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Abstract
We propose a model-free time delay signature (TDS) extraction method for optical chaos systems. The TDS can be identified from time series without prior knowledge of the actual physical processes. In optical chaos secure communication systems, the chaos carrier is usually generated by a laser diode subject to opto-electronic/all-optical time delayed feedback. One of the most important factors to security considerations is the concealment of the TDS. So far, statistical analysis methods such as autocorrelation function (ACF) and delayed mutual information (DMI) are usually used to unveil the TDS. However, the effectiveness of these methods will be reduced when increasing the nonlinearity of chaos systems. Meanwhile, certain TDS concealment strategies have been designed against statistical analysis. In our previous work, convolutional neural network shows its effectiveness on TDS extraction of chaos systems with high loop nonlinearity. However, this method relies on the knowledge of detailed structure of the chaos systems. In this work, we formulate a blind identification method based on long short-term memory neural network (LSTM-NN) model. The method is validated against the two major types of optical chaos systems, i.e. opto-electronic oscillator (OEO) chaos system and laser chaos system based on internal nonlinearity. Moreover, some security enhanced chaotic systems are also studied. The results show that the proposed method has high tolerance to additive noise. Meanwhile, the data amount needed is less than existing methods.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical or electro-optical time-delayed chaos system has unique advantages of high dimension, high dynamical complexity, and especially remarkable broadband spectrum, which offers the potential for data secure transmission at high rates. Since the early demonstrations of optical chaos secure communication (OCSC) [1], OCSC has been widely studied and become one of the most feasible hardware encryption methods [2–7]. In OCSC, without the matched devices or parameters settings, it is hard for an eavesdropper to recover the message embedded in chaotic carriers. At present, there are two major types of optical chaos system. One is built on the internal nonlinearity of laser [8]. Semiconductor lasers subjected to external perturbation, such as all-optical time-delayed feedback and external optical injection, can generate chaos. The other is called opto-electronic oscillator (OEO) chaos system [9,10], where chaos is generated through an external nonlinear opto-electronic time-delayed feedback. Generally, these systems can be characterized by a time delayed differential equation:
where $x(t )$ is the state variable, and $x({t - \tau } )$ is the time delayed variation. The loop delay parameter $\tau \; $(i.e., time delay signature, TDS) is usually chosen as a cryptographic key. Because the delay value can be varied over a wide range by changing the loop length of the system circuit. Meanwhile the delay values at the transmitter and receiver sides can be matched accurately using tunable optical delay lines. As a result, a large time domain key space can be obtained. Therefore, an important factor to security considerations is the concealment of TDS.So far, much research has focused on concealing the TDS [11–15]. However, it will take some time for its security analysis to mature. For a successful recovery of the TDS, one can use different strategies, such as permutation information analysis [16,17], RF spectrum analyzer [18,19], singular values fraction measure [20], optimal transformations [21], autocorrelation function (ACF), delayed mutual information (DMI) [14,15,22], local linear fitting in a low-dimensional subspace [23], extrema statistics [24], and so on. Out of the methods listed, ACF and DMI are most widely used due to their effectiveness and robustness [25]. If there is a certain degree of linear correlation between a chaotic signal $x(t )$ and its time delayed variation $x({t - s} )$ when s equals to the system loop time delay $\tau $, a sharp peak located at $s = \tau $ will exhibit in the ACF and DMI computed under variation of s. But the linear correlation will be reduced or even eliminated when increasing the nonlinearity of the loop transformation, then the peak will decrease or even disappear [26].
Recently, deep learning technique has attracted more and more attention due to its advantages in processing complex nonlinear time series analysis and nonlinear modeling [27,28]. In [29], neural network (NN) was explored for travel time prediction. Prediction of time delay chaos systems via Long Short-Term Memory neural network (LSTM-NN) was studied in [30]. Thusly deep learning is an attractive technology aiming at security analysis of OCSC systems. In [31], a TDS extraction method was proposed based on convolutional neural network (CNN). The method is effective for time-delayed systems with high loop nonlinearity, and the needed minimal data size is less than ACF and DMI. Meanwhile, the scheme shows a good robustness in terms of handling noise. Unfortunately, this kind of analysis method is only effective under the condition that the structure of the OCSC is completely public. This precondition coincides with the “Kerckhoff’s principle” in cryptanalysis. From the viewpoint of implementing complexity, repeatedly offline training of different CNN is essential for different chaos systems, and the training accuracy depends on the approximation degree between offline system and actual system. These factors reduce the applicability of the scheme.
Hence, chaotic communication cryptanalysis against the TDS is still a challenging open problem. In this work, we propose a data-driven TDS extraction method, with which any prior knowledge of the actual physical processes is no longer necessary. A LSTM-NN is employed to carry out blind inverse modeling. From experimental and simulated time series, successful TDS extraction is given to show the effectiveness of our strategy.
2. Methodology
In Eq. (1), there will be an exact functional relationship between the history and future, which can be expressed as:
and the relationship is established by the nonlinear function f and the time delay constant $\tau $. Assume that we can observe a time series $\{{{x_i}} \}$ from system (1), as shown in Fig. 1. We arbitrarily select two subsequences $\{{{x_i}} \}_n^m$ and $\{{{x_i}} \}_{n + s}^{m + s}$. Subsequence $\{{{x_i}} \}_n^m$ is used as the input of a NN, and $\{{{x_i}} \}_{n + s}^{m + s}$ is set as the expected output. Then the NN could be trained to finish a sequence-to-sequence regression, as described by Eq. (3).As a result, an approximation model of $\varphi $ could be obtained. Here ${\varphi _s}$ will approach to $\varphi $ only when $s = \tau $. Subsequently, the approximation function ${\varphi _{s = \tau }}$ will be able to predict any subsequence $\{{{x_i}} \}_{p + s}^{q + s}$ from its historical information $\{{{x_i}} \}_p^q$.
By contrast, if $s \ne \tau $, the fixed function will no longer exist. The relationship between $\{{{x_i}} \}_n^m$ and $\{{{x_i}} \}_{n + s}^{m + s}$ could be considered as random mapping, which will lead to the failure of prediction ability of model ${\varphi _s}$.
Based on these facts, a blind identification strategy for TDS is shown in Fig. 2. Without knowing the actual time delay parameter $\tau $, we assume some values of time delay candidate $s \in \{{{s_1}, \ldots ,{s_N}} \}$. The training process is shown in Fig. 2(a). For $s = {s_j}$ ($j = 1, \ldots ,N$), subsequence $\{{{x_i}} \}_n^m$ is used as the input of LSTM-NN, and $\{{{x_i}} \}_{n + {s_j}}^{m + {\; }{s_j}}$ is set as the expected output. Then the NN is trained to get a model ${\varphi _j}$. In the testing/predicting process as shown in Fig. 2(b), we put another subsequence $\{{{x_i}} \}_p^q$ into the trained NN model ${\varphi _j}$. A predicted sequence noted as $\{{{y_i}} \}_p^q$ will be obtained. The prediction reliability of model ${\varphi _j}$ is evaluated using normalized root mean square error (NRMSE) [30], which is given by Eq. (4).
Based on the strategy, the detailed implementation method is stated as follows. Equation (1) can be approximately written as the following difference equation,
where $\Delta t$ is the unit time step. In order to obtain enough training set from the time series $\{{{x_i}} \}$ with limited length, the training set are formatted asTo finish the sequence-to-sequence regression, we employ the LSTM-NN because its unique advantages in processing one dimensional time series with strong nonlinearity [30,32,33]. MATLAB2018a Neural Network Toolbox is used to carry out modelling and testing/ forecasting work in our study. A LSTM-NN is trained with the following layer specifications. The number of hidden units is set as 200, then a 50 size fully connected layer and a drop layer with a drop probability of 0.5 are created. The solver ‘Adam’ is used to train with mini-batch size of 20. Initial learning rate is set as 0.01. The gradient threshold is set as 1. The number of training iterations (MaxEpochs) is set as $\mathrm{\beth}$ given for different system.
3. TDS extraction from experimental OEO time series
Since the early laboratory demonstrations of OCSC based on OEO was shown in 2005 [1], this kind of chaotic laser has been widely studied and become one of the most feasible hardware chaotic carrier generator. Therefore, as an experimental example, OEO systems considering nonlinear intensity modulation and nonlinear phase modulation are used to generate chaotic time series to demonstrate the effectiveness of LSTM-NN based TDS extraction method.
3.1 OEO system under intensity modulation
The experimental setup of OEO system under intensity modulation is shown in Fig. 3(a). In the system, an output light from a laser diode (LD, TLG-200) is injected into a Mach-Zehnder intensity modulator (IM, EOSPACE AX-OMSS-20-PFA-PFA-LV) which works in nonlinear region. The normalized nonlinear relationship between input and output of IM can be formulated as $co{s^2}(\cdot )$. After an optical delay fiber line (DL) and a variable optical attenuator (VOA), the output light of IM is converted into an electric signal by a photodiode (PD, PD-18G-V). Then, the output of the PD is divided into two parts, where one signal is amplified by an amplifier (AMP, GT-PA-1MHz${\sim} $1 GHz) and sent back into IM, and the other signal is received and displayed on the oscilloscope (Tektronix, DSA 72504D Digital Serial Analyzer, 25 GHz, 100GS/s).
Fig. 3. (a) Schematic diagram of the OEO system, LD, laser diode; MZM, Mach-Zehnder modulator; AMP, broadband radio frequency amplifier; PD, photodiode; DL, delay line; VOA, variable optical attenuator. (b) experimental the time series $\{{{x_i}} \}$. (c) The Normalized Root Mean Square Error (NRMSE) function versus s for experimental time series $\{{{x_i}} \}$.
The loop time delay is set as $\tau = 19.2ns$ (960 points), the generated experimental waveform (50 GS/s) collected from PD is shown in Fig. 3(b). A time series $\{{{x_i}} \}$ of 5160 samples (103.2 ns) is generated for TDS extraction. Varying time delay candidate s in the range $[{0.7,\; 20.2} ]ns$ with a fixed step size $h = 0.5ns$, $E(s )$ is plotted in Fig. 3(c). It is found that when s is increased to 19.2 ns, a sharp peak will emerge from the background.
As a contrast, there is no obvious peak in ACF and DMI calculated using the same time series $\{{{x_i}} \}$, as shown in Figs. 4(a) and 4(b). When the length of time series $\{{{x_i}} \}$ increases to $2 \times {10^4}$, a peak at $s = 19.2ns$ can be clearly identified in ACF and DMI, as shown in Figs. 4(c) and 4(d). The results show that the time series $\{{{x_i}} \}$ of length $N = 5160$ is not enough for ACF and DMI, but it is enough for our method.
Fig. 4. (a) ACF and (b) DMI for $s \in [{0.7,20.2} ]ns$, and a 104.2 ns time series with 5210 data points was used. (c) ACF and (d) DMI for $s \in [{0.7,20.2} ]ns$, and a 400 ns time series with $2 \times {10^4}$ data points was used.
3.2 OEO system under phase modulation
The experimental setup is shown in Fig. 5(a). An output light from a laser diode (TSL-C89-P-FA-M) is modulated by a PM (KG-PM-15-10G). After an optical delay fiber line (DL), a phase-to-intensity conversion is carried out by a fiber Bragg grating (FBG, TDCMB-C000-NCBF01). The output signal of the FBG is divided into two portions after passing through an optical coupler (OC). One signal is converted into an electric signal by PD1 after passing through a VOA, and then amplified by an AMP (Centellax, OA4MVM3). The other signal is captured and displayed by the oscilloscope after PD2.
Fig. 5. (a) The time series of the laser chaos system with optical feedback and (b) $E(s )$ function versus s for TDS extraction of system (2), and a time series with 8660 data points ($86.6ns$) was used.
For time delay $\tau = 73.6ns$ (7360 points), the generated waveform (100 GS/s) is shown in Fig. 5(b). A time series of 8660 samples (86.6 ns) is used for TDS extraction. The parameters in (6)-(9) are given as $L = 200,\; L^{\prime} = 100,\delta = 10,\vartheta = 10,\; {M_1} = 90,{M_2} = 10$, and MaxEpochs is set as $\mathrm{\beth} = 20$. We vary time delay estimation $s \in [{3.6,75.6} ]ns$ to get $E(s ),$ as shown in Fig. 5(c). A sharp peak is obtained at the correct value of time delay when $s = 73.6ns$. As shown in Figs. 6(a) and (b), there is no obvious peak in ACF and DMI. But when the length of time series increases to $N = 1.2 \times {10^4}$, a successful TDS extraction via ACF and DMI is shown in Figs. 6(c) and 6(d).
Fig. 6. (a) ACF and (b) DMI for $s \in [{3.6,75.6} ]ns$, and a time series with 8660 data points (86.6 ns) was used. (c) ACF and (d) DMI for $s \in [{3.6,75.6} ]ns$, and a time series with $1.2 \times {10^4}$ data points ($120ns$) was used.
3.3 OEO system under strong nonlinearity
For OEO system, the nonlinearity can be measured by Lyapunov exponents, Kaplan-Yorke dimension and the Kolmogorov-Sinai entropy. These measures increase linearly with the feedback gain $\beta $ [26]. The effectiveness of mathematical or statistical based TDS extraction methods will be reduced when increasing the feedback gain $\beta $. Therefore, the identification performance of our method under strong nonlinear effect is studied in detail by numerical approach in this section.
OEO generator can be described by the following Ikeda equation [9]:
Table 1. The minimal data size ${M_{data}}$ needed to extract the TDS for different $\beta $
The ACF is defined as [19]:
The DMI is defined as [19]:
4. TDS extraction from simulation time series generated by a laser system
In this section, another major type of optical chaos system modeled by the Lang-Kobayashi equation is studied, including a laser system with single optical feedback, a laser chaos system with unidirectional coupled optical feedback and a chaotic laser system with a phase-modulated Sagnac loop.
4.1 Laser chaos system with single optical feedback
A laser chaos system with optical feedback which can be modeled as [31]:
The parameters are set as ${G_N} = 8.4 \times {10^{ - 13}}{m^3}{s^{ - 1}}$, ${N_0} = 1.4 \times {10^{24}}{m^{ - 3}}$, ${\tau _p} = 1.927 \times {10^{ - 12}}s$, $k = 2.75n{s^{ - 1}}$, $\alpha = 3.0$, $J = 1.098 \times {10^{33}}{m^{ - 3}}{s^{ - 1}}$, $\epsilon = 2.5 \times {10^{ - 23}}$, and ${\tau _0} = 100.8ns$. Equation (13) is carried out by fourth order Runge-Kutta algorithm with a time step of $6.25\; ps$. A time series$\; $of 21398 samples ($133.7ns$) is used for TDS extraction. The parameters in Eqs. (6)-(9) are given as $L = 60,\; L^{\prime} = 20,\delta = 100,\vartheta = 10,\; {M_1} = 49,{M_2} = 29$, and MaxEpochs is set as $\mathrm{\beth} = 25$. Figure 7(a) shows $E(s )$ when $s \in [{40.8,{\; }101.8} ]$, and a minimum is obtained at $s = 100.8ns$. Meanwhile, there is no obvious peak in ACF and DMI, as shown in Figs. 7(b) and 7(c).
4.2 Laser chaos system with unidirectional coupled optical feedback
A laser chaos system with unidirectional coupled optical feedback can be modeled as [34]:
Fig. 7. (a) $E(s )$ function versus s, (b) ACF and (c) DMI for $s \in [{40.8,\; 101.8} ]ns$. A time series with 21398 data points ($133.7ns$) was used.
The parameters are set as $\textrm{g} = 1.5 \times {10^4}$, $\varepsilon = 5.0 \times {10^{ - 7}}$, ${N_{0,d}} = 1.5 \times {10^8}$, ${N_{0,r}} = 1.5 \times {10^8}$, ${\tau _{p,d}} = 2.0 \times {10^{ - 12}}$, ${\tau _{p,r}} = 2.0 \times {10^{ - 12}}$, ${\tau _{s,d}} = 2.0 \times {10^{ - 9}}$, ${\tau _{s,r}} = 2.0 \times {10^{ - 9}}$, ${\tau _{inj}} = 2.0 \times {10^{ - 9}}$, ${\alpha _d} = 5.0$, ${\alpha _r} = 5.0$, ${f_d} = 193550 \times {10^9}$, ${f_r} = 193563.2 \times {10^9}$, ${\kappa _d} = 15.0 \times {10^9}$, ${\kappa _{inj}} = 20.0 \times {10^9}$, ${I_d} = 22.05 \times {10^{ - 3}}$, ${I_r} = 22.05 \times {10^{ - 3}}$, $q = 1.602 \times {10^{ - 19}}$, ${\tau _d} = 30.0 \times {10^{ - 9}}$, ${\tau _r} = 10.0 \times {10^{ - 9}}$. The system is numerically solved by the fourth order Runge-Kutta method with a time step of $10ps$. Usually, the time series of laser intensity ${I^d} = E_d^2$ and ${I^r} = E_r^2$ are available for observation and recording. A time series $\{{{I^r}} \}$ of 3102 samples (31.02 ns) is used for TDS extraction. The parameters in Eqs. (6)-(9) are given as $L = 100,\; L^{\prime} = 10,\delta = 10,\vartheta = 2,\; {M_1} = 199,{M_2} = 1$, and MaxEpochs is set as $\mathrm{\beth} = 20$. Varying $s \in [{0.5,\; 11} ]ns$, Fig. 8(a) shows the $E(s )$, and a minimum is obtained at $s = {\tau _r} = 10ns$. Meanwhile, there is no obvious peak in ACF and DMI, as shown in Figs. 8(b) and 8(c).
Fig. 8. (a) $E(s )$ function versus s for TDS extraction of system (14-15), (b) ACF and (c) DMI for $s \in [{0.5,{\; }11} ]ns$. A time series with 3102 data points ($31.02ns$) was used.
4.3 Chaotic laser system with a phase-modulated Sagnac loop
A chaotic laser system with a phase-modulated Sagnac loop [35] is given as:
5. Analysis on noise added to OEO time series
In this section, we focus on the influence of noise level on the recognition performance. In practical optical communication systems, optical and electrical noise should be considered simultaneously. For simplification, here we use additive white Gaussian noise to evaluate the robustness of our method.
Fig. 9. (a) $E(s )$ function versus s for TDS extraction of system (16), (b) ACF and (c) DMI for $s \in [{2,{\; }42} ]ns$. A time series with 14062 data points ($31.02ns$) was used.
Simulation is carried out for the OEO generator (10). For different signal-to-noise ratio (SNR), the comparisons are shown in Table 2 when $\beta = 10$ (a relatively large value considering the limited implementation capability of actual devices). It shows that all these methods have a strong ability of handling noise. And the required data size for LSTM-NN is still minimal, and it marginally increases as the noise level increases.
Table 2. The minimal data size ${M_{data}}$ needed to extract the TDS for different SNR, $\beta = 10$.
Generally, the effectiveness of both CNN and LSTM-NN based TDS extraction methods is built on the fact that there is an approximately fixed nonlinear functional relationship between the chaotic signal and its delayed version when the delay is equal to the system loop delay. In our previous work, CNN is used to identify such functional feature indirectly through the images which are generated by reconstructing the chaotic time series into a two-dimensional phase space. Therefore, prior knowledge about the chaotic system model is indispensable in the training process to generate the labelled image set. However, in our current research, LSTM-NN is used to directly model the delay function candidates ${\varphi _s}$ from the output time series, and TDS identification is established by evaluate the prediction ability of ${\varphi _s}$. As a result, model information of the chaotic system is no longer needed and blind identification can be established.
6. Conclusion
We conclude that a data driven LSTM-NN model can identify TDS of time-delayed optical chaotic systems. Compared with existing methods, including ACF, DMI, and CNN based method, LSTM-NN based method shows its advantages, which lies in the following aspects: 1) No prior knowledge about chaotic system structure or mathematical model is required; 2) Minimum data size requirements; 3) High tolerance to system nonlinearity and additive noise. In the case of chaotic OEO, the proposed method is successfully demonstrated under strong nonlinearity. The proposed strategy also has the potential to be used for detecting the vulnerability of other security enhanced OCSC systems as discussed in section 4, and this will be the focus of our research in the next stage. From the perspective of safety assessment, we suggest that the ability of resisting deep-learning-based analysis should be considered in designing physical layer secure communication schemes. Subsequently, this type of analysis method will help to promote the security level and further maturity of optical chaos secure communications.
Funding
Open Research Project of The Hubei Key Laboratory of Intelligent Geo-Information Processing (KLIGIP-2019B11); National Key Research and Development Program of China (2018YFB1801304).
Disclosures
The authors declare no conflicts of interest.
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