Open Access
Add to BibSonomy
Abstract
Chaotic encryption is a promising scheme for physical layer security. By solving the multi-dimensional chaotic equations and transforming the obtained results, both bit-level and symbol-level encryption can be realized. One of the mainstream symbol-level encryption solutions is the constellation shifting (CS) scheme, which treats the chaotic sequence as artificial noise and adds it to the QAM signal sequence to achieve encryption. However, this scheme has several technical flaws in practical application, in terms of computational complexity and coexistence with blind equalization algorithm and the probabilistic shaping (PS) technique. In this paper, we propose a novel symbol-level encryption scheme based on phase ambiguity (PA), which converts the two sequences originally used to generate artificial noise into a set of phase rotation keys and complex conjugate keys, so that the encrypted symbols are still on the ideal constellation point coordinates. Simulation verification is carried out in a discrete multi-tone (DMT) system with 64QAM modulation. Results show that the proposed scheme can fully retain the shaping gain brought by the PS technique and avoid the error convergence of the blind equalizer. Moreover, the time required to solve the chaotic equations is only 38% of the CS scheme. Experimental verification is carried out, and the obtained results once again prove the superiority of the proposed encryption algorithm, which is a practical alternative for future physical layer secure optical communications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the continuous development of global data services, the traffic pressure faced by optical fiber access systems is increasing rapidly. In order to meet the requirements of high-speed data transmission, a large number of access schemes based on orthogonal frequency division multiplexing (OFDM) have been demonstrated through theory and experiment [1–5]. Due to its high spectrum efficiency (SE) and its ability to effectively resist the chromatic dispersion in optical fibers, OFDM is regarded as one of the reliable solutions for high-speed access in the future [6]. As a real-valued variant of OFDM, discrete multi-tone (DMT) is widely used in cost-effective intensity modulation and direct detection (IM/DD) systems [3–5].
In addition to increasing the data rate of the access system, how to improve its security performance is also one of the focuses of the academic and industrial circles [6]. In recent years, the physical layer security communication scheme based on chaotic encryption has been one of the research hotspots [6–13]. Among them, a mainstream solution is the constellation shifting (CS) method, which was first proposed in [10]. The main idea is to use a 3-dimension (3D) hyper digital chaos to generate three independent chaotic sequences, the first one is applied for bit-level XOR encryption, and the other two are regarded as symbol-level artificial noise to realize the constellation shifting encryption of quadratic-amplitude modulation (QAM) signals. In 2019, Y. Huang et al. improved this scheme and proposed using multi scrolls Jerk chaos to replace the original hyper Chen, which can reduce the encryption complexity and increase the key space [11]. In 2020, F. Wang et al. experimentally demonstrated the feasibility of the CS encryption scheme in a high-speed DMT system for the first time, but the results also show that the encryption operations will cause a certain deterioration in the bit error rate (BER) performance [12]. In 2021, B. Zhu et al. used probabilistic shaping (PS) technique and chaotic encryption in cascade for the first time to improve the BER performance of encrypted DMT signals [13]. However, as we will introduce in detail in Section 2, the CS-based encryption scheme has a series of technical defects in practical applications. To start with, if the value of the time step is small when solving the chaotic equation, the artificial noise used for symbol-level encryption will be cracked by time-domain blind equalization algorithms such as direct-detection least mean square (DD-LMS). In order to ensure the effectiveness of encryption, it is necessary to increase the value of time step, or introduce additional random shuffling sequences as in [13], either way will lead to an increase in computational complexity. The second problem is that when the DD-LMS algorithm is applied for post-equalization of channel impairments, due to the artificial noise introduced by the CS scheme, the equalizer is prone to error convergence. The third problem is that when the PS technology is used in cascade, the CS scheme will cause the probability distribution to deviate from the well-known Maxwell-Boltzmann (MB) distribution. This deviation from the optimal distribution will affect the effectiveness of PS, which in turn will lead to the loss or even disappearance of the shaping gain.
In this paper, after analyzing the potential defects of the existing CS scheme in detail, we propose a new symbol-level encryption scheme based on phase ambiguity (PA). The main idea of the proposed scheme is to convert the two sequences originally used to generate artificial noise into a set of phase rotation keys and a set of conjugate keys, so that the encryption process introduces an artificial phase ambiguity to the symbol. Since the QAM constellation has four-fold symmetry, it can be ensured that the encrypted symbols are still located on the standard coordinates on the constellation diagram. The proposed scheme is verified by simulation and experiment in a DMT system with a symbol rate of 10GBaud and a transmission distance of 5km. The simulation results show that the proposed scheme has good compatibility with DD-LMS and PS, and can greatly reduce the computing time required to solve the chaotic equations.
2. Principles
2.1 Principle of the CS scheme
Taking the three-dimensional (3D) Jerk chaotic equation system as an example, the current mainstream constellation shifting (CS) encryption scheme for DMT signals can be represented by Fig. 1. First of all, the multi-scroll chaotic system based on Jerk model can be realized by the following ordinary differential equations [10]:
Although the above-mentioned encryption scheme has been proved to be effective in the literature [10–13], this scheme has the problem of being incompatible with DD-LMS algorithm and PS technique in practical applications.
2.2 Interplay of the CS scheme and DD-LMS
When using the CS scheme for encryption, the most fatal technical flaw is the with DD-LMS, which was pointed out for the first time in a recent document [13]. If the time step value set when generating the chaotic sequence used for shifting is too small, the noise-like $\{{k{x_i} + jk{y_i}} \}$ can be regarded as a slow changing process, and the artificial noise used for encryption can be cracked by blind equalization algorithms, resulting in the confidentiality to be compromised. However, aggressively increasing the value of $\varDelta T$ is not a practical solution. Under the premise that the total sequence length N is the same, a larger time step means more iterations in the integration solution process. This will result in a substantial increase in computing time, which is not conducive to practical applications and implementation. For the sake of intuition, we use time steps of different orders to solve the same Jerk equations on a completely consistent operation platform, and calculate the relative value of the computing time required, as shown in Fig. 2. It can be seen from the results that the calculation time increases with the increase of the time step. After the order of magnitude is higher than 10−1, the calculation time begins to increase substantially.
In the literature [12], the time step used is 10−2. Through our simulation, we have proved that a value of this order of magnitude does not guarantee effective symbol-level confidentiality. We have drawn the $\{{k{x_i} + jk{y_i}} \}$ constellation diagrams under the three conditions of time step from 10−2 to 1 as shown in Fig. 3(a)-(c). The smaller the value of time step is, the smaller the change of $\{{k{x_i} + jk{y_i}} \}$ between adjacent moments, which can be regarded as a slow changing process gradually. After the value of the time step is less than 10−3, the values of the chaotic sequences are almost unchanged for dozens or even hundreds of adjacent moments. In this case, for a relatively short period of time, the signal change realized by encryption can be regarded as a kind of quasi-static linear impairment, which can be easily cracked by blind equalization algorithm. We set the scale to k = 2.5 (which is sufficiently large for k), and apply it to the noise-free PS-64QAM signal, and the encrypted constellation diagrams obtained are shown in Fig. 3(d)-(f). After that, the DD-LMS algorithm with an order of 81 and a convergence coefficient of 10−4 is applied to crack the encrypted constellation diagrams respectively, and the equalized results are shown in Fig. 3(g)-(i). For the case where the time step is 10−2 and 10−1, the constellation diagram after DD-LMS cracking can clearly see 64 clusters, indicating that the symbol-level encryption is almost invalid. In contrast, after $\varDelta T$ is increased to 1, there is a significant improvement in reliability. However, it must be pointed out that compared with the setting in the literature [12], this setting requires 124.02% more time to solve the chaotic equations.
Fig. 3. Illustration of the cracking of the symbol-level encryption caused by the DD-LMS algorithm under different $\varDelta T$.
In short, for practical systems, in order to prevent eavesdroppers from cracking the encryption method of constellation shifting through the DD-LMS algorithm, it is necessary to ensure that the sequence used for symbol-level encryption is not a slow-changing process. But according to the conventional scheme, this means that the computational complexity of the chaotic equations solving process increases dramatically, which will affect the real-time performance of the communication system. The random interleaving scheme proposed in [13] can circumvent this problem to a certain extent, but an additional shuffle sequence shared by the receiving and sending ends is required. Even if this scheme is used, there is still another incompatibility between CS encryption and DD-LMS, which is the error propagation problem in the equalization process.
As shown in Fig. 4, for a symbol S1 whose original coordinates are (-5, -3), assuming that it moves to the Y1 after being affected by channel impairments such as chromatic dispersion [14], polarization mode dispersion [15], and nonlinearity [16], it will not be misjudged during the equalization process of DD-LMS. That is, the error calculated by the equalizer is consistent with the actual channel impairment. But this is not the case when chaotic encryption is introduced. Take the symbol S2 in the fourth quadrant as an example. It is first moved to the position of S2’ by the chaotic sequence before being transmitted into the channel. Although the channel impairment it suffers is exactly the same as S1, it will face the problem of incorrect decision in the receiving end. During the equalization process of DD-LMS, the symbol Y2 will be erroneously classified to the cluster corresponding to the (7, -1) constellation point, causing the calculated error (represented by the red solid arrow) to be completely different from the actual channel impairment (represented by the black dashed arrow). This will affect DD-LMS's estimation of the channel response, and cause the accumulation of equalization errors in continuous iterations, resulting in a serious error propagation phenomenon.
Fig. 4. Illustration of the problem that the channel impairment cannot be correctly estimated when the encrypted signal is equalized using DD-LMS.
2.3 Interplay of the CS scheme and PS
Probabilistic shaping technology has been a key enabling technology for many recent record-setting optical fiber communication experiments [17]. Its main idea is to make the appearance probability of each constellation point in the constellation diagram be generated according to the famous Maxwell-Boltzmann (MB) distribution, which can be expressed as
where $\lambda \ge 0$ is the shaping parameter, which controls the degree of shaping, and the function $Z(\lambda )$ is chosen to normalize the probability distribution [18].Since the MB distribution has been proven to maximize capacity under power-constrained conditions, it must be pointed out that when the CS encryption technology is used in cascade with PS, the encrypted symbol distribution will deviate from the desired MB distribution, thereby affecting the shaping gain that the PS technique should achieve. Taking the k = 0.8 used in [13] as an example, the probability distributions of the transmitted sequence before and after symbol-level chaotic encryption are shown in Fig. 5. It can be seen that after encryption, the probability distribution of constellation points deviates seriously from the original MB distribution. Many symbols that should have been concentrated in the inner constellation diagram points diverge to higher power coordinates. This is obviously contrary to the intention of PS to reduce the transmission probability of high-power points to save the overall energy.
Fig. 5. Illustration of the change in the probability distribution of the PS-64QAM signal due to chaotic encryption.
In order to analyze the interplay between the CS encryption and PS, Monte Carlo simulation is launched to determine the loss of the shaping gain under different values of k. The simulation setup is shown in Fig. 6. First, the PS-64QAM sequence that obeys the MB distribution is generated according to the probabilistic amplitude shaping (PAS) architecture by constant composition distribution matching (CCDM) and low-density parity-check (LDPC) encoding. The sequence length is 216, and the corresponding shaping parameter is taken as $\lambda = 0.05$. When the value of the shaping parameter is larger, the incompatibility between CS and PS will be more obvious. According to [19], for PS-64QAM, the appropriate value of the shaping parameter ranges from 0 to 0.05, so we choose the upper bound of this range as an example to clearly show the interplay between the CS scheme and PS. After that, the chaotic sequences of the same length are solved with a time step of $\varDelta T = 1$, and then scaled according to the value of k to be loaded on the PS sequence. Then the encrypted sequence is transmitted through the AWGN channel. At the receiving end, the chaotic shifting sequence consistent with the transmitting end is subtracted from the obtained signal, and mutual information (MI) calculation is performed between the decrypted symbol sequence and the original sequence. In addition, we also calculated the BER between the received signal before decryption and the original signal. This result is to measure the effectiveness of the encryption. The larger the measured BER, the more effective the symbol-level encryption is.
Fig. 6. Simulation setup. PS: probabilistic shaping, AWGN: additive white Gaussian noise, MI: mutual information, BER: bit error rate.
The MI results obtained by Monte Carlo simulation are shown in Fig. 7(a). According to the literature [19], PS-64QAM with a shaping parameter of 0.05 can achieve the greatest MI performance improvement when SNR ≈ 12 dB, so we will focus on analyzing the MI performance of different values of k under this condition. For the two values when k < 1, the gain that the PS technique should bring can be kept relatively intact. When k = 1, only a very weak MI performance improvement can be observed. And the MI performance corresponding to the remaining two scaling factors is even lower than the uniform 64QAM. This phenomenon indicates that as the value of k increases, the distribution of the encrypted signal will deviate more seriously from the optimal MB distribution, which will aggravate the incompatibility with the PS technology. However, it is not feasible to significantly reduce the scaling factor in order to preserve the shaping gain. As shown in Fig. 7(b), when the value of k is small, such as 0.5, although the shaping gain of the PS technology can be relatively well preserved, the effectiveness of encryption is obviously insufficient. When the SNR is large enough, even if the received signal is directly demodulated without matching decryption at the receiving end, the BER compared with the original data at the transmitting end is only in the order of 10−2, which is far from guaranteeing the confidentiality of communication. After further increasing the value of k, the effectiveness of encryption can be significantly improved, but for actual systems, if the CS scheme is adopted, the trade-off between system performance and encryption effectiveness must be considered. The above analysis shows that in order to better retain the shaping gain brought by the PS technique while ensuring the reliability of confidentiality, it is necessary to propose a new symbol-level encryption method compatible with PS.
Fig. 7. The influence of different values of scaling factors k on (a) MI performance and (b) encryption effectiveness of PS-64QAM signal.
2.4 Proposed encryption scheme based on phase ambiguity
The proposed novel symbol-level encryption method is inspired by the scheme in [6] and the concept of four-fold phase ambiguity in coherent receivers [20]. In the literature [6], each dimension in the high-order chaotic equation system is assigned to correspond to each concentric circle in the constellation diagram, and then the chaotic sequence of this dimension is converted into a phase rotation within the concentric circle through the remainder operation. Through such a scheme, the encrypted symbols will still be located on the standard point coordinates of the constellation diagram, which can effectively avoid the incompatibility problem with DD-LMS and PS. However, the disadvantage of this solution is that since the number of constellation points on each concentric circle in the QAM constellation diagram is inconsistent, as the order of QAM increases, the number of chaotic equations required will increase sharply. For example, for the 16QAM modulation format used in [6], a four-dimensional chaos is required, if the modulation order is increased to 64QAM, then a 10-dimensional chaos will be required. Obviously, this scheme has practical defects when it is extended to higher order modulation orders. The solution we propose is to use the four-fold symmetry of the QAM constellation to change the symbol-level encryption to a phase ambiguity. The encrypted signal sequence $\{{S_{_i}^{PA}} \}$ can be expressed as:
Fig. 8. Illustration of the proposed PA scheme for 64QAM, (a) constellation diagram before encryption, and constellation diagrams after encryption, when (b) ${K^{Conjugate}} = 0$, and (c) ${K^{Conjugate}} = \textrm{1}$.
The hardware implementation complexity comparison between the proposed encryption algorithm and the CS scheme is shown in Table 1. When performing CS encryption, 2N real multipliers and 2N real adders are required to scale $\{{{x_i}} \}$ and $\{{{y_i}} \}$, and add them to $Re ({S_i})$ and ${\mathop{\rm Im}\nolimits} ({S_i})$ respectively. In the PA algorithm, firstly, N real adders and N remainder takers are used to calculate $\{{K_i^{Rotation}} \}$, and N sign operators and N XOR operators are used to realize the calculation of $\{{K_i^{Conjugate}} \}$. It is worth pointing out that for $\{{{S_i}} \}$ and $\{{S_{_i}^{PA}} \}$, the sets formed by the absolute values of their real and imaginary parts are exactly the same. Therefore, after obtaining $\{{K_i^{Rotation}} \}$ and $\{{K_i^{Conjugate}} \}$, only 2N negators are required at most (corresponding to the situation where all real and imaginary parts need to be negated) to implement encryption according to the phase ambiguity scheme. In general, the hardware implementation complexity of the two schemes is similar, but as we pointed out in the simulation results in Section 3, the PA scheme has obvious advantages over CS in terms of time complexity.
Table 1. Hardware Complexity Comparison of CS algorithm and PA algorithm
3. Simulation analysis
In order to verify the effectiveness of the proposed encryption scheme, both numerical simulation and commercial software simulation were carried out. First, we analyze the encryption effectiveness of the proposed algorithm in the case of a low time step. Taking the case of $\varDelta T\textrm{ = 1}{\textrm{0}^{ - 5}}$ as an example, when $\{{{x_i}} \}$ and $\{{{y_i}} \}$ change slowly, the change of $\{{K_i^{Rotation}} \}$ is as shown in Fig. 9. It can be seen that even when the time step is very small, the encryption sequence generated by the proposed PA scheme will still change rapidly. This means that in practical applications, the proposed scheme can effectively remove the requirement for a large value of $\varDelta T$ in the traditional scheme, and greatly reduce the time required to solve the chaotic equations in the encryption and decryption process. In subsequent simulations, the values of $\varDelta T$ are all set to 10−5 for the proposed PA encryption method. Compared with the $\varDelta T\textrm{ = 1}$ required by the CS method, 62% of the computing time can be saved in the process of solving the chaotic equations.
Fig. 9. Time domain diagram of $\{{K_i^{Rotation}} \}$, $\{{{x_i}} \}$ and $\{{{y_i}} \}$, when $\varDelta T\textrm{ = 1}{\textrm{0}^{\textrm{ - 5}}}$.
Next, we use the simulation system in Fig. 6 again to verify the compatibility of the proposed scheme with probabilistic shaping through the Monte Carlo method. It can be seen from Fig. 10(a) that the proposed scheme avoids the problem that the probability distribution deviates from the MB distribution caused by the CS encryption method. The symbols after phase rotation and conjugate calculation are still located on the standard constellation point coordinates, so the shaping gain can be preserved intact. This advantage of the PA scheme is also applicable under other shaping parameters or modulation format orders. In addition, the encryption effectiveness comparison between the proposed scheme and the CS scheme is as shown in Fig. 10(b), which is demonstrated by comparing the BERs between the received undecrypted signals and the original signals. The value of k in the traditional CS scheme is set to 0.8, which is the optimal parameter selection suggested in [13]. It can be seen that the proposed PA scheme can achieve a BER higher than 2.9×10−1 in the entire tested SNR range, while the BER of the CS scheme has a significant drop as the SNR increases. After the SNR is higher than 13 dB, the effectiveness of the symbol-level encryption of the proposed scheme is more than twice that of the traditional scheme. In short, the proposed scheme not only fully retains the gains brought by the PS technique, but also improves the reliability of symbol-level encryption.
Fig. 10. Monte Carlo simulation results of (a) MI performance and (b) encryption effectiveness for PS-64QAM.
Furthermore, we built an IM/DD system in the VPItransmissionMaker software to simulate the transmission performance of the DMT signal using the proposed encryption scheme. The simulation setup is as shown in Fig. 11. At the transmitting end, we first use an offline MATLAB program to generate the DMT signal, the FFT length is 64, the cyclic prefix length is 16, and the number of OFDM symbols is 120. In order to achieve Hermitian symmetry, only half of the 64 subcarriers can be used to carry data, and the other half need to carry their corresponding complex conjugates. Among the 32 subcarriers that carry data, 5 are used as guard bands, 4 are used to carry training sequences, and the rest are used to carry encrypted information. After the DMT signal sequence is imported into the VPItransmissionMaker, it is first converted into an electrical signal through a digital-to-analog converter (DAC), and then amplified by an electrical amplifier (EA) to a peak-to-peak value of 2Vpp. After that, it is fed into the Mach-Zehnder modulator (MZM) for electro-optical conversion (E/O). The optical signal input to the MZM is generated by an external cavity laser (ECL) with a center wavelength of 1550 nm, a linewidth of 100kHz, and an output power of 14.5dBm. The modulated optical signal is first transmitted through a 5 km standard single-mode fiber (SSMF), and then is input into a variable optical attenuator (VOA). After the VOA adjusts the received optical power to an appropriate value, the optical signal is fed into the photodiode (PD) to complete the photoelectric conversion (O/E). The received DMT signal is collected and stored after being sampled by the ADC, and an offline digital signal processing (DSP) program is used to calculate the BER. The detailed parameter settings of each device in the system are shown in Table 2.
Table 2. Detailed Simulation Parameters
In the simulation process, we used the uniformly distributed 64QAM and the PS-64QAM with a shaping parameter of 0.05 for comparison. In order to ensure the same net rate, the baud rate of 64QAM-DMT is set to 10GBaud, while the baud rate of PS-64QAM-DMT is 11.5GBaud. The BER results of the two signals after using different encryption schemes are shown in Fig. 12. It can be seen that when chaotic encryption is not used, with the 20% soft-decision forward-error-correction (SD-FEC) threshold of 2×10−2 as a Ref. [21], the PS technique can achieve a sensitivity gain of approximately 1.24 dB. After introducing the chaotic encryption scheme based on constellation shifting, the performance of both 64QAM-DMT and PS-64QAM-DMT will deteriorate. For the uniformly distributed signal, the received optical power (ROP) cost increases by about 0.71 dB, while for the probabilistically shaped signal, it is about 0.79 dB. In contrast, the proposed encryption scheme based on phase ambiguity hardly causes any loss of receiving sensitivity. For the two modulation formats tested, the sensitivity changes before and after encryption are both less than 0.2 dB. This result once again confirms that the proposed scheme can better coexist with PS technology.
In addition, we have also studied the BER changes of the two encryption methods after using DD-LMS for equalization. As shown in Fig. 13, after using a DD-LMS equalizer with a taps order of 21 for equalization, the BER of the PA scheme can be improved to a small extent. However, under the same equalizer settings, the BER of the CS scheme will deteriorate drastically. The reason for the degraded BER performance after equalization in the CS scheme is erroneous convergence. Since the taps in the DD-LMS equalizer are continuously updated with iterations, once an incorrect estimation of the channel response as shown in Fig. 4 occurs, the error will continue to propagate in subsequent iterations. And the larger the update rate, the more prone the equalizer is to erroneous convergence. According to the simulation results, even if the update rate of the filter is reduced by an order of magnitude, the BER performance after equalization is still not as good as before. This phenomenon shows that the proposed scheme has better compatibility with DD-LMS in the post-equalization process than the traditional CS scheme, because it can effectively avoid the occurrence of erroneous convergence.
4. Experimental verification
An experimental verification is carried out to confirm the performance of the proposed encryption algorithm. At the transmitting end, an arbitrary waveform generator (AWG) with a sampling rate of 64GSa/s is used to perform the digital-to-analog conversion of the DMT signal, and the peak-to-peak value of the output radio frequency (RF) signal is 0.25Vpp. A linear wideband amplifier with a maximum gain of 23dB was used to amplify the peak-to-peak value to 2Vpp before feeding the RF signal into the MZM. The parameters of the laser, MZM, fiber, and PD are the same as the settings in the previous section. For the received signal detected by PD, a digital phosphor oscilloscope (DPO) with a sampling rate of 100GSa/s was used for analog-to-digital conversion.
The results obtained from the experiments are shown in Fig. 14. The BER performance of the six tested signals under different ROPs is shown in Fig. 14(a). The performance of the signals using PS technique is better than that of the uniformly shaped signals. Among the three signals using PS-64QAM-DMT, the performance of the unencrypted signal is the best, the performance of the proposed PA encryption scheme is slightly inferior, and the performance of the CS scheme is significantly deteriorated. Taking the 20% SD-FEC threshold as a reference, in the experimental results, the sensitivity cost before and after encryption by the proposed scheme is about 0.3dB, while the cost of the traditional CS scheme is as high as 1dB. This result again proves the advantage of the proposed scheme in compatibility with PS signals. Similarly, for the two encrypted signals using the 64QAM-DMT modulation format, the sensitivity penalty caused by using the PA and CS schemes is about 0.4dB and 1.3dB, respectively.
Fig. 14. Experimental results of (a) BER versus ROP for 10GBaud uniform 64QAM-DMT and 11.5GBaud PS-64QAM-DMT, and (b) BER before and after equalization of DD-LMS.
The BER changes before and after DD-LMS equalization in the experimental results are shown in Fig. 14(b). When the update rate of the equalizer is large, using DD-LMS for signals based on the CS scheme will result in serious erroneous convergence. As a result, the BER over the entire measurement range has deteriorated to an intolerable level. After reducing the update rate by an order of magnitude, for the case where the ROP is lower than -4dBm, the use of DD-LMS can improve the performance, but for the data points with the ROP of -4dBm and -3dBm, the equalizer still fails to converge correctly. In contrast, for the signal using the PA scheme, the performance improvement can already be achieved by equalization when the update rate is 1×10−4. The above results again verify the superiority of the proposed encryption algorithm in compatibility with DD-LMS equalizers.
5. Conclusion
In this paper, a chaotic encryption scheme based on phase ambiguity is proposed. Compared with the current mainstream constellation shifting encryption scheme, the proposed symbol-level encryption method has the following advantages: 1. When the chaotic equation is solved in a very small time step, the reliability of encryption can still be guaranteed, so the complexity in practical applications can be reduced; 2. When the probability shaping technology is used in cascade, the encryption process will not cause the probability distribution of the signal to deviate from the MB distribution, so the shaping gain brought by the PS technology can be completely retained; 3. When the DD-LMS algorithm is used for post-equalization, the proposed encryption scheme will not cause the error convergence of the equalizer. In summary, the proposed solution is a feasible solution for future security-enhanced optical fiber communications.
Funding
National Key Research and Development Program of China (2019YFB1803701); National Natural Science Foundation of China (61727817, 61835002, 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.
References
1. K. Wang, W. Zhou, L. Zhao, F. Zhao, and J. Yu, “Bi-Directional OFDM Truncated PS-4096QAM Signals Transmission in a Full-Duplex MMW-RoF System at E-Band,” J. Lightwave Technol. 39(11), 3412–3419 (2021). [CrossRef]
2. Y. Chen, S. Shen, Q. Zhou, S. Yao, R. Zhang, S. Omar, and G. Chang, “A Reliable OFDM-Based MMW Mobile Fronthaul With DSP-Aided Sub-Band Spreading and Time-Confined Windowing,” J. Lightwave Technol. 37(13), 3236–3243 (2019). [CrossRef]
3. Z. Dong, Y. Chen, D. Zou, X. Zhao, L. Zhou, and F. Li, “DMT Transmission in Short-Reach Optical Interconnection Employing a Novel Bit-Class Probabilistic Shaping Scheme,” J. Lightwave Technol. 39(1), 98–104 (2021). [CrossRef]
4. J. Ding, J. Zhang, Y. Wei, F. Zhao, C. Li, and J. Yu, “Comparison of Geometrically Shaped 32-QAM and Probabilistically Shaped 32-QAM in a Bandwidth-Limited IM-DD System,” J. Lightwave Technol. 38(16), 4352–4358 (2020). [CrossRef]
5. Z. Zhang, J. Li, Y. Wang, and Y. Qin, “Direct Detection of Pilot Carrier-Assisted DMT Signals With Pre-Phase Compensation and Imaginary Noise Suppression,” J. Lightwave Technol. 39(6), 1611–1618 (2021). [CrossRef]
6. Z. Wang, Y. Xiao, S. Wang, Y. Yan, B. Wang, Y. Chen, Z. Zhou, J. He, and L. Yang, “Probabilistic shaping based constellation encryption for physical layer security in OFDM RoF system,” Opt. Express 29(12), 17890–17901 (2021). [CrossRef]
7. J. Ren, B. Liu, D. Zhao, S. Han, S. Chen, Y. Mao, Y. Wu, X. Song, J. Zhao, X. Liu, and X. Xin, “Chaotic constant composition distribution matching for physical layer security in a PS-OFDM-PON,” Opt. Express 28(26), 39266–39276 (2020). [CrossRef]
8. J. Shen, B. Liu, Y. Mao, R. Ullah, J. Ren, J. Zhao, and S. Chen, “Enhancing the Reliability and Security of OFDM-PON Using Modified Lorenz Chaos Based on the Linear Properties of FFT,” J. Lightwave Technol. 39(13), 4294–4299 (2021). [CrossRef]
9. S. Chen, B. Liu, J. Ren, Y. Mao, R. Ullah, X. Song, Y. Bai, L. Jiang, S. Han, J. Zhao, Y. Wan, X. Zhu, and J. Shen, “A 7D Cellular Neural Network Based OQAM-FBMC Encryption Scheme for Seven Core Fiber,” J. Lightwave Technol. 39(22), 7191–7198 (2021). [CrossRef]
10. A. Sultan, X. Yang, A. A. E. Hajomer, and W. Hu, “Chaotic Constellation Mapping for Physical-Layer Data Encryption in OFDM-PON,” IEEE Photonics Technol. Lett. 30(4), 339–342 (2018). [CrossRef]
11. Y. Huang, Y. Chen, K. Li, Y. Li, J. Ma, and J. Yu, “Multi Scrolls Chaotic Encryption for Physical Layer Security in CO-OFDM,” in Optical Fiber Communication Conference (OFC) 2019, OSA Technical Digest (Optical Society of America, 2019), paper Th1J.8.
12. F. Wang, B. Zhu, K. Wang, M. Zhao, L. Zhao, and J. Yu, “Physical Layer Encryption in DMT Based on Digital Multi-Scroll Chaotic System,” IEEE Photonics Technol. Lett. 32(20), 1303–1306 (2020). [CrossRef]
13. B. Zhu, F. Wang, and J. Yu, “A Chaotic Encryption Scheme in DMT for IM/DD Intra-Datacenter Interconnects,” IEEE Photonics Technol. Lett. 33(8), 383–386 (2021). [CrossRef]
14. L. Sun, J. Du, C. Wang, Z. Li, K. Xu, and Z. He, “Frequency-resolved adaptive probabilistic shaping for DMT-modulated IM-DD optical interconnects,” Opt. Express 27(9), 12241–12254 (2019). [CrossRef]
15. F. Li, X. Li, J. Yu, and L. Chen, “Optimization of training sequence for DFT-spread DMT signal in optical access network with direct detection utilizing DML,” Opt. Express 22(19), 22962–22967 (2014). [CrossRef]
16. J. Wei, T. Rahman, S. Calabrò, N. Stojanovic, L. Zhang, C. Xie, Z. Ye, and M. Kuschnerov, “Experimental demonstration of advanced modulation formats for data center networks on 200 Gb/s lane rate IMDD links,” Opt. Express 28(23), 35240–35250 (2020). [CrossRef]
17. J. Cho and P. J. Winzer, “Probabilistic Constellation Shaping for Optical Fiber Communications,” J. Lightwave Technol. 37(6), 1590–1607 (2019). [CrossRef]
18. M. N. Tehrani, M. Torbatian, H. Sun, P. Mertz, and K. Wu, “A Novel Nonlinearity Tolerant Super-Gaussian Distribution for Probabilistically Shaped Modulation,” in Proceedings of IEEE European Conference on Optical Communications (IEEE, 2018), pp. 1–4.
19. F. A. Barbosa, S. M. Rossi, and D. A. A. Mello, “Phase and Frequency Recovery Algorithms for Probabilistically Shaped Transmission,” J. Lightwave Technol. 38(7), 1827–1835 (2020). [CrossRef]
20. G. Lu, T. Sakamoto, and T. Kawanishi, “Flexible high-order QAM transmitter using tandem IQ modulators for generating 16/32/36/64-QAM with balanced complexity in electronics and optics,” Opt. Express 21(5), 6213–6223 (2013). [CrossRef]
21. D. Che, C. Sun, and W. Shieh, “Maximizing the spectral efficiency of Stokes vector receiver with optical field recovery,” Opt. Express 26(22), 28976–28981 (2018). [CrossRef]
