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Abstract
In this paper, a novel electro-optic chaotic system with enhanced nonlinearity by deep learning (ENDL) is proposed to achieve time-delay signature (TDS) elimination. A long-short term memory network (LSTM) is trained by a specially designed loss function to enhance the nonlinear effect that can hide the TDS of the system. For the first time, the trained deep learning module is put into a single feedback loop to participate in chaos generation. Simulation results show that the ENDL system can eliminate TDS and increase the bandwidth to more than 31GHz when the feedback intensity is very low (α = 4V). Moreover, the complexity of the chaotic output can be improved with permutation entropy (PE) reaching 0.9941. The synchronization result shows that the ENDL system has high sensitivity to TDS but has low sensitivity to the feedback intensity, thus the system has both high security and high robustness. This system has an uncomplicated synchronization structure and high flexibility, and it opens up a new direction for high-quality chaos generation.
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1. Introduction
In recent years, with the development of optical communication, security has become more and more important. Optical chaotic systems have drawn considerable attention [1,2] due to their advantages of high bandwidth, noise-like, and sensitivity to parameters. Optical chaotic sources are mainly divided into two categories, all-optical (AO) and electro-optical (EO) sources. AO sources chiefly introduce disturbances through optical feedback, optical injection, and photoelectric feedback, which causes the laser to exhibit nonlinear behavior [3]. However, AO systems have high requirements on customized devices, and it is difficult to synchronize for AO systems [4,5]. EO chaotic systems use a semiconductor laser as the light source and use external devices such as Mach-Zehnder modulator (MZM) to introduce nonlinear feedback to generate chaos. EO systems are described by the Ikeda equation [4]. This kind of EO system has the advantages of low cost and high synchronization robustness, so it is widely researched [6–9].
There are two focal points in the study of EO chaotic systems: time-delay signature (TDS) hiding and bandwidth enhancement. Since the delayed feedback loop of an EO system can cause periodic correlation of the output waveform, TDS is introduced into the output. TDS can be obtained through copious statistical analysis and deep learning methods [10–13]. TDS is an essential key of the EO chaotic systems. Studies have shown that if the cracker knows the TDS of a chaotic system, the system can be easily reconstructed [14]. Therefore, whether TDS concealment can be achieved is an important evaluation criterion for chaotic security systems. By now, a variety of TDS concealing schemes have been reported [6–9,15–22], such as multi-level chaotic encryption [7,15], multiple feedback coupling [6,8,16–18], DSP processing [9,19,20], nonlinear function [21] and the time-domain fractional Fourier transform for post-processing [22]. Most of the above schemes introduce many additional physical devices and complex physical structures to enhance the nonlinear effect of the feedback loop. This means that these schemes require a large number of matched physical devices during chaotic synchronization, which is hard to apply in practice. As a result, the synchronization structures of the schemes above are complicated, and a large number of physical devices bring high costs, which hinder the promotion of these schemes. In addition, for chaotic secure communication, a waveform with higher bandwidth can carry a higher information rate, so generating broadband chaotic waveforms is also a crucial research point. Up to now, the bandwidth enhancement schemes of the EO system are reported in [23–25]. In [23], photoelectric XOR logic gates and dual delay feedback are used to generate 29.1GHz broadband chaos, and [24] uses photoelectric hybrid feedback to achieve 20GHz broadband chaos synchronization. Meanwhile, [25] uses self-phase modulation and dual optical feedback, obtaining a 25.3GHz chaotic waveform. The above-mentioned bandwidth-enhancing schemes also have a large number of physical devices and complex physical structures. At the same time, they are not flexible.
Until now, with the development of deep learning, there have been many pieces of research on the combination of deep learning and optical chaotic secure communication. In [26,27], deep learning is applied to replace hardware modules to achieve chaotic synchronization, as a result, 32Gbps high-speed chaotic communication is achieved. In addition, a neural network is used to compensate for errors caused by parameter mismatch during chaotic synchronization [28]. Meanwhile, [12,13] use deep learning to crack TDS, and [29–31] predict chaotic systems by deep learning. The above research proves the broad prospects of the application of deep learning to optical chaotic secure communication. However, the above schemes have not combined deep learning into chaotic generation, in which the nonlinear effect of deep learning can be used to produce high-performance chaotic waveforms.
In this paper, a phase-moduled EO chaotic system with enhanced nonlinearity by deep learning (ENDL) is proposed to eliminate TDS and enhance the bandwidth. In the ENDL system, a dedicated loss function is designed to train an LSTM network to provide the expected nonlinear effect. After training, the deep learning module is added into the EO feedback loop for chaos generation. Simulation results show that the ENDL system can generate chaos with TDS hidden, high complexity and broadband when the feedback intensity $\alpha$ is very low ($\alpha$ = 4V). We design the synchronization scheme of the ENDL system, simulation results show that the system is highly sensitive to TDS and robust to $\alpha$. The ENDL system uses only one feedback loop to generate high-quality chaos, involving fewer physical devices, therefore, it is more flexible. This scheme opens up a new direction for high-quality chaos generation.
2. Model of chaos generation
The schematic diagram of the phase-modulated electro-optic (PMEO) chaotic system with enhanced nonlinearity by deep learning (ENDL) is shown in Fig. 1. There are two steps in the proposed schematic diagram: the first step is the training process of the deep learning module, and the second step is the chaos generating process. The components of the system are introduced as follows. In the first step, a conventional electro-optic (CEO) chaos system is composed, which has been studied in depth by [4]. A continuous wave (CW) laser light from LD is modulated by PM, then the output is transmitted through the time-delay fiber and Mach Zehnder interferometer (MZI), then it is detected by a photodiode (PD) and amplified by a radio-frequency amplifier (RF). The output of RF is fed back to PM. After the CEO system is constructed, the output of RF is used as the data set to train the deep learning module, containing an LSTM network. After the training process is complete, the second step starts. The trained deep learning module is added into the feedback loop to participate as a nonlinear transformation. The output of RF is sent to the trained deep learning module, then the output is fed back to the modulation signal terminal of PM. Other part of the chaos generating system is the same as the CEO system in the first step. Both the first step and the second step use the same physical devices. Based on CEO phase chaotic system equation [4]:
Define the nonlinear transformation introduced by the deep learning module be $F(\cdot )$, then
where $V(t)$ is the output of ${V}_{2}({t})$ through the deep learning module. Let $x(t)=\frac {\pi V_{4}(t)}{2 V_{\pi }}$, $\beta =\frac {G A g P \pi }{2 V_{\pi }}$, $\alpha =G A g P$, $\gamma =\frac {\pi }{2 V_{\pi }}$, then Eqs. (5) and (6) can be simplified as:In order to guide the deep learning module to achieve the expected nonlinear transformation, we propose the idea of defining a specific loss function to evaluate the performance of the neural network and guide the neural network to learn and evolve in the defined direction. Based on this idea, a loss function for LSTM training is designed. Because the autocorrelation function (ACF) of the output of the EO phase chaotic system has TDS at the feedback delay $T$, the interference delay ${\delta T}$ and their integer multiples, the key of the system is exposed. In order to hide the TDS of EO system, we take the sum of the absolute values of TDS easily exposed by the CEO system as the loss function of LSTM. For a segment of sequence $n$, the loss function Loss is defined as:
Fig. 1. Schematic diagram of the proposed ENDL system. LD: laser diode, PM: phase modulator, DL: delay line, MZI: Mach-Zehnder interferometer, PD: photodiode, RF: radio-frequency amplifier, LSTM: long-short term memory network, BPF: Band-pass Filter.
During the training of the deep learning module, the chaotic sequence of the CEO system is used as the input of the LSTM in each iteration. Then the loss of the LSTM output sequence is calculated. Subsequently, the loss value is back propagated according to the loss value, and the gradient descent algorithm is used to update the parameters of the network, so as to make the network evolve in the direction of loss reduction (the direction of TDS peak reduction) in the iteration. Finally, the ultimate goal is to enable the network to produce a nonlinear transformation that can eliminate TDS peak of the ENDL system.
3. Numerical results and discussion
We use PyTorch to construct an LSTM model. The fourth-order Runge-Kutta algorithm is used to solve equations Eqs. (7)–(9), where the calculation result of Eq. (9) is the CEO chaotic sequence, which is used as the training data, and the result of Eq. (7) is the output of the ENDL system. The major parameters in the simulation are $T$ = 1 ns, $\delta T$ = 0.4 ns, $\tau$ = 25 ps, $\theta =5 \mu \mathrm {s}$, $\phi _0=\pi / 4$. The simulation step is 10ps, and the simulation length for generating chaotic output is 1000000.
3.1 Deep learning model selection and proof of the validity of the loss function
3.1.1 Comparison of different deep learning model
Since the chaotic sequence is a one-dimensional time series, we attempt to use a fully connected artificial neural network (ANN) and an LSTM as the network in the deep learning module, as shown in Fig. 2. Since the loss function is processed on a sequence, but the ANN has no memory effect, we add two registers to store data for the ANN so that the data is input into it in segments. The ANN in our simulation contains one 10000-dimensional input layer, one 21000-dimensional hidden layer, and one 10000-dimensional output layer, as shown in Fig. 2(${\rm a}$). On the contrary, LSTM is a kind of recurrent neural network, which has memory effect in the time dimension, so no additional registers are required. Data can be input and output one by one during training and using. We use a 4-layer 500-dimensional LSTM, as shown in Fig. 2(${\rm b}$), then a 500-dimensional linear layer and a 1-dimensional output layer are added after the LSTM layer. Both the ANN and LSTM are trained by the data generated from the first step in Fig. 1, using the loss function described in Eq. (10). After training, the deep learning module is put into the feedback loop in the second step of Fig. 1 for chaos generation, then chaotic sequences with a length of 100,000 are obtained.
Fig. 2. Schematic diagram of the network in the deep learning module. (a): the schematic diagram of ANN, (b): the schematic diagram of LSTM.
For the ANN after training, the data is input and output in segments, therefore, a certain correlation between each segment of output is caused, which greatly affects the randomness and security of the final output. Figure 3 shows the autocorrelation function (ACF) of the output sequence from the ANN system and the LSTM system. From Figs. 3(${\rm a}$) and 3(${\rm b}$), it can be seen that the ACF curves of the output from both systems have no TDS at the feedback delay of 1ns, but the ACF from the ANN system has peaks at 100ns and its integer multiples. These peaks are caused by the correlation introduced by every segment of output. On contrary, the LSTM system does not have the above-mentioned peaks because the data is input and output one by one. At the same time, the LSTM system does not need to add additional registers, so the achievability is better. Therefore, the LSTM system is selected as the deep learning model in this paper.
3.1.2 Proof of the validity of the loss function
In the ENDL system, the deep learning module is added in the feedback loop to provide nonlinear transformations that can achieve TDS hiding and bandwidth enhancement. However, the direction of the nonlinear effect of LSTM itself is random. Therefore, we define Eq. (10) as the loss function during the training of LSTM, and the loss function guides the network to evolve in the direction of the effective nonlinear transformation. Since the loss function proposed in this paper is self-designed, it is necessary to prove whether the loss function works expectedly in the LSTM training and whether the trained LSTM plays the expected role in reducing TDS.
The absolute value of the correlation coefficient (CC) between the input and output of the LSTM is used to measure the strength of the nonlinear effect of LSTM. The smaller the $\left | {\rm CC} \right |$, the stronger the nonlinear effect of the LSTM. CC is defined as [9]:
Fig. 4. The signal waveform: (a): The trend of the values of Loss during the training of LSTM, (b): The trend of the values of $\left | {\rm CC} \right |$ during the training of LSTM, (c): The relationship between $\left | {\rm CC} \right |$ and Loss.
Figure 4(${\rm a}$) shows the trend of Loss during the LSTM training. The overall Loss has a declining trend and can quickly converge to below 0.01. It can be seen that the training process based on the loss function Eq. (10) is optimizable, and the network can evolve in the correct direction. The trend of $\left | {\rm CC} \right |$ during the LSTM training is shown in Fig. 4(${\rm b}$). $\left | {\rm CC} \right |$ shows a trend of declining and converging, which is similar to the trend of Loss. The declining of $\left | {\rm CC} \right |$ indicates that the nonlinear effect of LSTM is increasing during the training process. Figure 4(${\rm c}$) shows the relationship between $\left | {\rm CC} \right |$ and Loss. $\left | {\rm CC} \right |$ and Loss are positively correlated, which indicates that the nonlinear effect of trained LSTM makes the Loss decrease.
Fig. 5. Time series of the output generated by (a): trained LSTM and (c): untrained LSTM; ACF curves of the output generated by (b): trained LSTM and (d): untrained LSTM.
Figures 5(a) and 5(c) is the output waveform based on the structure of the second step in Fig. 1 with the trained LSTM and the untrained LSTM, and Figs. 5(b) and 5(d) are the corresponding ACFs. It can be seen that the trained LSTM participates in the feedback to produce a chaotic waveform, and its ACF has no obvious peak at the time of $\delta T$ = 0.4ns and $T$ = 1ns. While the untrained LSTM cannot produce the required chaotic waveform. Therefore, the effect of TDS concealment is caused by the nonlinear effect of the LSTM guided by the loss function. This fact can prove that the enhanced nonlinearity exists and it is generated by the training process with Eq. (10).
3.2 Dynamic complexity analysis
In this section, we compare the complexity of the time series waveforms generated by the ENDL and CEO systems under different feedback strength $\alpha$. We analyze the chaotic time traces and the corresponding RF spectra. Moreover, we quantitatively analyze the complexity through the maximum Lyapunov exponent $\lambda _{\max }$, permutation entropy (PE), and effective bandwidth (BW). Figs. 6 (a$_1$)–6(a$_3$) are the chaotic time waveform comparisons generated by the ENDL and CEO systems under $\alpha$ equaling 4V, 7V, and 8.5V, respectively. In addition, the corresponding RF spectra of the ENDL system and the CEO system are shown in Figs. 6(b$_1$)–6(b$_3$), and Figs. 6(c$_1$)–6(c$_3$), respectively. The RF spectra of the chaotic waveform generated by the ENDL system are flatter than that of the CEO system under the three $\alpha$s, which means that the ENDL system has higher bandwidth and a better carrier performance than the CEO system. We use the effective bandwidth BW as the span between direct current (DC) and the frequency containing 80% of the energy in the RF spectrum, the calculation results are marked in Figs. 6(b$_1$)–6(c$_3$). The bandwidth of the ENDL under three $\alpha$s is greater than 31GHz, while the bandwidth of the CEO system is less than 20GHz. It can be seen that the enhanced nonlinearity introduced by the deep learning module has achieved the effect of increasing the bandwidth of the chaotic carrier.
Fig. 6. Time series (first row), RF spectra of the output of the ENDL system (second row), and RF spectra of the output of the CEO system (third row) when $\alpha$ = 4V (first column), 7V (second column), and 8.5V (third column).
After that, the maximum Lyapunov exponent $\lambda _{\max }$ is used to quantitatively measure whether the chaotic system enters a chaotic state. If the maximum Lyapunov exponent of the system is positive, it is considered to be in a chaotic state [32]. The Lyapunov exponent $\lambda _{\max }$ is defined as:
where $d(t)$ is the average divergence at time $t$, and $C$ is the initial separation vector. For the discrete-time system $xn+1=f(xn)$, the above formula can be changed toFig. 7. The signal waveform: (a): the maximum Lyapunov exponents, (b): PE, (c): the bandwidth of the output from the ENDL and CEO system when the feedback gain $\alpha$ increases.
Permutation entropy (PE) is used to quantitatively measure the time complexity of the system through phase space reconstruction. It is generally considered that the system enters a high-complexity chaotic state when PE is greater than 0.9 [7]. For the sequence $\{u(i), i=1,2, \ldots, k\}$, embed it in the m-dimensional space ${U}_{{i}}=[ {u}({i}), {u}({i}+{L}), \ldots, {u}({i}+( {m}-1) {L})]$, $m$ is the embedding dimension, $L$ is the embedding delay time, and $U_i$ is the ascending order. Then its permutation entropy PE is defined as [33]:
Usually $m$ = 6, $L$ = 2 [8], and $P_{j}$ is the probability distribution of the symbol sequence $j$. We calculate the PE trend of the output sequence from the ENDL and CEO system when the $\alpha$ changes from 0V to 12.9V. As shown in Fig. 7(b), the ENDL system enters a complex chaotic state when $\alpha$=0.6V and the corresponding PE equals 0.9218. The largest PE reaches 0.9941 when $\alpha$ = 1.7V. In comparison, the CEO system has a PE greater than 0.9 when $\alpha$ = 8.1V, and the largest PE reaches 0.9524 when $\alpha$ = 12.8V. It can be seen from Fig. 7 that the rising rate of PE with the increase of $\alpha$ of the ENDL system is significantly faster than that of the CEO system, and PE of the ENDL system is higher than that of the CEO system under the same $\alpha$.
In addition, we plot the trend of effective bandwidth BW of the RF spectra of the ENDL system and CEO system when $\alpha$ alters from 0V to 12.9V. As shown in Fig. 7(c), the bandwidth of the ENDL system increases much faster than the CEO system. Moreover, the bandwidth of the ENDL is always greater than 30GHz when $\alpha$ > 0.9V, and it reaches 36.23GHz when $\alpha$ = 0.9V. It can be seen that the ENDL system has achieved bandwidth enhancement compared with the CEO system.
3.3 Security of chaotic waveforms analysis
Time-delay signature(TDS) is the key of a chaotic security system. Therefore, whether the TDS can be hidden is a key indicator to measure the security of the system. The autocorrelation function (ACF) and delayed mutual information (DMI) curve are usually used to analyze the TDS of a chaotic system. ACF is defined as Eq. (11), and DMI is defined as [8]:
where $p[x(t)]$ and $p[x(t), x(t+s)]$ are the marginal probability distribution and the joint probability distribution, respectively.Figures 8(a)–8(c) is the ACF of the ENDL and CEO output under different $\alpha$, and Figs. 8(d)–8(f) is the DMI. It can be seen from Fig. 8 that the TDS of CEO are all exposed at the time of $\delta T$ = 0.4ns, $T$ = 1ns, and $T+\delta T$ = 1.4ns under three $\alpha$s. On contrary, there is no TDS exposure in ACF or DMI of the ENDL system. Therefore, the ENDL system realizes TDS elimination starting from a very low $\alpha$.
Fig. 8. ACF (first row) and DMI (second row) of the output from the ENDL and CEO system at different $\alpha$s. (a) and (d): $\alpha$ = 4V, (b) and (e): $\alpha$ = 7V, (c) and (f): $\alpha$ = 8.5V.
After that, the influence of feedback intensity $\alpha$ on TDS suppression is analyzed. Define environment variables background$\_$low (BL) and background$\_$high (BH) as follows:
Fig. 9. Value of the peaks of output in (a): ACF and (b): DMI for increasing the feedback gain $\alpha$ at 1 ns and 0.4 ns.
Figure 10 shows the comparison of the TDS trend of the ENDL and CEO system when $\alpha$ alters from 0V to 12.9V. Figures 10(a) and 10(b) are the ACF trends at t = 1 ns and t = 0.4 ns, while Figs. 10(c) and 10(d) are the DMI trends at t = 1 ns and t = 0.4 ns. It can be seen that the TDS values of the ENDL system decline significantly faster than that of the CEO. Moreover, the absolute values of TDS of the ENDL are much lower than that of the CEO when $\alpha$ is low.
Fig. 10. Value of the peaks of output in ACF (first row) and DMI (second row) of the output from the ENDL and CEO system at different TDSs for increasing the feedback gain $\alpha$. (a) and (c): t = 1ns, (b) and (d): t = 0.4 ns.
3.4 Chaos synchronization
Figure 11 shows a schematic diagram of the synchronization structure of the ENDL system. Both transmitter and receiver use the same deep learning module, and the remaining parameters are matched. The chaotic output is drawn from PM$_1$ and sent to the receiver. At the receiving end, the signal is divided into two channels, one is passed into PM$_2$, and the other is passed through the delay line DL$_2$, MZI$_2$, PD$_2$, RF$_2$, and deep learning module, which are all matched with the transmitter, then the signal is inverted as modulation signal of PM$_2$. The equation of the system is as follows:
Based on the above synchronization structure, Fig. 12 shows the influence of parameter $T$ and $\alpha$ mismatch on chaotic synchronization. First, when the mismatch between $T$ and $\alpha$ is 0V, CC = 1, which shows that the ENDL system can be strictly synchronized when the system parameters are perfectly matched. Figure 12(a) shows the effect of $T$ mismatch on chaotic synchronization, CC decreases rapidly with the mismatch of $T$ increases. When $T$ mismatch equals 3ps, CC = 0.9427, which shows that CC is less than 0.95 when $T$ mismatch is greater than 2ps. This proves that the system is very sensitive to the key $T$. Figure 12(b) shows the effect of $\alpha$ mismatch on chaotic synchronization. Within 50% of alpha mismatch, the value of CC is greater than 0.95. This shows that the synchronization of the ENDL system is not sensitive to $\alpha$. Since $\alpha$ is not the key of the ENDL system, the robustness of synchronization is high.
Fig. 12. The cross-correlation coefficients of the transmitter and receiver when (a): T and (b): $\alpha$ are mismatched.
4. Conclusion
In summary, this paper proposes an electro-optic chaotic system based on deep learning that introduces enhanced nonlinear feedback to eliminate TDS and increase the bandwidth. The deep learning module is trained by designing a reasonable loss function to guide its nonlinear evolution direction. For the first time, the trained deep learning module is added to the feedback loop of electro-optic chaos. The deep learning module provides a wealth of nonlinearity in the feedback loop, and a chaotic carrier with stronger security and higher complexity is generated. When the feedback intensity $\alpha$ is very low ($\alpha$ = 4V), chaos with a bandwidth exceeding 31 GHz and TDS hidden can be generated. The synchronization structure of ENDL is designed, and the simulation results show that the system is sensitive to the key $T$ but robust to the non-key $\alpha$. Since this system only involves single-loop feedback and does not have a complex physical structure, it has low requirements for physical device parameter matching and has higher flexibility.
Funding
National Natural Science Foundation of China (61831003, 62021005).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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