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Abstract
In this paper, an optical transmitter authentication method using hardware fingerprints based on the characteristic of electro-optic chaos is proposed. By means of phase space reconstruction of chaotic time series generated by an electro-optic feedback loop, the largest Lyapunov exponent spectrum (LLES) is defined and used as the hardware fingerprint for secure authentication. The time division multiplexing (TDM) module and the optical temporal encryption (OTE) module are introduced to combine chaotic signal and the message to ensure the security of the fingerprint. Support vector machine (SVM) models are trained to recognize legal and illegal optical transmitters at the receiver. Simulation results show that LLES of chaos has the fingerprint characteristic and is highly sensitive to the time delay of the electro-optic feedback loop. The trained SVM models can distinguish electro-optic chaos generated by different feedback loops with a time delay difference of only 0.03ns and have a good anti-noise ability. Experimental results show that the recognition accuracy of the authentication module based on LLES can reach 98.20% for both legal and illegal transmitters. Our strategy can improve the defense ability of optical networks against active injection attacks and has high flexibility.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, optical communication occupies a dominant position in communication networks, and optical networks play an important role in information transmission in the access network and backbone network. A large amount of personal data, financial data, and national defense data are transmitted in optical networks, which puts forward higher requirements for communication security. Due to certain vulnerabilities, optical networks are easily threatened by eavesdropping and active injection attacks [1]. Traditional security protection methods, such as information encryption [2–4] and key distribution [5], can realize the encryption of information and effectively resist illegal eavesdropping, but cannot prevent attackers from carrying out active injection attacks in optical networks. The security of optical networks will be greatly threatened if it is accessed by illegal attackers [6,7], which means the legitimacy of the information in communication is difficult to be correctly identified and all the information encryption methods will become invalid. Therefore, identity authentication is required in optical networks to protect against active injection attacks.
Identity authentication is a process to verify whether the authenticated object is true or effective [8,9]. By verifying the attributes of the authenticated object, identity authentication is able to verify the legitimacy of the authenticated user, and its main function is to establish secure communication for legal users. Among several kinds of identity authentication methods, using hardware fingerprints to construct an authentication system can effectively enhance the security of optical networks due to their uniqueness, heritability, and invariance. Up to now, many kinds of hardware fingerprints in optical networks have been proposed [1,10–12]. [1] uses principal component analysis (PCA) to extract optical spectral features and realize the detection of unauthorized signals in optical networks. In [10], a hardware fingerprint authentication method for OFDM-PON is proposed and experimentally demonstrated. In [11], a security authentication technique based on optical network channel characteristics is proposed, and the dynamic changes of bit error rates (BER) and signal-to-noise ratios (SNR) measured in the fiber channel are regarded as hardware fingerprints. [12] proposes a novel deep hardware fingerprint identification method based on deep learning for optical fiber ethernet devices. All the above methods can enhance the security of optical networks, however, there are still some issues. Whether the spectral features, hardware fingerprints of the ONU device, and fingerprint of the fiber ethernet device all depend on the device itself and are vulnerable to the aging of devices. At the same time, it is difficult to control the differences between devices, which leads to the poor flexibility and portability of the above scheme. In addition, the complexity of the hardware fingerprint used in the above scheme is not high, which has the risk of being reconstructed by illegal users using various physical devices.
Chaos is a common phenomenon in nature and widely exists in nonlinear systems, which is also one of the important concepts in nonlinear science. Chaos generated by nonlinear systems means long-term unpredictability and sensitivity to initial conditions. This feature is inherent to biological entities alike nerve cells, electronic networks, and security and it is widespread in a wide class of nonlinear systems including complex optical fiber networks with modulation of optical gain, nonlinearity, and other control parameters [13,14]. At the same time the analysis [15–20] and evaluation [21–23] of chaotic time series is also of considerable interest. The commonly recognized techniques are the usage of the correlation exponents [21] for measurement of the statistics of strange attractors, Lyapunov exponents to measure the attractor dimensions [22], evaluation of the coefficient vectors [23] to distinguish different chaos scenarios [24] and reservoir computing in time-delay autonomous Boolean ring networks [25]. However, all the above methods are concerned mainly around low-dimensional deterministic chaos including delayed feedback oscillators [26] and this requires the investigation of the high-complexity optical chaos. Nowadays, there is a lot of research on optical chaos in the field of information encryption [27–30]. The research on using a neural network to improve the performance of chaotic sources is increasing gradually and good results have been achieved [29,31,32]. Optical chaos has high complexity and broad bandwidth, so it is widely used as the optical carrier to protect messages in optical networks from eavesdropping. But there is little research on the application of optical chaos to defend against active injection attacks in optical networks. The optical chaos is generated by the deterministic hardware structure, so the chaotic characteristics theoretically have invariance and uniqueness and have the potential to be used as the hardware fingerprint of optical transmitter systems in optical networks and be applied in the identity authentication system. Since optical chaos is able to be controlled by adjusting hardware parameters, the flexibility of the identity authentication system based on chaotic fingerprints is greatly improved. Meanwhile, thanks to the development of reversible optical transformations and optical encryption methods based on time lens, such as fractional Fourier transform [33] and optical temporal encryption (OTE) [34], hardware fingerprints based on optical chaos can be safely combined with signals at optical transmitters.
In this paper, a fingerprint construction method based on the largest Lyapunov exponent spectrum (LLES) of different electro-optic chaos is proposed. At the transmitter, electro-optic chaotic feedback loops with different hardware parameters are configured. The time division multiplexing (TDM) module and the optical temporal encryption (OTE) module are introduced to ensure the security of chaotic signals. At the receiver, the received signal is decrypted by the optical temporal decryption (OTD) module and the time division demultiplexing (TDD) module. The authentication system calculates the largest Lyapunov exponent spectrum from the received chaotic time series and then uses the trained support vector machine (SVM) models to realize the classification of illegal electro-optic chaos and legal electro-optic chaos. Therefore, the security authentication of different optical transmitters is realized. Simulation and experimental results indicate that the largest Lyapunov exponent spectrum is sensitive to the time delay $T$ of the electro-optic feedback loop and can be easily classified by the SVM models. In electro-optic feedback loops, time delay $T$ can be easily adjusted, which means that our scheme has high flexibility. Our method opens up a new direction for the application of optical chaos to authentication technology.
2. Principle and models
The optical transmitter authentication system based on the characteristic of electro-optic chaos is shown in Fig. 1. The traditional electro-optic intensity delayed feedback loop in the transmitter is constructed in order to combine the chaotic signal and the message with encryption inside the TDM and OTE module. Then the encrypted time division multiplexing signal is transmitted to the receiver. After signal decryption and demultiplexing, the chaotic signal is separated and its fingerprint is extracted so that the characteristics of the electro-optic chaos can be obtained. Support vector machine (SVM) models which have been trained based on these characteristics are used to recognize whether the received chaotic signal is from a legal transmitter.
Fig. 1. The principle of the authentication system based on chaotic hardware fingerprints in optical networks. LD, laser diode. PC, polarization controller. EDFA, erbium-doped optical fiber amplifier. MZM, Mach-Zehnder modulator. OC, optical coupler. DL, delay line. PD, photodiode. RF, radio frequency amplifier. TDM, time division multiplexer. PM, phase modulator. $\textrm{D}$, Dispersion component. TDD, time division demultiplexer.
2.1 Optical chaos generated by electro-optic feedback loops
Optical chaotic sources are mainly divided into all-optic sources [35] and electro-optic sources. Among them, electro-optic chaotic sources have the advantages of high complexity and low cost. A traditional electro-optic intensity feedback loop has been shown as the transmitter in Fig. 1 [36]. Chaotic signals can be obtained when the feedback strength in the loop is large enough based on the nonlinear effect of the Mach-Zehnder modulator (MZM). The dynamics of the electro-optic intensity feedback loop can be expressed as Eq. (1)
Eq. (1) indicates that an electro-optic intensity feedback loop mainly has two controllable hardware parameters, the feedback strength $\beta$ and the time delay $T$. When these two parameters change, the characteristics of the output signal of the feedback loop will also change. However, the feedback strength $\beta$ has a great influence on the complexity of chaotic signals generated by the electro-optic feedback loop. Figure 2(a) shows the change of permutation entropy (PE) of $x(t)$ versus $\beta$ and $T$. PE is usually used to measure the complexity of timeseries [37], and it is calculated as follows: for a sequence, $\boldsymbol {u(i)}=\left \{u(1), u(2) \ldots u(K)\right \}$, reconstruct it into $K-dL+L$ subsequences with embedding dimension $d$ and embedding delay time $L$, then arrange these subsequences in ascending order $\pi$. Then PE of $\boldsymbol {u(i)}$ can be expressed as Eq. (2).
where $p\left (\pi \right )$ is the occurring probability of order $\pi$.Fig. 2. Analysis of the traditional electro-optic intensity feedback loop. (a) PE of the output signal varies with $\beta$ and $T$. (b) The output signal of the electro-optic intensity feedback loop when $\beta$=4.5, $T$=5ns
As shown in Fig. 2(a), when $\beta$ of the electro-optic feedback loop is low, the PE of the output signal is also low. As $\beta$ increases, the output signal gradually becomes chaotic, and its PE of it also increases. When $\beta \geq 4.5$, regardless of the value of feedback delay $T$, the PE of the output signal is close to 0.9, which means the complexity of the output signal is very high. The temporal waveform of the chaotic output signal is shown in Fig. 2(b). Figure 2 shows the following result. Despite the complexity of the output optical signal, the feedback loop has a strong dependence on feedback strength $\beta$. On the other hand, it has little dependence on time delay $T$ under conditions of the deep modulation of chaotic signal in the current authentication system and the feedback strength $\beta$ variations are very small. There will be few values of $\beta$ values available for electro-optic feedback loops at the legal transmitter, leading to the number of devices that can access the authentication system will be extremely limited if $\beta$ is used to distinguish different optical transmitters. Therefore, to ensure both safety and practicality, it is more reasonable to construct different electro-optic feedback loops by changing the time delay $T$. Moreover, $\beta$ is determined by multiple parameters in an electro-optic loop, which means the attacker can approach the legal $\beta$ by a variety of means. As a result, the probability of the electro-optic chaos being reconstructed is greatly increased and the security of the authentication system is affected.
2.2 Theory of OTE module
OTE module consists of a phase modulator (PM) and a dispersion compoent [34]. The PM can be mathematically described as Eq. (3)
Where $E_{ {out }}(t)$ and $E_{ {in }}(t)$ is the output and the input of the PM. $V_{\pi p}$ is the half-wave voltage of the PM and $V_{key}(t)$ is a radio frequency sinusoidal signal that is mathematically described as $V_{key}(t)$=$A_0cos(2\pi f_0t)$, where $A_0$ and $f_0$ stand for the amplitude and the frequency of $V_{key}(t)$, respectively. The dispersive component $D_E$ can be constructed by a dispersive fiber. When excluding the higher-order dispersions, the transfer function of the dispersive fiber in the frequency domain can be written as Eq. (4) [34] Where $K_{1}$ is a constant, $\beta _{2E}=-D_{E}\lambda ^{2}/2\pi c$ is the group velocity dispersion. $\lambda$ is the wavelength of LD and $c$ is velocity of light in vacuum. $L_{E}$ is the length of the dispersion fiber. When the impact of the transmission fiber link on the signal is excluded, to realize the decryption of the signal, the dispersion coefficient ($D_D$) and the length ($L_D$) of the dispersive fiber in the OTD module should satisfy the following condition:Meanwhile, the ${V_{key}}^{,}(t)$ should be inverse to $V_{key}(t)$. OTE module can encrypt chaotic signals and messages after time division multiplexing, ensuring that illegal attackers cannot separate chaotic signals and messages, and thus ensuring the security of fingerprints.
2.3 Calculation of LLES
Phase space reconstruction is an effective tool for analyzing nonlinear time series [18,20]. The parameters of phase space reconstruction, embedding dimension $m$, and embedding delay $\tau$ can greatly affect the degree to which the reconstructed phase space can reflect the properties of the original chaos [20,38–40]. However, due to the very high complexity of electro-optic chaos, limitation of the calculation accuracy, and noise in the sequence, the reconstructed phase space of an electro-optic chaotic time series is difficult to perfectly reflect the properties of the original electro-optic feedback loop.
Therefore, in order to reflect the maximum properties of the original chaos, referring to the vectorization method of PE in [41], for time series $\{\boldsymbol {c}(\boldsymbol {n})\}$ calculated from a chaotic map $c(n+1)=F_c[c\left (n\right )]$, we construct $N$ phase spaces by traversing the embedding delay $\tau$ from 1 to $N$ as Eq. (6) and Eq. (7).
Among the measurement indicators of chaos, the Lyapunov exponent is widely used to describe the properties of a chaotic dynamic system and has been proven to be characteristic and independent of initial conditions [22,42]. For the discrete-time system $c(n+1)=F_c[c\left (n\right )]$, the Lyapunov exponent $\lambda$ is calculated as Eq. (8):
For a $N$-dimensional chaos, there are n Lyapunov exponents theoretically. Among them, the largest Lyapunov exponent (LLE) is often used to measure whether a system is in a chaotic state. In [43], Wolf gave a method to calculate the largest Lyapunov exponent based on time series. Using Wolf‘s method, we can calculate the largest Lyapunov exponent for each reconstructed phase space $\boldsymbol {c_{R: \tau }}$, and finally obtain the largest Lyapunov exponent spectrum (LLES) as Eq. (9):
The whole calculation process can be simply illustrated as Fig. 3. When $\{\boldsymbol {c}(\boldsymbol {n})\}$ is reconstructed into $N$ phase spaces $\boldsymbol {c_{R}}(\boldsymbol {N})$, each $\boldsymbol {c_{R: \tau }}$ in $\boldsymbol {c_{R}}(\boldsymbol {N})$ can be thought to subject to a new m-dimensional map $F_{c\tau }$ as Eq. (10)
Fig. 3. Calculation process of largest Lyapunov exponent spectrum(LLES). (a) time series of $c(n+1)=F_{c}[c(n)]$. (b) different phase spaces reconstructed from $\{\mathbf {c}(\mathbf {n})\}$ when m=3 for example.
According to Eq. (8), we can know that the calculation of the Lyapunov exponent only depends on the map $F_c$. Although we can hardly reconstruct a phase space that perfectly reflects the original map $F_c$, especially when the dimension of the original attractor is high [18], we can still use the map $F_{c\tau }$ to partly characterize the properties of it. For an electro-optic intensity feedback loop shown in Fig. 1, the output chaotic time series $x(n)$ which follows the map $x\left (n+1\right )=F_x[x\left (n\right )]$ can also be thought of as a sample series of $x(t)$ in Eq. (1), which means the map $F_x$ will change when the parameters in Eq. (1) change, such as the feedback strength $\beta$ and the time delay $T$. Then this change is able to be reflected on the LLES calculated from chaotic time series. Each $\lambda _{x\tau }$ of the LLES is determined by the map $F_{x\tau }$ which is closely related to the map $F_x$. And each LLES has $N$ elements, so the probability of two LLES being identical is extremely low. Therefore, the LLES can be theoretically used to realize the authentication of different electro-optic intensity feedback loops.
3. Numerical results and discussion
In this section, the fourth-order Runge-Kutta algorithm is used to solve Eq. (1) to model the electro-optic intensity feedback loop. The major parameters in the simulation are $\upsilon =25ps$, $\theta =5\mu s$, $\phi =\pi /4$, $\lambda =1550 nm$, $L_E=L_D=3km$, $D_E=500ps/nm/km$, $K_1=1$, $A_0=V_\pi p=4V$, $f_0=53.451GHz$. The simulation step is 10ps, and the simulation length for generating chaotic output is 4000000.
3.1 Numerical calculation and analysis of LLES
In phase space reconstruction theory, when the embedding dimension $m\geq 2m^{\prime }+1$ (where $m^{\prime }$ is the dimension of the original chaos), the reconstructed phase space can be regarded as fully reflecting the properties of the original phase space [18,20]. However, the dimension of the time-delay chaos is infinite [44–46], and the increase of $m$ will significantly increase the complexity of computation. When m=7, it takes several hours to calculate single LLES.
In order to further analyze the influence of the parameters of phase space reconstruction on the LLES, two electro-optic feedback loops EOFL1 and EOFL2 are set for the following analysis. The time delay $T$ of EOFL1 is set as 5ns while the time delay of EOFL2 is set as 7.5ns, and the feedback strength $\beta$ of EOFL1 and EOFL2 are both 4.5. After calculating several LLES corresponding to the chaotic signal generated by EOFL1 and EOFL2, we respectively calculate the correlation coefficient (CC) between LLES of the same chaos and CC between LLES of different chaos when $m$ and $N$ change. For each LLES, 20000 points of the chaotic time series are used. CC is often used to identify the correlation between time series and it is defined as Eq. (11) [30]:
Next, we analyze the average and standard deviation (Std) of these CC. The results are shown in Table 1 and Table 2. According to Table 1, no matter what the value of $m$ is, CC between LLES of the same chaos is always higher than CC between LLES of different chaos. However, as $m$ goes up, the Std of CC goes down. LLES of chaos become more stable, which has advantages for the subsequent authentication process.
Table 1. Influence of embedding dimension $m$ on LLES when $N$=1000
Table 2. Influence of maximum embedding delay $N$ on LLES when $m$=6
As for the maximum embedding delay $N$, according to Table 2, with the increase of the maximum embedding dimension $N$, although the average CC between LLES of the same chaos decreases, the average CC between LLES of different chaos decreases, which means the invariance of LLES decreases slightly, but its uniqueness increases, and the stability of LLES has not changed much. Meanwhile, during the authentication process, theoretically, a larger $N$ means a more comprehensive representation of the dynamic characteristics of the original chaos and a larger fingerprint space.
Therefore, we select $m=6, N=1000$ to achieve phase space reconstruction and calculate LLES using Wolf‘s method. Figure 4(a1-a4) show the LLES of different electro-optic chaos when the time delays $T$ of the electro-optic intensity feedback loop are different while the feedback strength $\beta$ are both 4.5. Obviously, when the time delays $T$ of the feedback loop changes, the LLES of the corresponding output chaotic signals are also different, which means LLES possesses the uniqueness of fingerprint characteristics. When the hardware parameters of the feedback loop remain unchanged, the LLES of the output electro-optic chaos when selecting different time periods of the time series for calculation and giving different initial conditions are respectively shown in Fig. 4(b1-b4) and Fig. 4(c1-c4). It is easy to see that changes in initial conditions and time periods have little effect on LLES, which means LLES has the invariance of fingerprint characteristics.
Fig. 4. (a1-a4) When $\beta =4.5$, $T$ are respectively 2.5ns, 5ns, 7.5ns, and 10ns, LLES of the chaos generated by different elector-optic feedback loops. (b1-b4) When $\beta =4.5$, $T=5ns$, LLES calculated from different time periods of the chaos generated by the same elector-optic feedback loop. (c1-c4) When $\beta =4.5$, $T=5ns$, LLES of the chaos generated by the electro-optic feedback loop with initial conditions of Eq. (1) are respectively $0.8\times 10^{-6}$, $0.9\times 10^{-6}$, $1\times 10^{-6}$, and $1.1\times 10^{-6}$.
After obtaining the chaotic time series generated by EOFL1 and EOFL2 with different initial conditions, we randomly selected 20000 points of them to calculate multiple LLES of chaos generated by EOFL1 and EOFL2. A separating hyperplane is used to further verify the above conclusions. The LLES is essentially a $N$ dimension vector, by calculating the distance $\gamma$ of LLES from a hyperplane, it is easy to distinguish LLES of electro-optic chaos generated by different feedback loops, as shown in Fig. 5(a). The hyperplane is defined as Eq. (12)
and $\gamma$ is defined as Eq. (13): Where $x_i$ is the $i$th LLES, $\omega$ and $b$ are confirmed by solving the corresponding convex quadratic programming problem.Fig. 5. Separate LLES of chaos generated by different electro-optic feedback loops. (a) Using hyperplane to separate LLES. (b) CC between one LLES of chaos generated by EOFL1 and multiple LLES of chaos generated by both EOFL1 and EOFL2. (c) CC between one LLES of chaos generated by EOFL2 and multiple LLES of chaos generated by both EOFL1 and EOFL2.
We use CC to further verify that LLES has the characteristic of the fingerprint. CC between one LLES of chaos generated by EOFL1 and multiple LLES of chaos generated by both EOFL1 and EOFL2 are shown in Fig. 5(b), while CC between one LLES of chaos generated by EOFL2 and multiple LLES of chaos generated by both EOFL1 and EOFL2 are shown in Fig. 5(c). The results indicate that LLES of the electro-optic chaos generated by the same feedback loop is more relevant than those generated by different feedback loops, and the CC between LLES of the same chaos has no intersection with the CC between LLES of different chaos, which means LLES do reflect the variation of time delays $T$ in the electro-optic intensity feedback loop and can be used as the fingerprint in authentication systems.
We also investigate the effect of slight changes in parameters $T$ and $\beta$ on the calculated LLES. As shown in Fig. 6(a) and Fig. 6(b), the time delays $T$ of the electro-optic feedback loop are respectively 5ns and 7.5ns when $\beta$ is changed in the range of $\pm$0.1, the value of CC between LLES of chaos generated by the original feedback loop and the changed feedback loop decreases slightly, but it is difficult to be lower than 0.41. However, when $\beta$ is 4.5 and the time delay $T$ is changed in the range of $\pm$0.1ns, as shown in Fig. 6(c) and Fig. 6(d), the value of CC between LLES of chaos generated by the original feedback loop and the changed feedback loop drops below 0.27 after only changing $\pm$0.06ns.
Fig. 6. CC between LLES of chaos generated by the original feedback loop and the changed feedback loop. (a) T=5ns, $\beta$ is changed in the range of $\pm$0.1. (b) T=7.5ns, $\beta$ is changed in the range of $\pm$0.1. (c) and (d) $\beta$=4.5,T is changed in the range of $\pm$0.1ns,
This result indicates that LLES is sensitive to the change of time delay $T$ of the feedback loop, but has less sensitivity to the change of feedback strength $\beta$, which means using different time delay $T$ to construct different optical transmitters has higher security, and small adaptation of feedback strength $\beta$ of the electro-optic feedback loop will not influence the LLES of the output chaotic signal too much. The relative insensitivity of LLES to the feedback strength makes the authentication system more practical due to $\beta$ is determined by the parameters of various hardware devices, such as gain coefficients of the RF, PD, or the attenuation of the loop. Compared with the time delay $T$, the adjustment of $\beta$ is less flexible and controllable.
3.2 Performance of OTE module
In order to verify that the OTE module can effectively ensure the security of the chaotic signal, the time division multiplexing signal combined with the on-off keying (OOK) signal and the chaotic signal generated by the electro-optic feedback loop when $\beta =4.5, T=5ns$ is injected into the OTE module. The waveforms and power spectrums of the signals before and after encryption are shown in Fig. 7(a1-a2) and Fig. 7(b1-b2). Obviously, the temporal waveform of the encrypted signal has great changes compared with the original signal, the boundary between chaotic signals and messages is no longer clear, and the power spectrum also changes. To further analyze the performance of the OTE module, CC is used to analyze the correlation between the original signal and the decryption signal when there is a parameter mismatch between the OTE module and the OTD module.
Fig. 7. Performance of OTE module. (a1-a2) Temporal waveforms of the time division multiplexing signal and the encrypted signal. (b1-b2) Power spectrum of the time division multiplexing signal and the encrypted signal. (c1-c2) CC between the original signal and the decrypted signal with parameter mismatch between the OTE module and OTD module
Figure 7(c1-c2) respectively indicate that when the dispersion coefficient $D$ of the OTD module and frequency $f_0$ of $V_{key}(t)$ are different from that of the OTE module, the correlation between the decrypted signal and the original signal decreased significantly, which proves that OTE module can encrypt chaotic signals effectively. Illegal attackers will not be able to obtain the legal chaotic signals for authentication and insert their own signal to replace the original message unless they obtain the keys of the OTE module accurately.
Since the change of time series cannot strictly represent the change of the corresponding LLES, we further analyze the influence of the OTE module on the LLES of the chaotic signal. The results are shown in Fig. 8.
Fig. 8. The performance of optical time encryption module for LLES. (a1), LLES calculated from the original chaotic signal. (a2) LLES calculated from the encrypted signal. (b1) When the dispersion coefficient $D_E$ of OTD is mismatched with that of the OTE module, CC between the LLES of the decryption signal and the original chaotic signal. (b2) When the frequency $f_0$ of $V_{key}(t)$ injected in OTD is mismatched with that of the OTE module, CC between the LLES of the decryption signal and the original chaotic signal.
Figure 8(a1-a2) indicate that LLES calculated from the encrypted time series has changed both in range and trend. This proves that the OTE module can encrypt LLES, and illegal attackers cannot obtain the fingerprint of the original chaotic sequence by analyzing the encrypted signal, so the security of the fingerprint is guaranteed. At the same time, Fig. 8(b1-b2) indicate that no matter whether the dispersion coefficient or the frequency of RF signal $V_{key}(t)$ is mismatched, LLES calculated from OTD output signal is not correlated with the LLES of the original chaotic signal, and the value of CC between them drops rapidly to about 0.1 to 0.2. This proves that LLES of the chaotic signal is highly sensitive to both hardware parameters of the OTE module. In order to obtain LLES of legal transmitters, the illegal attacker must match the two hardware parameters strictly simultaneously, and the key space of the fingerprint is greatly enhanced.
3.3 Construction of authentication module and performance analysis
We propose a recognition method of LLES based on support vector machine (SVM) models. SVM is an effective binary classification model in the field of machine learning. When the data is linearly inseparable, the SVM transform them to a high-dimensional space through a nonlinear transformation (also called a kernel function) so that they are separable in this high-dimensional space. The training process of SVM is to find and optimize a hyperplane in this high-dimensional space to realize the classification of different data features [47].
Therefore, we introduce multiple SVM modules to jointly complete the establishment of the authentication system, and use the loss function of SVM to solve the problems encountered during the training. We select different electro-optic feedback loops with feedback delay $T$=2.5ns, $T$=5ns, and $T$=7.5ns to correspond to legal optical transmitters, and respectively calculate multiple LLES of them as the SVM training set. During the training, we first regard the LLES corresponding to the electro-optic feedback loop with $T$=2.5ns as legal, and set their labels as "1", while the LLES corresponding to electro-optic feedback loops with $T$=5ns and $T$=7.5ns are regarded illegal and their labels are set as "-1", thus completing the first SVM training. When training the second SVM, the LLES corresponding to the electro-optic feedback loop with $T$=5ns are regarded as legal and their labels are set as "1", while the LLES corresponding to electro-optic feedback loops with $T$=2.5ns and $T$=7.5ns are regarded as illegal and their labels are set as "-1". When training the third SVM, we regard the LLES corresponding to the electro-optic feedback loop with $T$=7.5ns as legal and set their labels as "1", while the LLES corresponding to electro-optic feedback loops with $T$=2.5ns and $T$=5ns are regarded as illegal and their labels are set as "-1". Thus, we obtain three SVM models which can recognize the legal optical transmitters with feedback delay $T$=2.5ns, $T$=5ns, and $T$=7.5ns, respectively.
Next, we use the loss function to further realize the identification of illegal transmitters. The loss function can quantify the error in the classification process. For illegal transmitters, even if SVM models could classify them into one certain class, their loss function will be much larger than that of legal transmitters. The loss function of SVM can be shown as Eq. (14): [1,47]:
Where $K$ is the number of SVM, $k$ means the $k$ th SVM, $\boldsymbol {x_i}$ is the vector to be sorted, $y_{ki}$ $\in$ −1,1 is the label corresponding to $\boldsymbol {x_i}$, when $\boldsymbol {x_i}$ is regarded as legitimate by the trained k th binary SVM, the corresponding $y_{ki}$ is 1; otherwise, it is −1. $g_{ki}$ is the classification score of $\boldsymbol {x_i}$, it is determined by Eq. (15): Where $\omega _k$ and $b_k$ is the parameter of the hyperplane of the $k$ th trained binary SVM.Since SVM cannot learn the characteristics of illegal transmitters during training, if illegal transmitters are wrongly classified as legal, their loss function will be much larger than that of legal transmitters. By setting a threshold, we can further judge the legitimacy of the received signals:
Therefore, we obtain a general recognition method for an optical transmitter: if its LLES is recognized as "-1" by all three SVM, or "1" by two or more SVM, then it is certainly illegal; When the LLES is recognized as "1" by one SVM and "-1" by the other two SVM, then the loss function of this classification is calculated and Eq. (16) is used to further determine how to classify the transmitter. To simplify the analysis, we assume that the OTE module and the OTD module are ideal in the following analysis. In the testing phase, electro-optic feedback loops with $\Delta \mathrm {T}\leq {0.05}$ns are set at the illegal transmitter to analyze the accuracy of the authentication module, where $\Delta \mathrm {T}$ means the difference of time delay $T$ of electro-optic feedback loops set at illegal and legal transmitters.
The recognition accuracy of SVM models combined with the loss function is shown in Fig. 9. When the threshold is small, even for legal transmitters, the loss function will be smaller than the threshold, and the authentication module will judge legal transmitters as illegal. When the threshold is too large, illegal transmitters cannot be correctly judged because although the value of their loss function is higher than that of legal transmitters, it is still less than the threshold. Therefore, in order to ensure the feasibility of the authentication module, it is important to select the threshold which can make the authentication module have higher recognition accuracy for both legal transmitters and illegal transmitters. Figure 9 shows that by selecting the appropriate threshold, the recognition accuracy of the authentication module for illegal transmitters with $\Delta \mathrm {T}$ of the electro-optic loops is 0.01ns and 0.02ns is able to respectively reach 95.96$\%$ and 96.97$\%$, while the when $\Delta \mathrm {T}\geq$0.03ns can reach 100$\%$, which means the authentication system based on LLES has high precision.
Fig. 9. Recognition accuracy of the authentication module for legal transmitters and illegal transmitters with different $\Delta \mathrm {T}$ and different thresholds.
We also analyze the recognition accuracy of the authentication module under different noise environments. The results are shown in Table 3. By selecting the appropriate threshold, when SNR=18dB, under the condition that the difference of $T$ of electro-optic feedback loops at legal and illegal transmitters is only less than 0.05ns, the overall recognition accuracy of the authentication module is close to that under the noiseless condition, which proves that LLES has an anti-noise ability.
Table 3. Recognition accuracy of the authentication module under different noise environments
In addition, during the analysis process, we also introduce some all-optic chaotic sources whose time delays are the same as those of electro-optic intensity feedback loops at legal transmitters. By virtue of the Lang-Kobayashi differential-delay rate equations [41] instead of delayed feedback maps used for modelling of the all optical cavities [48], one may generate the time series of all-optic chaos and calculate their LLES for authentication. The Lang-Kobayashi rate equations are described as [41]:
Table 4. Simulation parameters of all-optic chaos
The results show that the recognition accuracy of such illegal chaos is 100$\%$, which means that LLES not only reflects the change of time delay but also reflects the overall properties of the whole dynamics of chaos. Attackers cannot implement the imitation of LLES by arbitrarily constructing time series with certain time delay signatures, which further enhances the security of the authentication system.
4. Experimental setup and performance analysis
In order to verify that LLES can be used to recognize actual chaotic signals, we simplify the authentication system as shown in Fig. 1 and set up the experiment as shown in Fig. 10, which means in our experiment, in order to focus on the analysis of the performance of the authentication module for recognizing LLES of chaos generated by electro-optic feedback loops at legal and illegal transmitters, the TDM and the OTE module are considered as ideal. At the transmitter, the laser diode (LD, FLS-2800, EXFO) works at 1550nm and emits the continuous wave (CW) laser which is amplified by the erbium-doped fiber amplifier (EDFA, AEDFA-23-B-FA, Amonics) to enhance the nonlinearity of the loop. Then the output of EDFA inject into the Mach-Zehnder modulator (MZM, Mach-40, Thorlabs) and divided into two parts by a 50:50 optical coupler (OC). One of the outputs is transmitted through the delay line (DL), detected by a photodiode (PD1, RXM25AF, Thorlabs), and amplified by a radio-frequency amplifier (RF, SHF-S126A, SHF). At the receiver, PD2 (XPDV2320R-VF-FP, Finisar) is used to detect another output of OC, and the output is sampled by an oscilloscope (OSC, DPO 72004C, Tektronix) with the sample rate of 25Gb/s. In the feedback loop, the half-wave voltage of MZM is 4V, the responsivity of PD1 is 0.75A/W, the gain of the RF amplifier is 29dB, and the responsivity of PD2 is 0.65A/W.
Fig. 10. Experimental setup. LD, laser diode. PC, polarization controller. EDFA, erbium-doped optical fiber amplifier. MZM, Mach-Zehnder modulator. OC, optical coupler. DL, delay line. PD, photodiode. RF, radio frequency amplifier. OSC, oscilloscope.
By increasing the output power of EDFA, the feedback strength of the electro-optic feedback loop will be increased and lead to the generation of chaotic signals. When the output signal becomes chaotic, we measured the voltage injected into the RF port of the MZM and deduced that the feedback strength in the loop is between 3.9 and 4.4. In the experiment, we construct four electro-optic intensity feedback loops, A, B, C, and D, with different time delays by selecting delay lines of different lengths. The time delay $T$ of electro-optic feedback loops A, B, C, and D are respectively 29.8ns, 33.28ns, 34.56ns, and 37.48ns. When the output of the four feedback loops is chaotic, we sampled the chaotic signals of each feedback loop several times. For each feedback loop, 252 LLES with different time periods and initial conditions are calculated. Figure 11 shows the LLES of electro-optic chaos generated by the four feedback loops.
Fig. 11. (a1-a4) LLES of chaos with different time periods and initial conditions generated by A. (b1-b4) LLES of chaos with different time periods and initial conditions generated by B. (c1-c4) LLES of chaos with different time periods and initial conditions generated by C. (d1-d4) LLES of chaos with different time periods and initial conditions generated by D.
Due to the influence of noise and the instability of feedback strength, the fingerprint characteristics of LLES are not very clear in Fig. 11. Next, we respectively set A, B, C, and D at the illegal transmitter to test the recognition accuracy of the authentication module. 80$\%$ of all LLES are used as training sets and 20$\%$ are used as tests to train the SVM models. The authentication results are shown in Fig. 12. No matter which electro-optic feedback loop is selected at the illegal transmitter, under the appropriate threshold, the identification accuracy of the authentication module to the legal transmitters and the illegal transmitters can be higher than 98.20$\%$, and has the potential to reach 100$\%$. In addition, due to the EDFA being introduced in the experiment, the noise in the loop is also relatively high. The experimental results show that the authentication module combined with SVM models and loss function can effectively recognize LLES of different electro-optic chaos, and has a good anti-noise ability.
Fig. 12. Recognition results of the experiment when the illegal transmitter is respectively constructed by (a) A, (b) B, (c) C, (d) D.
5. Conclusion
We propose a hardware fingerprint authentication method based on the characteristics of electro-optic chaos. Based on the phase space reconstruction theory, LLES can be used as the hardware fingerprint of an electro-optic feedback loop. TDM and OTE modules are introduced to ensure the security of chaotic fingerprints. By training multiple SVM models and introducing the loss function, the authentication module based on LLES can effectively classify legal transmitters and illegal transmitters with different electro-optic feedback loops. Compared to existing hardware fingerprint methods, the authentication method based on LLES of chaos generated by electro-optic feedback loops is highly flexible. Simulation results show that the trained SVM models can effectively distinguish illegal transmitters when the time delay difference of the electro-optic feedback loops set at legal and illegal transmitters is only 0.03ns under the condition of SNR=18dB. Experimental results show that the recognition accuracy of legal transmitters and illegal transmitters can both reach 98.20$\%$. Our work provides a more flexible scheme for secure authentication and has the potential to be combined with more existing encryption methods to realize the integration of encryption and authentication in optical networks.
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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