Abstract

A chaotic-shift-keying (CSK) scheme is designed based on a chaos system with electro-optical hybrid time delayed feedback structure. By switching the time delay parameter as a message feeding method, the generated chaotic signal is no longer suffered from return map attack, which is an innate vulnerability of traditional CSK. When the coupling of the seed electrical chaotic system and the nonlinear optical time delay feedback loop is carefully weighed, this CSK scheme shows a good robustness in terms of handling noise for transmitting digital signals. By demodulating the digital signals with the chaotic coherent detection method, a bit error rate of 6×10−4 is achieved at the signal-to-noise ratio of 10dB in the simulation. The proposed method has a promising application prospect in some harsh environments.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Chaotic systems have some attractive features including sensitivity to initial conditions and system parameters, noise-like behaviors and long-time unpredictability [1]. In concept, these properties coincide with the “confusion” and “diffusion” strategies of cryptography. As such, chaotic systems have been drawing considerable interest in secure communication systems [2,3]. Starting from the early work of chaos synchronization [4], chaos secure communication based on the concept of chaotic hardware key has been considered as a promising solution for secure data transmission.

Chaos secure communication systems are originally designed and experimentally realized based on nonlinear analogue circuits [5,6]. Subsequently, optical chaos systems have been designed and gained much research attention in the area of physical layer secure communication [7,8]. Since optical chaos secure communication at the rate of 2.4 Gb/s was experimentally achieved over 120km of optical fiber in the metropolitan area network of Athens in 2005 [9], high-speed optical chaos secure communication with single wavelength [10] or wavelength division multiplexing scheme [11] over long-distance (longer than 100km) optical fibers has been experimentally demonstrated in laboratories. It is worth noting that these high-speed schemes are conducted over low-noise optical fiber channel. However, in some practical communication environments, including wireless communication, underwater communication, and free space communication, worse signal deterioration is inevitable. In fact, reliable transmission of information has always been a challenging open problem in these non-ideal communication environments. Therefore, the demand for chaos secure communication under bad channel conditions needs to be considered.

To overcome this issue, a direct idea is decontaminating chaotic signals. Some noise cleaning algorithms including Discrete-time Kalman filtering, Extended Kalman filters and Wiener filter are useful for this task [1214]. In [15,16], the effect of noise cleaning on the bit-error-rate (BER) performance of differential chaos shift keying (DCSK) with strong anti-noise ability is analyzed in detail. Simulation results show that just when signal-to-noise ratio (SNR) exceeds a certain threshold, a slight performance improvement can be achieved by these noise cleaning algorithms.

Non-coherent chaotic demodulation strategy is another method to resist noise. The receiver does not need to regenerate the chaotic carrier via synchronization. Two classes of non-coherent chaos communication systems have been studied and developed: 1) differential chaos shift keying (DCSK) was proposed in 1996 [17]. It has considerable robustness against signal pollution (BER of 10−4 at the SNR of 16 dB) but low security. 2) correlation delay shift keying (CDSK) was proposed in 2002 [18]. Compared to DCSK, it increases the security at the cost of BER performance degradation (demodulated BER of 10−3 at the SNR of 18 dB). In subsequent researches, various improved variants of DCSK and CDSK are designed aim to solve one or two drawbacks of them: bad BER performance, doubtful security, low spectral efficiency, and low transmission rate. For example, based on reference modulated (RM)-DCSK, BER of 10−4 can be obtained at the SNR of 16dB [19]. Using DCSK scheme in [20], BER of 10−4 can be obtained at the SNR of 14dB. Frequency Modulated (FM)-DCSK and generalized (G)-CDSK effectively increase the security at the cost of BER performance degradation [21]. Recently, a new DCSK modulation scheme based on reverse time chaos is proposed in [22], BER of 10−5 can be obtained at the SNR of 15dB. In these modulation/demodulation strategies, chaotic synchronization is no longer a vital process, which means that the initial conditions and parameters sensitivity of chaotic system is no longer necessarily considered in demodulation process. As such, chaos communication based on non-coherent detection can increase the BER performance at the cost of security degradation [21].

For coherent chaos secure communication, there are mainly three types of methods, namely chaos masking (CM), chaos-shift-keying (CSK), and chaotic parameters modulation (CPM). Since a CM system is easy to implement, it is widely used for physical layer optical secure communication [9]. The main idea of CM is to superimpose the information signal on the chaotic carrier. Because power change in the resultant signal will lead to weak security, the power of the information signal must be much less than that of the chaotic carrier. Hence, CM is relatively sensitive to channel noise. CPM has higher degree of security since that information signal is injected into the chaotic system to change its trajectories unpredictably. However, when design a CPM system, a specific adaptive controller with a certain mathematical form needs to be designed for a given chaotic system, which weakens the practicability of this type of method in some cases. In concept, CSK essentially belongs to the class of coherent spread spectrum modulation. Although it is vulnerable to return map regression [23,24], there still exist many possibilities and great significances in improving it for transmission of confidential information in non-ideal channels.

In this paper, a coherent CSK communication scheme is proposed based on a hybrid chaotic system. To design the hybrid chaotic system, a three-dimensional (3D) electrical chaotic system is chosen as the seed system, and a nonlinear optical time delayed feedback loop is constructed by using an electro-optical Mach-Zehnder modulator (MZM). Simulation results show that the system has good robustness in terms of handling additive channel noise and high level of security when considering the return map attack.

2. Principle and system setup

The schematic of the electro-optical hybrid chaotic system is illustrated in Fig. 1. The system is composed of an electrical seed system and an electro-optical time delayed feedback loop. In our current setup, three-dimensional Lorenz system [25] is employed as the seed system, as shown in the dashed box, and the mathematical model is written as

$$\left\{ \begin{array}{l} \dot{x} = a(y - x)\\ \dot{y} = cx - y - xz\\ \dot{z} = xy - bz \end{array} \right.,$$
where a, b and c are system parameters.

 figure: Fig. 1.

Fig. 1. Schematic of the hybrid electro-optical chaotic system

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The electrical output x of the seed system is sent into the nonlinear feedback loop, which is composed of a radio frequency amplifier (AMP), a laser diode (LD), a Mach-Zehnder modulator (MZM) and a photodetector (PD). The electric input of MZM is amplified by the AMP which means that the amplitude of the input can span several Vpi (half-wave voltage) of the modulator’s transfer function. As a result, MZM is working in nonlinear region. For simplicity, the normalized relationship between input signal x(t) and output of PD x1(t) can be denote by ${x_1}(t )= co{s^2}({Gx(t )+ \varphi } )$ where G is defined as the nonlinear coefficient of the feedback loop, φ is the offset phase. The output x1(t) is then fed back into the seed system.

Consider the time delay induced by the whole feedback loop, the overall mathematical model of the hybrid system can be expressed as

$$\left\{ \begin{array}{l} \dot{x} = a(y - x)\\ \dot{y} = cx - y - xz\\ \dot{z} = xy - bz + p{\cos^2}(Gx(t - \tau ) + \varphi ) \end{array} \right.,$$
where p is defined as the coupling coefficient.

Then we design a coherent CSK method based on system (2). The system structure is illustrated in Fig. 2. At the emitter side, the binary message m(t) is embedded in the chaotic carrier by changing the loop delay through a switch controller. Two delay values ${\tau _1}$ and ${\tau _2}$ are used to encode ‘0’ bit and ‘1’ bit of the binary message, respectively. Then the mathematical model of the transmitter can be written as

$$\left\{ \begin{array}{l} \dot{x} = a(y - x)\\ \dot{y} = cx - y - xz\\ \dot{z} = xy - bz + p{\cos^2}(Gx(t - (1 - m(t)){\tau_1} - m(t){\tau_\textrm{2}}) + \varphi ) \end{array} \right.,$$
where m(t)∈{0,1}. Chaotic signal x(t) is send out as the carrier.

 figure: Fig. 2.

Fig. 2. Architecture of CSK system based on the proposed hybrid chaotic system

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After transmitted in the noisy channel, the captured signal at the receiver side is written as

$$s(t) = x(t) + n(t),$$
where n(t) is the additive Gaussian noise. The received signal s(t) is split into two portions, one is directly sent into the electrical part, and the other is sent into a MZM, which is working under the same condition with that in the transmitter. After a PD, the output s1(t) can be denote as ${s_1}(t )= co{s^2}({Gs(t )+ \varphi } )$, which is also sent into the electrical part with a fixed time delay $\tau ^{\prime}$. Then, the mathematical model of the receiver system is designed as
$$\left\{ \begin{array}{l} {{\dot{x}}_r} = a({y_r} - {x_r})\\ {{\dot{y}}_r} = cs - {y_r} - s{z_r}\\ {{\dot{z}}_r} = s{y_r} - b{z_r} + p{\cos^2}(Gs(t - \tau^{\prime}) + \varphi ) \end{array} \right..$$

Note that in Eq. (5), the system parameters are mainly identical to those in the transmitter except the time delay. The value of $\tau ^{\prime}$ will influence the synchronization state between the received signal s(t) and response signal xr(t). When $\tau ^{\prime} = \tau $, the intended synchronization between system (5) and system (2) can be reached. Otherwise the synchronization will be destroyed. We consider Eq. (3) and Eq. (5), and set $\tau ^{\prime} = {\tau _1}$. Then, the synchronization can be established for m(t) = 0, which means the error signal e(t)=xr(t)-s(t) will be very small. While for m(t) = 1, the synchronization error will be very large due to the unmatched delay ${\tau _2}$. Then the demodulation can be reliably realized by low pass filtering e(t), and followed by threshold decision.

Theoretically, the electrical part (dashed boxes in Fig. 2) of the system can be established by an analog electric circuit or implemented in a digital signal processing device with analog digital (AD) and digital analog (DA) interfaces. For digital way, its robustness shows a better potential in noisy scenario. The main restrictions lie in the following two aspects. One is that a digitalized chaos system may be suffered by the dynamical degradation phenomenon, which could significantly affect the security performance of the communication. In our scheme, with the participation of analog optical feedback, the degradation can be improved by the hybrid structure [26]. The other issue is that digital computation may limit the bandwidth of the generated chaotic signal and low the data rate of communication. This could be solved by parallel computing strategy in a certain degree. For instance, we can construct multiple digital seed systems in a field-programmable gate array (FPGA), and share one high speed optical nonlinear feedback loop using time division multiplex strategy.

3. Main results

First, we demonstrate the dynamical characteristic of the hybrid chaotic system. According to Eq. (2), system parameters are set as a = 10, b = 8/3, c = 20. In the numerical simulation, the peak-to-peak value of x(t) is about 32, the nonlinear coefficient is appropriately given as G = 0.2. As a result, the peak-to-peak value of Gx(t) is about 6.4, thus can span about 4Vpi of the MZM, which is feasible for the MZM in real world [27]. Other parameters are set as offset phase φ=0, and time delay τ=40ns. System (2) is solved numerically using the fourth order Runge-Kutta algorithm with a step size h = 0.1. When the initial conditions are (x, y, z)=(1, 2, -5), x(t) = 0 for $t \in [{ - \tau ,{\; }0} )$, the three largest Lyapunov exponents ${{\lambda }_{1,2,3}}$ of all the spectrum versus the coupling coefficient p is shown in Fig. 3(a). The system can enter a permanent chaotic zone for $p \in [{ - 70, - 20} ]$, which is visualized in the bifurcation diagram as shown in Fig. 3(b).

 figure: Fig. 3.

Fig. 3. (a) The largest lyapunov exponents ${{\lambda }_{1,2,3}}$ versus the coupling coefficient p, (b) Bifurcation diagram showing local maxima of the coordinate x(t) for p∈[-70, 55].

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The three largest Lyapunov exponents ${{\lambda }_{1,2,3}}$ of all the spectrum versus the time delay $\tau $ is shown in Fig. 4(a), which reveals that, in the region $\tau \in [{0.2,{\; \; }100} ]ns$, there is a permanent chaos range. This result observed in Fig. 4(a) is visualized in the bifurcation diagram as shown in Fig. 4(b). The system attractor in phase planes (xy), (xz) and (yz) are shown in Figs. 4(c), 4(d) and 4(e) for p=-50 and $\tau = 40ns$, respectively. This system is similar to the Lorenz system, but the dynamical property in the phase space is more complicated.

 figure: Fig. 4.

Fig. 4. (a) The largest lyapunov exponents ${{\lambda }_{1,2,3}}$ versus the time delay, (b) Bifurcation diagram showing local maxima of the coordinate x(t) for $\tau \in [{0.2,{\; \; }100} ]ns$. The system attractor in phase planes xy (c), xz (d) and yz (e) for p=-50 and $\tau = 40ns$.

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Based on the chaotic system setup, we demonstrate the modulation and demodulation performance of the CSK transmission according to Eqs. (3)–(5). It is assumed that the electrical parts are implemented in digital signal processing devices, and the clock frequency of iteration is assumed to be 500MHz. The two delay values are set as ${\tau _1} = 20ns$ and ${\tau _2} = 40ns$. The data rate is set as 25kbps, and the signal-to-noise ratio (SNR) for the transmission channel is set as 15dB. Figure 5 shows the performance of the CSK strategy. The waveform of message m(t) is plotted in Fig. 5(a). After the encryption strategy, the message signal is significantly distorted, as shown in Fig. 5(b). Since the system parameters in the emitter and receiver are well matched, the message can be reliably recovered, as illustrated in Fig. 5(c). Moreover, the corresponding spectrum of the encrypted signal s(t) is shown in Fig. 5(d). No clear signature is observed in the encrypted signal spectrum. The narrow band digital signal (25kbps) is spread to the wideband analog chaotic signal (several MHz), which forms a generalized spread-spectrum manner. For a legitimate receiver, the eye-diagram is fully distinct and open after low pass filtering, as shown in Fig. 5(e). Error free demodulation can be established. As for the intruder, the eye-diagram is shown in Fig. 5(f), and the BER is calculated to be 0.5152.

 figure: Fig. 5.

Fig. 5. Message encoding/decoding, SNR = 15 dB. (a) An example of a 25kbps digital message, (b) Transmitted signal s(t) and (d) corresponding spectrum, (c) recovered message (dashed line shows the digital message signal m(t)), (e) eye diagram of the decrypted signal at the output of low pass filter, (f) eye diagram of the encrypted signal s(t).

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4. BER analysis

In this Section, the BER performance of the proposed CSK communication system is analysed in detail. When system (2) enters the permanent chaotic regime, the BER versus the coupling coefficient is shown in Fig. 6, and the SNR is set as 25dB. The figure clearly shows that the BER performance has a positive correlation with the increase of coupling strength.

 figure: Fig. 6.

Fig. 6. BER versus coupling coefficient p

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Moreover, when the coupling coefficient is set as p=-50, the BER performance under different SNR is studied. Figure 7 shows the BER performance under different circumstances. Error free transmission can be maintained when SNR > 12dB for data transmission rate up to 50kbps. When the data rate is set as 25kbps, the BER is 6.0006×10−4 at 10dB SNR. The result indicates that the demodulation performance can still accord with the requirements of communication standards even under harsh channel environment.

 figure: Fig. 7.

Fig. 7. BER versus SNR for different data rate

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5. Return map attack

The return map attack, which was proposed in 1995 [28], is a major challenge for traditional CSK communication systems. In traditional chaos shift keying communication system, the message signals are used to modulate (change) the parameters of chaotic transmitters such that the transmitters work in different chaotic attractors. Since the local maximum and local minimum can reveal the amplitude information of attractors, any change in the sizes of the attractors causes shifts or some structural deformations in return map [29].

As a contrast, we demonstrate the return map analysis of a conventional CSK communication. By modulating the system parameter b of Lorenz system (1) with binary message m(t), a simple CSK can be established:

$$\left\{ \begin{array}{l} b = {b_1},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} m(t) = 0\\ b = {b_2},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} m(t) = 1 \end{array} \right..$$
Assuming that ${Q_n}$ and ${P_n}$ are the n-th maxima and n-th minima of the transmitted signal x(t), two variables are defined as following:
$$\left\{ \begin{array}{l} {A_n} = ({Q_n} + {P_n})/2\\ {B_n} = {Q_n} - {P_n} \end{array} \right..$$

When parameters a = 10, c = 28, b1 = 8/3 and b2 = 2.2, we can construct a return map, (An vs Bn), which is plotted in Fig. 8(a). Note that there are three segments in the return map, and each segment is further split into two strips. It is obvious that the split of the map is caused by the switching of the value of b between b­1 and b2. On the contrary, the proposed system (2) does not suffer from this deficiency. The return map (An vs Bn) for ${\tau _1}$ and ${\tau _2}$ is illustrated in Fig. 8(b). The attacker cannot find two distinct trajectories for the two binary states of the digital message signal m(t), instead a diffused pattern is obtained.

 figure: Fig. 8.

Fig. 8. (a) The return map of system (1) for parameters a = 10, c = 28, b1 = 8/3 and b2 = 2.2, b is modulated by digital message signal m(t); (b) The return map of the proposed system (2) for parameters a = 10, b = 8/3, c = 20 and ${\tau _1} = 20ns$, , time delay $\tau $ is modulated by m(t) digital message signal.

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6. Time-delay signature

TDS issue is an important concern in time delayed feedback systems. Therefore, we study the performance of TDS suppression of system (2) using autocorrelation function (ACF) and delay mutual information DMI, which are widely used methods with considerable robustness. For a time series $\omega (t )$, the ACF C(s) is defined as

$$C(s) = \frac{{\left\langle {[\omega (t) - \left\langle {\omega (t)} \right\rangle ][\omega (t - s) - \left\langle {\omega (t)} \right\rangle ]} \right\rangle }}{{{{[\left\langle {\omega (t) - \left\langle {\omega (t)} \right\rangle } \right\rangle ]}^2}}}.$$
where <·> stands for time average. The DMI D(s) is defined as
$$D(s) = \sum\limits_{\omega (t),{\kern 1pt} \omega (t - s)} {P(\omega (t),{\kern 1pt} {\kern 1pt} {\kern 1pt} \omega (t - s)){\textrm{ln}}} \frac{{P(\omega (t),{\kern 1pt} {\kern 1pt} {\kern 1pt} \omega (t - s))}}{{P(\omega (t)){\kern 1pt} {\kern 1pt} {\kern 1pt} P(\omega (t - s))}}.$$
where $P({\omega (t )} )$ is the probability distribution function of $\omega (t )$, and $P({\omega (t ),\omega ({t - s} )} )$ is the joint probability distribution function. Simulation results are shown in Fig. 9, there are no obvious extremums at s = 80ns in both ACF and DMI curves for time delay $\tau = 80ns$. This means that the TDS of system (2) can be concealed.

 figure: Fig. 9.

Fig. 9. (a) Autocorrelation function C(s) and (b) delay mutual information D(s) of chaotic output signal x(t) of system (2), a = 10, b = 8/3, c = 20, p=-50, G = 0.2, $\varphi = 0$ and $\tau = 40ns$.

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In what follows, the background QACF and QDMI in the C(s) and D(s) are defined as

$$\begin{array}{cc} \begin{array}{l} {Q_{ACF}} = [{{\underline{P} }_{ACF}},{{\bar{P}}_{ACF}}]\\ {{\underline{P} }_{ACF}} = mean\{ C(s)\} - \sigma (C(s))\\ {{\bar{P}}_{ACF}} = mean\{ C(s)\} + \sigma (C(s))\\ {Q_{DMI}} = [{{\underline{P} }_{DMI}},{{\bar{P}}_{DMI}}]\\ {{\underline{P} }_{DMI}} = \frac{{mean\{ D(s)\} - \sigma (D(s))}}{{mean\{ D(s)\} - \sigma (D(s))}}\\ {{\bar{P}}_{DMI}} = \frac{{mean\{ D(s)\} + \sigma (D(s))}}{{mean\{ D(s)\} - \sigma (D(s))}} \end{array}&{,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} for{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} s \in [0,{\kern 1pt} {\kern 1pt} {\kern 1pt} 100]} \end{array}ns,$$
and the peaks size at the time delays are defined as
$$\begin{array}{l} {P_{ACF}} = {C_{s = \tau }}(s)\\ {P_{DMI}} = \frac{{{D_{s = \tau }}(s)}}{{mean\{ D(s)\} - \sigma (D(s))}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} . \end{array}$$
where ${\sigma }$ is the standard deviation.

Figure 10(a) shows the peak size in the C(s) at the time delay $\tau = 80ns$ as a function of the coupling coefficient p. In this simulation, the coupling coefficient p is varying from -30 to -60. If the peak size is in the range of the background (the area between the green lines in Fig. 10), the time delay information can be concealed. For $\textrm{p} \in [{ - 60,{\; \; } - 30} ]$, there is no peak can be distinguished from the background in C(s). Similarly, Fig. 10(b) shows the peak size found in the D(s) at the time delay $\tau = 80ns$ as a function of the coupling coefficient. There is no peak in D(s). It can be speculated that the delayed correlation of the feedback loop is concealed in the multidimensional nonlinear dynamics.

 figure: Fig. 10.

Fig. 10. (a) Value of the peak in ACF and (b) Value of the peak in DMI at $\tau = 80ns$ for increasing the coupling coefficient p.

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7. Sensitivity of parameter mismatch

In real world communication system under harsh environment, a mismatch of the parameters is highly probable. It may lead to the degradation of decryption performance. From the security point of view, the sensitivity to the keys is also a great concern. Both the robustness and the sensitivity to key parameters is discussed when data rate is set as 200kbps.

Here we consider the system parameters a, b, c, and two analog parameters in the nonlinear feedback loop, the time delay $\tau $ and the nonlinear coefficient G. The parameter sensitivity is evaluated by calculating BER of the decrypted data when certain mismatch is intentionally introduced. In Fig. 11(a), for parameters a, b, and c (blue lines), even if the mismatch grows to 50%, usable BER can still be obtained (BER$< $ 1×10−3). And the red curve shows the BER of the decrypted data respect to mismatch of nonlinear coefficient G. When the mismatch is under 4%, no error code occurs. When the mismatch is under 16%, BER is less than 1×10−3. Even if detuning of G grows to 28%, usable BER can still be obtained (BER = 2×10−3). These results indicate that the CSK method is very robust to system parameters a, b, c, and nonlinear coefficient G detuning. As for the time-delay, the BER versus time delay mismatch is depicted in Fig. 11(b). When the mismatch is under 2ns, no error code occurs. A mismatch of 12ns will cause the BER performance degrade seriously. For conventional optical chaotic systems, the frequency spectrum can span to several GHz or even tens of GHz, thus the time delay sensitivity is usually at ps level. In our scheme, the bandwidth of the chaotic signal is limited by the digital signal processing (DSP) device, therefore the sensitivity is also limited. However, the adjustable range of time delay could also be very large in a DSP device, and the key space could be maintained. Meanwhile, the time-delay can also be matched accurately by adjusting the length of optical path. Therefore, the CSK scheme is both secure and feasible in real-world applications.

 figure: Fig. 11.

Fig. 11. (a) Influence of system parameters a, b, c, and nonlinear coefficient G mismatch on the BER, (b) BER variation of the decrypted signal with time delay mismatch.

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8. Conclusions

In order to design a CSK communication system under the condition of low signal-to-noise ratio (SNR), we construct a new hybrid electro-optical time delay system by introducing a sine nonlinear feedback into a three-dimensional seed electrical chaotic system. Simulation results show that the proposed system does not suffer from return map attack and TDS extraction. Moreover, the proposed scheme exhibits promising performance based on BER test. Moreover, both the robustness and the sensitivity to key parameters have been considered and analyzed. Simulation results show the feasibility and the security of the scheme. The proposed scheme has the potential to be used in some harsh communication environments.

Funding

National Key Research and Development Program of China (2018YFB1801304); Key Project of R&D Program of Hubei Province (2017AAA046).

Acknowledgments

The authors are grateful to those who are fighting against COVID-19 on the frontlines, especially in Wuhan. It is because of their selflessness, bravery and perseverance that this paper could be finally finished. We firmly believe that the epidemic will come to an end soon.

Disclosures

The authors declare no conflicts of interest.

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18. M. Sushchik, L. S. Tsimring, and A. R. Volkovskii, “Performance analysis of correlation-based communication schemes utilizing chaos,” IEEE Trans. Circuits Syst. I 47(12), 1684–1691 (2000). [CrossRef]  

19. H. Yang and G. P. Jiang, “High-Efficiency Differential-Chaos-Shift-Keying Scheme for Chaos-Based Noncoherent Communication,” IEEE Transactions on Circuits and Systems II 59(5), 312–316 (2012). [CrossRef]  

20. C. Bai, H. P. Ren, and C. Grebogi, “Experimental Phase Separation Differential Chaos Shift Keying Wireless Communication Based on Matched Filter,” IEEE Access 7, 25274–25287 (2019). [CrossRef]  

21. Kaddoum and Georges, “Wireless Chaos-Based Communication Systems: A Comprehensive Survey,” IEEE Access 4, 2621–2648 (2016). [CrossRef]  

22. C. Bai, H. P. Ren, M. S. Baptista, and C. Grebogi, “Digital underwater communication with chaos,” Communications in Nonlinear Science and Numerical Simulation 73, 14–24 (2019). [CrossRef]  

23. Y. Tao, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998). [CrossRef]  

24. S. J. Li, G. Chen, and G. álvarez, “Return-map cryptanalysis revisited,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16(5), 1557–1568 (2006). [CrossRef]  

25. R. Gao, “A novel track control for Lorenz system with single state feedback,” Chaos, Solitons Fractals 122, 236–244 (2019). [CrossRef]  

26. M. F. Cheng, C. K. Luo, X. X. Jiang, L. Deng, M. M. Zhang, C. J. Ke, S. N. Fu, M. Tang, P. Shum, and D. Liu, “An Electro-Optic Chaotic System Based on a Hybrid Feedback Loop,” J. Lightwave Technol. 36(19), 4259–4266 (2018). [CrossRef]  

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

28. G. Pérez and H. A. Cerdeira, “Extracting messages masked by chaos,” Phys. Rev. Lett. 74(11), 1970–1973 (1995). [CrossRef]  

29. T. Yang, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998). [CrossRef]  

References

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  1. N. Jiang, Y. J. Wang, A. K. Zhao, S. Q. Liu, Y. Q. Zhang, L. Chen, B. C. Li, and K. Qiu, “Simultaneous bandwidth-enhanced and time delay signature-suppressed chaos generation in semiconductor laser subject to feedback from parallel coupling ring resonators,” Opt. Express 28(2), 1999–2009 (2020).
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  2. D. Wang, L. Wang, Y. Guo, Y. Wang, and A. Wang, “Key space enhancement of optical chaos secure communication: chirped FBG feedback semiconductor laser,” Opt. Express 27(3), 3065–3073 (2019).
    [Crossref]
  3. Y. Ma, S. Xiang, X. Guo, Z. Song, A. Wen, and Y. Hao, “Time-delay signature concealment of chaos and ultrafast decision making in mutually coupled semiconductor lasers with a phase-modulated Sagnac loop,” Opt. Express 28(2), 1665–1678 (2020).
    [Crossref]
  4. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phy. Rev. Lett. 64(8), 821–824 (1990).
    [Crossref]
  5. T. Yang and L. O. Chua, “Secure communication via chaotic parameter modulation,” IEEE Trans. Circuits Syst. I 43(9), 817–819 (1996).
    [Crossref]
  6. K. M. Cuomo and A. V. Oppenheim, “Paper 11 – Circuit implementation of synchronized chaos with applications to communications,” Controlling Chaos 71, 153–156 (1996).
  7. N. Jiang, A. K. Zhao, C. P. Xue, J. M. Tang, and K. Qiu, “Physical secure optical communication based on private chaotic spectral phase encryption/decryption,” Opt. Lett. 44(7), 1536–1539 (2019).
    [Crossref]
  8. W. Zhang, C. Zhang, C. Chen, H. Zhang, W. Jin, and K. Qiu, “Hybrid chaotic confusion and diffusion for physical layer security in OFDM-PON,” IEEE Photonics J. 9(2), 1–10 (2017).
    [Crossref]
  9. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
    [Crossref]
  10. J. Ke, L. Yi, Z. Yang, Y. Yang, Q. Zhuge, Y. Chen, and W. Hu, “32 Gb/s chaotic optical communications by deep-learning-based chaos synchronization,” Opt. Lett. 44(23), 5776–5779 (2019).
    [Crossref]
  11. Y. Fu, M. Cheng, X. Jiang, Q. Yu, L. Huang, L. Deng, and D. Liu, “High-speed optical secure communication with an external noise source and an internal time-delayed feedback loop,” Photonics Res. 7(11), 1306–1313 (2019).
    [Crossref]
  12. M. B. Luca, S. Azou, G. Burel, and A. Serbanescu, “On exact Kalman filtering of polynomial systems,” IEEE Trans. Circuits Syst. I 53(6), 1329–1340 (2006).
    [Crossref]
  13. S. Wang, Z. Long, J. Wang, and J. Guo, “A noise eduction method for discrete chaotic signals and its application in communication,” in Proceedings of IEEE International Congress on Image and Signal Processing (IEEE, 2011), pp. 2303–2307.
  14. D. C. Soriano, R. Attux, J. M. T. Romano, M. B. Loiola, and R. Suyama, “Denoising chaotic time series using an evolutionary state estimation approach,” in Proceedings of IEEE Computational Intelligence in Control and Automation (IEEE, 2011), pp. 116–122.
  15. Z. Jako and G. Kis, “Application of noise reduction to chaotic communications: a case study,” IEEE Trans. Circuits Syst. I 47(12), 1720–1725 (2000).
    [Crossref]
  16. Z. Jako and G. Kis, “On the effectiveness of noise reduction methods in DCSK systems,” in Proceedings of IEEE International Symposium on Circuits and Systems (IEEE, 2002), pp. 437–440.
  17. G. Kolumbán, G. K. Vizvári, W. Schwarz, and A. Abel, “Differential chaos shift keying: A robust coding for chaos communication,” in Proceedings of IEEE Workshop on Nonlinear Dynamics of Electronic System (IEEE, 1996), pp. 87–92.
  18. M. Sushchik, L. S. Tsimring, and A. R. Volkovskii, “Performance analysis of correlation-based communication schemes utilizing chaos,” IEEE Trans. Circuits Syst. I 47(12), 1684–1691 (2000).
    [Crossref]
  19. H. Yang and G. P. Jiang, “High-Efficiency Differential-Chaos-Shift-Keying Scheme for Chaos-Based Noncoherent Communication,” IEEE Transactions on Circuits and Systems II 59(5), 312–316 (2012).
    [Crossref]
  20. C. Bai, H. P. Ren, and C. Grebogi, “Experimental Phase Separation Differential Chaos Shift Keying Wireless Communication Based on Matched Filter,” IEEE Access 7, 25274–25287 (2019).
    [Crossref]
  21. Kaddoum and Georges, “Wireless Chaos-Based Communication Systems: A Comprehensive Survey,” IEEE Access 4, 2621–2648 (2016).
    [Crossref]
  22. C. Bai, H. P. Ren, M. S. Baptista, and C. Grebogi, “Digital underwater communication with chaos,” Communications in Nonlinear Science and Numerical Simulation 73, 14–24 (2019).
    [Crossref]
  23. Y. Tao, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998).
    [Crossref]
  24. S. J. Li, G. Chen, and G. álvarez, “Return-map cryptanalysis revisited,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16(5), 1557–1568 (2006).
    [Crossref]
  25. R. Gao, “A novel track control for Lorenz system with single state feedback,” Chaos, Solitons Fractals 122, 236–244 (2019).
    [Crossref]
  26. M. F. Cheng, C. K. Luo, X. X. Jiang, L. Deng, M. M. Zhang, C. J. Ke, S. N. Fu, M. Tang, P. Shum, and D. Liu, “An Electro-Optic Chaotic System Based on a Hybrid Feedback Loop,” J. Lightwave Technol. 36(19), 4259–4266 (2018).
    [Crossref]
  27. J. J. Suárez-Vargas, B. A. Márquez, and J. A. González, “Highly complex optical signal generation using electro-optical systems with non-linear, non-invertible transmission functions,” Appl. Phys. Lett. 101(7), 071115 (2012).
    [Crossref]
  28. G. Pérez and H. A. Cerdeira, “Extracting messages masked by chaos,” Phys. Rev. Lett. 74(11), 1970–1973 (1995).
    [Crossref]
  29. T. Yang, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998).
    [Crossref]

2020 (2)

2019 (7)

D. Wang, L. Wang, Y. Guo, Y. Wang, and A. Wang, “Key space enhancement of optical chaos secure communication: chirped FBG feedback semiconductor laser,” Opt. Express 27(3), 3065–3073 (2019).
[Crossref]

N. Jiang, A. K. Zhao, C. P. Xue, J. M. Tang, and K. Qiu, “Physical secure optical communication based on private chaotic spectral phase encryption/decryption,” Opt. Lett. 44(7), 1536–1539 (2019).
[Crossref]

J. Ke, L. Yi, Z. Yang, Y. Yang, Q. Zhuge, Y. Chen, and W. Hu, “32 Gb/s chaotic optical communications by deep-learning-based chaos synchronization,” Opt. Lett. 44(23), 5776–5779 (2019).
[Crossref]

Y. Fu, M. Cheng, X. Jiang, Q. Yu, L. Huang, L. Deng, and D. Liu, “High-speed optical secure communication with an external noise source and an internal time-delayed feedback loop,” Photonics Res. 7(11), 1306–1313 (2019).
[Crossref]

C. Bai, H. P. Ren, and C. Grebogi, “Experimental Phase Separation Differential Chaos Shift Keying Wireless Communication Based on Matched Filter,” IEEE Access 7, 25274–25287 (2019).
[Crossref]

C. Bai, H. P. Ren, M. S. Baptista, and C. Grebogi, “Digital underwater communication with chaos,” Communications in Nonlinear Science and Numerical Simulation 73, 14–24 (2019).
[Crossref]

R. Gao, “A novel track control for Lorenz system with single state feedback,” Chaos, Solitons Fractals 122, 236–244 (2019).
[Crossref]

2018 (1)

2017 (1)

W. Zhang, C. Zhang, C. Chen, H. Zhang, W. Jin, and K. Qiu, “Hybrid chaotic confusion and diffusion for physical layer security in OFDM-PON,” IEEE Photonics J. 9(2), 1–10 (2017).
[Crossref]

2016 (1)

Kaddoum and Georges, “Wireless Chaos-Based Communication Systems: A Comprehensive Survey,” IEEE Access 4, 2621–2648 (2016).
[Crossref]

2012 (2)

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

H. Yang and G. P. Jiang, “High-Efficiency Differential-Chaos-Shift-Keying Scheme for Chaos-Based Noncoherent Communication,” IEEE Transactions on Circuits and Systems II 59(5), 312–316 (2012).
[Crossref]

2006 (2)

S. J. Li, G. Chen, and G. álvarez, “Return-map cryptanalysis revisited,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16(5), 1557–1568 (2006).
[Crossref]

M. B. Luca, S. Azou, G. Burel, and A. Serbanescu, “On exact Kalman filtering of polynomial systems,” IEEE Trans. Circuits Syst. I 53(6), 1329–1340 (2006).
[Crossref]

2005 (1)

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

2000 (2)

Z. Jako and G. Kis, “Application of noise reduction to chaotic communications: a case study,” IEEE Trans. Circuits Syst. I 47(12), 1720–1725 (2000).
[Crossref]

M. Sushchik, L. S. Tsimring, and A. R. Volkovskii, “Performance analysis of correlation-based communication schemes utilizing chaos,” IEEE Trans. Circuits Syst. I 47(12), 1684–1691 (2000).
[Crossref]

1998 (2)

Y. Tao, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998).
[Crossref]

T. Yang, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998).
[Crossref]

1996 (2)

T. Yang and L. O. Chua, “Secure communication via chaotic parameter modulation,” IEEE Trans. Circuits Syst. I 43(9), 817–819 (1996).
[Crossref]

K. M. Cuomo and A. V. Oppenheim, “Paper 11 – Circuit implementation of synchronized chaos with applications to communications,” Controlling Chaos 71, 153–156 (1996).

1995 (1)

G. Pérez and H. A. Cerdeira, “Extracting messages masked by chaos,” Phys. Rev. Lett. 74(11), 1970–1973 (1995).
[Crossref]

1990 (1)

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phy. Rev. Lett. 64(8), 821–824 (1990).
[Crossref]

Abel, A.

G. Kolumbán, G. K. Vizvári, W. Schwarz, and A. Abel, “Differential chaos shift keying: A robust coding for chaos communication,” in Proceedings of IEEE Workshop on Nonlinear Dynamics of Electronic System (IEEE, 1996), pp. 87–92.

álvarez, G.

S. J. Li, G. Chen, and G. álvarez, “Return-map cryptanalysis revisited,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16(5), 1557–1568 (2006).
[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, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Argyris, A.

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

Attux, R.

D. C. Soriano, R. Attux, J. M. T. Romano, M. B. Loiola, and R. Suyama, “Denoising chaotic time series using an evolutionary state estimation approach,” in Proceedings of IEEE Computational Intelligence in Control and Automation (IEEE, 2011), pp. 116–122.

Azou, S.

M. B. Luca, S. Azou, G. Burel, and A. Serbanescu, “On exact Kalman filtering of polynomial systems,” IEEE Trans. Circuits Syst. I 53(6), 1329–1340 (2006).
[Crossref]

Bai, C.

C. Bai, H. P. Ren, M. S. Baptista, and C. Grebogi, “Digital underwater communication with chaos,” Communications in Nonlinear Science and Numerical Simulation 73, 14–24 (2019).
[Crossref]

C. Bai, H. P. Ren, and C. Grebogi, “Experimental Phase Separation Differential Chaos Shift Keying Wireless Communication Based on Matched Filter,” IEEE Access 7, 25274–25287 (2019).
[Crossref]

Baptista, M. S.

C. Bai, H. P. Ren, M. S. Baptista, and C. Grebogi, “Digital underwater communication with chaos,” Communications in Nonlinear Science and Numerical Simulation 73, 14–24 (2019).
[Crossref]

Burel, G.

M. B. Luca, S. Azou, G. Burel, and A. Serbanescu, “On exact Kalman filtering of polynomial systems,” IEEE Trans. Circuits Syst. I 53(6), 1329–1340 (2006).
[Crossref]

Carroll, T. L.

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phy. Rev. Lett. 64(8), 821–824 (1990).
[Crossref]

Cerdeira, H. A.

G. Pérez and H. A. Cerdeira, “Extracting messages masked by chaos,” Phys. Rev. Lett. 74(11), 1970–1973 (1995).
[Crossref]

Chen, C.

W. Zhang, C. Zhang, C. Chen, H. Zhang, W. Jin, and K. Qiu, “Hybrid chaotic confusion and diffusion for physical layer security in OFDM-PON,” IEEE Photonics J. 9(2), 1–10 (2017).
[Crossref]

Chen, G.

S. J. Li, G. Chen, and G. álvarez, “Return-map cryptanalysis revisited,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16(5), 1557–1568 (2006).
[Crossref]

Chen, L.

Chen, Y.

Cheng, M.

Y. Fu, M. Cheng, X. Jiang, Q. Yu, L. Huang, L. Deng, and D. Liu, “High-speed optical secure communication with an external noise source and an internal time-delayed feedback loop,” Photonics Res. 7(11), 1306–1313 (2019).
[Crossref]

Cheng, M. F.

Chua, L. O.

T. Yang and L. O. Chua, “Secure communication via chaotic parameter modulation,” IEEE Trans. Circuits Syst. I 43(9), 817–819 (1996).
[Crossref]

Colet, P.

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

Cuomo, K. M.

K. M. Cuomo and A. V. Oppenheim, “Paper 11 – Circuit implementation of synchronized chaos with applications to communications,” Controlling Chaos 71, 153–156 (1996).

Deng, L.

Y. Fu, M. Cheng, X. Jiang, Q. Yu, L. Huang, L. Deng, and D. Liu, “High-speed optical secure communication with an external noise source and an internal time-delayed feedback loop,” Photonics Res. 7(11), 1306–1313 (2019).
[Crossref]

M. F. Cheng, C. K. Luo, X. X. Jiang, L. Deng, M. M. Zhang, C. J. Ke, S. N. Fu, M. Tang, P. Shum, and D. Liu, “An Electro-Optic Chaotic System Based on a Hybrid Feedback Loop,” J. Lightwave Technol. 36(19), 4259–4266 (2018).
[Crossref]

Fischer, I.

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

Fu, S. N.

Fu, Y.

Y. Fu, M. Cheng, X. Jiang, Q. Yu, L. Huang, L. Deng, and D. Liu, “High-speed optical secure communication with an external noise source and an internal time-delayed feedback loop,” Photonics Res. 7(11), 1306–1313 (2019).
[Crossref]

Gao, R.

R. Gao, “A novel track control for Lorenz system with single state feedback,” Chaos, Solitons Fractals 122, 236–244 (2019).
[Crossref]

García-Ojalvo, J.

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

Georges,

Kaddoum and Georges, “Wireless Chaos-Based Communication Systems: A Comprehensive Survey,” IEEE Access 4, 2621–2648 (2016).
[Crossref]

González, J. A.

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

Grebogi, C.

C. Bai, H. P. Ren, M. S. Baptista, and C. Grebogi, “Digital underwater communication with chaos,” Communications in Nonlinear Science and Numerical Simulation 73, 14–24 (2019).
[Crossref]

C. Bai, H. P. Ren, and C. Grebogi, “Experimental Phase Separation Differential Chaos Shift Keying Wireless Communication Based on Matched Filter,” IEEE Access 7, 25274–25287 (2019).
[Crossref]

Guo, J.

S. Wang, Z. Long, J. Wang, and J. Guo, “A noise eduction method for discrete chaotic signals and its application in communication,” in Proceedings of IEEE International Congress on Image and Signal Processing (IEEE, 2011), pp. 2303–2307.

Guo, X.

Guo, Y.

Hao, Y.

Hu, W.

Huang, L.

Y. Fu, M. Cheng, X. Jiang, Q. Yu, L. Huang, L. Deng, and D. Liu, “High-speed optical secure communication with an external noise source and an internal time-delayed feedback loop,” Photonics Res. 7(11), 1306–1313 (2019).
[Crossref]

Jako, Z.

Z. Jako and G. Kis, “Application of noise reduction to chaotic communications: a case study,” IEEE Trans. Circuits Syst. I 47(12), 1720–1725 (2000).
[Crossref]

Z. Jako and G. Kis, “On the effectiveness of noise reduction methods in DCSK systems,” in Proceedings of IEEE International Symposium on Circuits and Systems (IEEE, 2002), pp. 437–440.

Jiang, G. P.

H. Yang and G. P. Jiang, “High-Efficiency Differential-Chaos-Shift-Keying Scheme for Chaos-Based Noncoherent Communication,” IEEE Transactions on Circuits and Systems II 59(5), 312–316 (2012).
[Crossref]

Jiang, N.

Jiang, X.

Y. Fu, M. Cheng, X. Jiang, Q. Yu, L. Huang, L. Deng, and D. Liu, “High-speed optical secure communication with an external noise source and an internal time-delayed feedback loop,” Photonics Res. 7(11), 1306–1313 (2019).
[Crossref]

Jiang, X. X.

Jin, W.

W. Zhang, C. Zhang, C. Chen, H. Zhang, W. Jin, and K. Qiu, “Hybrid chaotic confusion and diffusion for physical layer security in OFDM-PON,” IEEE Photonics J. 9(2), 1–10 (2017).
[Crossref]

Kaddoum,

Kaddoum and Georges, “Wireless Chaos-Based Communication Systems: A Comprehensive Survey,” IEEE Access 4, 2621–2648 (2016).
[Crossref]

Ke, C. J.

Ke, J.

Kis, G.

Z. Jako and G. Kis, “Application of noise reduction to chaotic communications: a case study,” IEEE Trans. Circuits Syst. I 47(12), 1720–1725 (2000).
[Crossref]

Z. Jako and G. Kis, “On the effectiveness of noise reduction methods in DCSK systems,” in Proceedings of IEEE International Symposium on Circuits and Systems (IEEE, 2002), pp. 437–440.

Kolumbán, G.

G. Kolumbán, G. K. Vizvári, W. Schwarz, and A. Abel, “Differential chaos shift keying: A robust coding for chaos communication,” in Proceedings of IEEE Workshop on Nonlinear Dynamics of Electronic System (IEEE, 1996), pp. 87–92.

Larger, L.

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

Li, B. C.

Li, S. J.

S. J. Li, G. Chen, and G. álvarez, “Return-map cryptanalysis revisited,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16(5), 1557–1568 (2006).
[Crossref]

Liu, D.

Y. Fu, M. Cheng, X. Jiang, Q. Yu, L. Huang, L. Deng, and D. Liu, “High-speed optical secure communication with an external noise source and an internal time-delayed feedback loop,” Photonics Res. 7(11), 1306–1313 (2019).
[Crossref]

M. F. Cheng, C. K. Luo, X. X. Jiang, L. Deng, M. M. Zhang, C. J. Ke, S. N. Fu, M. Tang, P. Shum, and D. Liu, “An Electro-Optic Chaotic System Based on a Hybrid Feedback Loop,” J. Lightwave Technol. 36(19), 4259–4266 (2018).
[Crossref]

Liu, S. Q.

Loiola, M. B.

D. C. Soriano, R. Attux, J. M. T. Romano, M. B. Loiola, and R. Suyama, “Denoising chaotic time series using an evolutionary state estimation approach,” in Proceedings of IEEE Computational Intelligence in Control and Automation (IEEE, 2011), pp. 116–122.

Long, Z.

S. Wang, Z. Long, J. Wang, and J. Guo, “A noise eduction method for discrete chaotic signals and its application in communication,” in Proceedings of IEEE International Congress on Image and Signal Processing (IEEE, 2011), pp. 2303–2307.

Luca, M. B.

M. B. Luca, S. Azou, G. Burel, and A. Serbanescu, “On exact Kalman filtering of polynomial systems,” IEEE Trans. Circuits Syst. I 53(6), 1329–1340 (2006).
[Crossref]

Luo, C. K.

Ma, Y.

Márquez, B. A.

J. J. Suárez-Vargas, B. A. Márquez, and J. A. González, “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.

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

Oppenheim, A. V.

K. M. Cuomo and A. V. Oppenheim, “Paper 11 – Circuit implementation of synchronized chaos with applications to communications,” Controlling Chaos 71, 153–156 (1996).

Pecora, L. M.

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phy. Rev. Lett. 64(8), 821–824 (1990).
[Crossref]

Pérez, G.

G. Pérez and H. A. Cerdeira, “Extracting messages masked by chaos,” Phys. Rev. Lett. 74(11), 1970–1973 (1995).
[Crossref]

Pesquera, L.

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

Qiu, K.

Ren, H. P.

C. Bai, H. P. Ren, M. S. Baptista, and C. Grebogi, “Digital underwater communication with chaos,” Communications in Nonlinear Science and Numerical Simulation 73, 14–24 (2019).
[Crossref]

C. Bai, H. P. Ren, and C. Grebogi, “Experimental Phase Separation Differential Chaos Shift Keying Wireless Communication Based on Matched Filter,” IEEE Access 7, 25274–25287 (2019).
[Crossref]

Romano, J. M. T.

D. C. Soriano, R. Attux, J. M. T. Romano, M. B. Loiola, and R. Suyama, “Denoising chaotic time series using an evolutionary state estimation approach,” in Proceedings of IEEE Computational Intelligence in Control and Automation (IEEE, 2011), pp. 116–122.

Schwarz, W.

G. Kolumbán, G. K. Vizvári, W. Schwarz, and A. Abel, “Differential chaos shift keying: A robust coding for chaos communication,” in Proceedings of IEEE Workshop on Nonlinear Dynamics of Electronic System (IEEE, 1996), pp. 87–92.

Serbanescu, A.

M. B. Luca, S. Azou, G. Burel, and A. Serbanescu, “On exact Kalman filtering of polynomial systems,” IEEE Trans. Circuits Syst. I 53(6), 1329–1340 (2006).
[Crossref]

Shore, K. A.

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

Shum, P.

Song, Z.

Soriano, D. C.

D. C. Soriano, R. Attux, J. M. T. Romano, M. B. Loiola, and R. Suyama, “Denoising chaotic time series using an evolutionary state estimation approach,” in Proceedings of IEEE Computational Intelligence in Control and Automation (IEEE, 2011), pp. 116–122.

Suárez-Vargas, J. J.

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

Sushchik, M.

M. Sushchik, L. S. Tsimring, and A. R. Volkovskii, “Performance analysis of correlation-based communication schemes utilizing chaos,” IEEE Trans. Circuits Syst. I 47(12), 1684–1691 (2000).
[Crossref]

Suyama, R.

D. C. Soriano, R. Attux, J. M. T. Romano, M. B. Loiola, and R. Suyama, “Denoising chaotic time series using an evolutionary state estimation approach,” in Proceedings of IEEE Computational Intelligence in Control and Automation (IEEE, 2011), pp. 116–122.

Syvridis, D.

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

Tang, J. M.

Tang, M.

Tao, Y.

Y. Tao, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998).
[Crossref]

Tsimring, L. S.

M. Sushchik, L. S. Tsimring, and A. R. Volkovskii, “Performance analysis of correlation-based communication schemes utilizing chaos,” IEEE Trans. Circuits Syst. I 47(12), 1684–1691 (2000).
[Crossref]

Vizvári, G. K.

G. Kolumbán, G. K. Vizvári, W. Schwarz, and A. Abel, “Differential chaos shift keying: A robust coding for chaos communication,” in Proceedings of IEEE Workshop on Nonlinear Dynamics of Electronic System (IEEE, 1996), pp. 87–92.

Volkovskii, A. R.

M. Sushchik, L. S. Tsimring, and A. R. Volkovskii, “Performance analysis of correlation-based communication schemes utilizing chaos,” IEEE Trans. Circuits Syst. I 47(12), 1684–1691 (2000).
[Crossref]

Wang, A.

Wang, D.

Wang, J.

S. Wang, Z. Long, J. Wang, and J. Guo, “A noise eduction method for discrete chaotic signals and its application in communication,” in Proceedings of IEEE International Congress on Image and Signal Processing (IEEE, 2011), pp. 2303–2307.

Wang, L.

Wang, S.

S. Wang, Z. Long, J. Wang, and J. Guo, “A noise eduction method for discrete chaotic signals and its application in communication,” in Proceedings of IEEE International Congress on Image and Signal Processing (IEEE, 2011), pp. 2303–2307.

Wang, Y.

Wang, Y. J.

Wen, A.

Xiang, S.

Xue, C. P.

Yang, C. M.

Y. Tao, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998).
[Crossref]

T. Yang, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998).
[Crossref]

Yang, H.

H. Yang and G. P. Jiang, “High-Efficiency Differential-Chaos-Shift-Keying Scheme for Chaos-Based Noncoherent Communication,” IEEE Transactions on Circuits and Systems II 59(5), 312–316 (2012).
[Crossref]

Yang, L. B.

Y. Tao, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998).
[Crossref]

T. Yang, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998).
[Crossref]

Yang, T.

T. Yang, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998).
[Crossref]

T. Yang and L. O. Chua, “Secure communication via chaotic parameter modulation,” IEEE Trans. Circuits Syst. I 43(9), 817–819 (1996).
[Crossref]

Yang, Y.

Yang, Z.

Yi, L.

Yu, Q.

Y. Fu, M. Cheng, X. Jiang, Q. Yu, L. Huang, L. Deng, and D. Liu, “High-speed optical secure communication with an external noise source and an internal time-delayed feedback loop,” Photonics Res. 7(11), 1306–1313 (2019).
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Zhang, C.

W. Zhang, C. Zhang, C. Chen, H. Zhang, W. Jin, and K. Qiu, “Hybrid chaotic confusion and diffusion for physical layer security in OFDM-PON,” IEEE Photonics J. 9(2), 1–10 (2017).
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Zhang, H.

W. Zhang, C. Zhang, C. Chen, H. Zhang, W. Jin, and K. Qiu, “Hybrid chaotic confusion and diffusion for physical layer security in OFDM-PON,” IEEE Photonics J. 9(2), 1–10 (2017).
[Crossref]

Zhang, M. M.

Zhang, W.

W. Zhang, C. Zhang, C. Chen, H. Zhang, W. Jin, and K. Qiu, “Hybrid chaotic confusion and diffusion for physical layer security in OFDM-PON,” IEEE Photonics J. 9(2), 1–10 (2017).
[Crossref]

Zhang, Y. Q.

Zhao, A. K.

Zhuge, Q.

Appl. Phys. Lett. (1)

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

Chaos, Solitons Fractals (1)

R. Gao, “A novel track control for Lorenz system with single state feedback,” Chaos, Solitons Fractals 122, 236–244 (2019).
[Crossref]

Communications in Nonlinear Science and Numerical Simulation (1)

C. Bai, H. P. Ren, M. S. Baptista, and C. Grebogi, “Digital underwater communication with chaos,” Communications in Nonlinear Science and Numerical Simulation 73, 14–24 (2019).
[Crossref]

Controlling Chaos (1)

K. M. Cuomo and A. V. Oppenheim, “Paper 11 – Circuit implementation of synchronized chaos with applications to communications,” Controlling Chaos 71, 153–156 (1996).

IEEE Access (2)

C. Bai, H. P. Ren, and C. Grebogi, “Experimental Phase Separation Differential Chaos Shift Keying Wireless Communication Based on Matched Filter,” IEEE Access 7, 25274–25287 (2019).
[Crossref]

Kaddoum and Georges, “Wireless Chaos-Based Communication Systems: A Comprehensive Survey,” IEEE Access 4, 2621–2648 (2016).
[Crossref]

IEEE Photonics J. (1)

W. Zhang, C. Zhang, C. Chen, H. Zhang, W. Jin, and K. Qiu, “Hybrid chaotic confusion and diffusion for physical layer security in OFDM-PON,” IEEE Photonics J. 9(2), 1–10 (2017).
[Crossref]

IEEE Trans. Circuits Syst. I (4)

M. B. Luca, S. Azou, G. Burel, and A. Serbanescu, “On exact Kalman filtering of polynomial systems,” IEEE Trans. Circuits Syst. I 53(6), 1329–1340 (2006).
[Crossref]

Z. Jako and G. Kis, “Application of noise reduction to chaotic communications: a case study,” IEEE Trans. Circuits Syst. I 47(12), 1720–1725 (2000).
[Crossref]

M. Sushchik, L. S. Tsimring, and A. R. Volkovskii, “Performance analysis of correlation-based communication schemes utilizing chaos,” IEEE Trans. Circuits Syst. I 47(12), 1684–1691 (2000).
[Crossref]

T. Yang and L. O. Chua, “Secure communication via chaotic parameter modulation,” IEEE Trans. Circuits Syst. I 43(9), 817–819 (1996).
[Crossref]

IEEE Transactions on Circuits and Systems II (1)

H. Yang and G. P. Jiang, “High-Efficiency Differential-Chaos-Shift-Keying Scheme for Chaos-Based Noncoherent Communication,” IEEE Transactions on Circuits and Systems II 59(5), 312–316 (2012).
[Crossref]

Int. J. Bifurcation Chaos Appl. Sci. Eng. (1)

S. J. Li, G. Chen, and G. álvarez, “Return-map cryptanalysis revisited,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16(5), 1557–1568 (2006).
[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, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Photonics Res. (1)

Y. Fu, M. Cheng, X. Jiang, Q. Yu, L. Huang, L. Deng, and D. Liu, “High-speed optical secure communication with an external noise source and an internal time-delayed feedback loop,” Photonics Res. 7(11), 1306–1313 (2019).
[Crossref]

Phy. Rev. Lett. (1)

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phy. Rev. Lett. 64(8), 821–824 (1990).
[Crossref]

Phys. Lett. A (2)

Y. Tao, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998).
[Crossref]

T. Yang, L. B. Yang, and C. M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998).
[Crossref]

Phys. Rev. Lett. (1)

G. Pérez and H. A. Cerdeira, “Extracting messages masked by chaos,” Phys. Rev. Lett. 74(11), 1970–1973 (1995).
[Crossref]

Other (4)

S. Wang, Z. Long, J. Wang, and J. Guo, “A noise eduction method for discrete chaotic signals and its application in communication,” in Proceedings of IEEE International Congress on Image and Signal Processing (IEEE, 2011), pp. 2303–2307.

D. C. Soriano, R. Attux, J. M. T. Romano, M. B. Loiola, and R. Suyama, “Denoising chaotic time series using an evolutionary state estimation approach,” in Proceedings of IEEE Computational Intelligence in Control and Automation (IEEE, 2011), pp. 116–122.

Z. Jako and G. Kis, “On the effectiveness of noise reduction methods in DCSK systems,” in Proceedings of IEEE International Symposium on Circuits and Systems (IEEE, 2002), pp. 437–440.

G. Kolumbán, G. K. Vizvári, W. Schwarz, and A. Abel, “Differential chaos shift keying: A robust coding for chaos communication,” in Proceedings of IEEE Workshop on Nonlinear Dynamics of Electronic System (IEEE, 1996), pp. 87–92.

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Figures (11)

Fig. 1.
Fig. 1. Schematic of the hybrid electro-optical chaotic system
Fig. 2.
Fig. 2. Architecture of CSK system based on the proposed hybrid chaotic system
Fig. 3.
Fig. 3. (a) The largest lyapunov exponents ${{\lambda }_{1,2,3}}$ versus the coupling coefficient p, (b) Bifurcation diagram showing local maxima of the coordinate x(t) for p∈[-70, 55].
Fig. 4.
Fig. 4. (a) The largest lyapunov exponents ${{\lambda }_{1,2,3}}$ versus the time delay, (b) Bifurcation diagram showing local maxima of the coordinate x(t) for $\tau \in [{0.2,{\; \; }100} ]ns$ . The system attractor in phase planes xy (c), xz (d) and yz (e) for p=-50 and $\tau = 40ns$ .
Fig. 5.
Fig. 5. Message encoding/decoding, SNR = 15 dB. (a) An example of a 25kbps digital message, (b) Transmitted signal s(t) and (d) corresponding spectrum, (c) recovered message (dashed line shows the digital message signal m(t)), (e) eye diagram of the decrypted signal at the output of low pass filter, (f) eye diagram of the encrypted signal s(t).
Fig. 6.
Fig. 6. BER versus coupling coefficient p
Fig. 7.
Fig. 7. BER versus SNR for different data rate
Fig. 8.
Fig. 8. (a) The return map of system (1) for parameters a = 10, c = 28, b1 = 8/3 and b2 = 2.2, b is modulated by digital message signal m(t); (b) The return map of the proposed system (2) for parameters a = 10, b = 8/3, c = 20 and ${\tau _1} = 20ns$ , , time delay $\tau $ is modulated by m(t) digital message signal.
Fig. 9.
Fig. 9. (a) Autocorrelation function C(s) and (b) delay mutual information D(s) of chaotic output signal x(t) of system (2), a = 10, b = 8/3, c = 20, p=-50, G = 0.2, $\varphi = 0$ and $\tau = 40ns$ .
Fig. 10.
Fig. 10. (a) Value of the peak in ACF and (b) Value of the peak in DMI at $\tau = 80ns$ for increasing the coupling coefficient p.
Fig. 11.
Fig. 11. (a) Influence of system parameters a, b, c, and nonlinear coefficient G mismatch on the BER, (b) BER variation of the decrypted signal with time delay mismatch.

Equations (11)

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{ x ˙ = a ( y x ) y ˙ = c x y x z z ˙ = x y b z ,
{ x ˙ = a ( y x ) y ˙ = c x y x z z ˙ = x y b z + p cos 2 ( G x ( t τ ) + φ ) ,
{ x ˙ = a ( y x ) y ˙ = c x y x z z ˙ = x y b z + p cos 2 ( G x ( t ( 1 m ( t ) ) τ 1 m ( t ) τ 2 ) + φ ) ,
s ( t ) = x ( t ) + n ( t ) ,
{ x ˙ r = a ( y r x r ) y ˙ r = c s y r s z r z ˙ r = s y r b z r + p cos 2 ( G s ( t τ ) + φ ) .
{ b = b 1 , m ( t ) = 0 b = b 2 , m ( t ) = 1 .
{ A n = ( Q n + P n ) / 2 B n = Q n P n .
C ( s ) = [ ω ( t ) ω ( t ) ] [ ω ( t s ) ω ( t ) ] [ ω ( t ) ω ( t ) ] 2 .
D ( s ) = ω ( t ) , ω ( t s ) P ( ω ( t ) , ω ( t s ) ) ln P ( ω ( t ) , ω ( t s ) ) P ( ω ( t ) ) P ( ω ( t s ) ) .
Q A C F = [ P _ A C F , P ¯ A C F ] P _ A C F = m e a n { C ( s ) } σ ( C ( s ) ) P ¯ A C F = m e a n { C ( s ) } + σ ( C ( s ) ) Q D M I = [ P _ D M I , P ¯ D M I ] P _ D M I = m e a n { D ( s ) } σ ( D ( s ) ) m e a n { D ( s ) } σ ( D ( s ) ) P ¯ D M I = m e a n { D ( s ) } + σ ( D ( s ) ) m e a n { D ( s ) } σ ( D ( s ) ) , f o r s [ 0 , 100 ] n s ,
P A C F = C s = τ ( s ) P D M I = D s = τ ( s ) m e a n { D ( s ) } σ ( D ( s ) ) .

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