can be concealed for b = 5 as the peak sizes for these b are all located in the background QDMI.

[1] Argyris A, Syvridis D. Chaos-based communications at high bit rates using commercial fiber-optic links. Nature 2005;438:343–6. [2] Xiang SY, Pan W, Zhang LY, Wen AJ, Shang L, Zhang H, Lin L. Phase-modulated dual-path feedback for time delay signature suppression from intensity and phase chaos in semiconductor laser. Opt Commun 2014;324:38–48. [3] Li N et al. Chaotic optical cryptographic communication using a threesemiconductor-laser scheme. J Opt Soc Am B 2012;29(1):101–8. [4] Jiang N et al. Isochronal chaos synchronization of semiconductor lasers with multiple time-delayed couplings. J Opt Soc Am B 2011;28(5):1139–45. [5] Ohtsubo J. Dynamics in semiconductor lasers with optical injection. In: Semiconductor Lasers, Springer Ser. Opt. Sci. 2013;111:169–204. [6] Rajesh S, Nandankumaran VM. Control of bistability in a directly modulated semiconductor laser using delayed optoelectronic feedback. Physica D 2006;213:113–20. [7] Lavrov R, Jacquot M, Larger L. Nonlocal nonlinear electro-optic phase dynamics demonstrating 10 Gb/s chaos communications. IEEE J Quantum Electron 2010;46(10):1430–5. [8] Kouomou YC, Colet P, Larger L, Gastaud N. Chaotic breathers in delayed electro-optical systems. Phys Rev Lett 2005;95(20):203903. [9] Larger L. Complexity in electro-optic delay dynamics: modeling, design and applications. Philos Trans R Soc A Math Phys Eng Sci 2013;371:20120464. [10] Udaltsov VS et al. Cracking chaos-based encryption system ruled by nonlinear time dealy differential equations. Phys Lett A 2003;308:54–60. [11] Udaltsov VS, Larger L, Geodgebuer JP. Time delay identification in chaotic cryptosystems ruled by delay-differential equations. J Opt Technol 2005;72:373–7. [12] Romain RM, Colet P, Larger L, et al. Digital key for chaos communication performing time delay concealment. Phys Rev Lett 2011;107(3):034103. [13] Cheng M, Deng L, Li H, Liu D. Enhanced secure strategy for electro-optic chaotic systems with delayed dynamics by using fractional Fourier transformation. Opt Exp 2014;22:5241–51. [14] Nguimdo RM, Verschaffelt G, Danckaert J, Van der Sande G. Loss of time-delay signature in chaotic semiconductor ring lasers. Opt Lett 2012;37:2541–3. [15] Soriano MC, Zunino L, Rosso OA, Fischer I, Mirasso CR. Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis. IEEE J Quantum Electron 2011;47(2):252–61. [16] Prokhorov M, Ponomarenko V, Karavaev A, Bezruchko B. Reconstruction of time-delayed feedback systems from time series. Physica D 2005;203:209–23. [17] Bünner M, Meyer T, Kittel A, Parisi J. Recovery of the time-evolution equation of time-delay systems from time series. Phys Rev E 1997;56:5083. [18] Ortin S, Guitierrez J, Pesquera L, Vasquez H. Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction. Phys A 2005;351:133–41. [19] Leonora U, Santagiustina M, Annovazzi-Lodi V. Enhancing chaotic communication performances by Manchester coding. IEEE Phot Tech Lett 2008;20:401–3. [20] Tronciu VZ et al. Chaos generation and synchronization using an integrated source with an air gap. IEEE J Quantum Electron 2010;46:1840–6. [21] Hu HP, Su W. L. f. Liu, Z. L. Yu. Electro-optic intensity chaotic system with varying parameters. Phys Lett A 2014;378(3):184–90. [22] Gao XJ, Xie FL, Hu HP. Enhancing the security of electro-optic delayed chaotic system with intermittent time-delay modulation and digital chaos. Optic Commun 2015;352:77–83.

L. Liu, M. Luo / Results in Physics 7 (2017) 4123–4129 [23] Liu LF, Miao SX, Cheng MF, Gao XJ. A new switching parameter varying optoelectronic delayed feedback model with computer simulation. Sci Rep 2016;6:22295. [24] Hizanidis J, Deligiannidis S, Bogris A, Syvridis D. Enhancement of chaos encryption potential by combining all-optical and electrooptical chaos generators. IEEE J Quantum Electron 2010;46 (11–12):1642–9. [25] Bogris A, Rizomiliotis P, Chlouverakis KE, Agyris A, Syvridis D. Feedback phase in optically generated chaos: A secret key for cryptographic applications. IEEE J Quantum Electron 2008;44(2):119–24.

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[26] Cheng M, Gao X, Deng L, Liu L, Deng Y, Fu S, Zhang M, Liu D. Time delay concealment in a three-dimensional electro-optic chaos system. IEEE Photon Technol Lett 2015;27(9):1030–3. [27] Liu LF, Miao SX, Cheng MF, Gao XJ. Improving the security of optoelectronic delayed feedback system by parameter modulation and system coupling. Opt Eng 2016;55(2):026101. [28] Pincus SM, Kalman RE. Not all (possibly) ‘random’ sequences are created equal. Proc Nat Acad Sci (PNAS) 1997;94(6):3513–8. [29] Bandt C, Pompe B. Permutation entropy: a natural complexity measure for time series. Phys Rev Lett 2002;88(17):174102.