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Abstract
In this paper, we propose a multi-dimensional multiplexing scheme for space division multiplexing optical transmission systems based on quaternion chaotic encryption. A constellation compression shaping mapping method is designed to replace the traditional 2n mapping scheme, which leads to flexible encoding modulation. In order to achieve orthogonality between data symbols and effectively suppress crosstalk, the spectrally superposed transmission of three-dimensional (3D) constellation data is carried out by code division multiplexing and 4D carrier-less amplitude phase joint modulation techniques based on orthogonal basis. The Chen's chaotic model is utilized to generate the rotation angle of the constellation points, which enables data encryption without changing the transmitted power, enhancing system’s security. The feasibility and superiority of the proposed scheme are successfully verified by constructing an experimental platform for a seven-core fiber transmission system.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The large-scale use of 5 G has spawned many burgeoning applications and industries, such as the meta-universe, intelligent industrial parks, and Internet of Things (IoT). The smart industry, smart society and smart home of the future cannot be separated from the support of fiber backbone networks with higher speeds and larger capacity. In recent years, space division multiplexing (SDM) optical transmission has been regarded as the second revolution of optical fiber transmission technology after wavelength division multiplexing (WDM) technology, which has been accorded great importance by various national scientific research institutions [1,2]. Theoretical models, coding modulation techniques, system hardware, optoelectronic materials, system algorithms, and other areas have seen significant advancements in the study up to this point [1–6]. While the advent of new sectors like IoT has simplified people’s lives, its rapid growth has raised some complex data security concerns. How to prevent the theft or destruction of the data generated, shared, and collected by more than 10 billion smart devices is, however, a complex and challenging problem to resolve.
To further facilitate the transmission capacity of space division multiplexing optical transmission system, only in terms of coding modulation, it is necessary to use high-order modulation format, orthogonal multiplexing schemes, and high-dimension coding as many as possible to apply all available degrees of freedom on optical fiber transmission systems [7]. The best way to improve optical transmission rate, capacity, or spectral efficiency of space division multiplexing is to employ multi-dimensional multiplexing and high-dimensional constellations. To transmit data at Tbit/s, 16-ary quadrature amplitude and phase modulation (16-QAM) was implemented in a THz-over-Fiber wireless communication system that combines WDM and SDM [8]. Chen et al. reported a 25-channel WDM system with polarization division multiplexing 16-QAM transmitting over 800 km standard single-mode fiber, utilizing nonlinear frequency division multiplexing and artificial neural network-based nonlinear spectrum equalization [9]. A four-dimensional constellation was established by gradually decomposing and superimposing the low-dimensional constellation, which was successfully demonstrated in seven-core optical fiber transmission [10]. However, to data, there has been no consummate design scheme for high-dimensional or multi-dimensional modulation format for space division multiplexing optical transmission systems. Therefore, the multi-dimensional multiplexing and high-dimensional constellation-coded modulation schemes must be further investigated.
Information resources are a source of strength to promote the high-quality development of the social digital economy. The safe and efficient circulation and use of information resources can stimulate the potential of information elements, energize the real economy and comprehensively promote the progress of various industries. Thus, the secure optical transmission of data is a major challenge facing the world [11]. Physical layer encryption is often utilized as a high-security optical transmission scheme in optical fiber transmission systems. Chaotic encryption is the most commonly used data encryption method due to its superior characteristics such as noise-like, high parameter key based sensitivity, and large bandwidth [12–18]. In most cases, it immediately affects either the phase or amplitude. One is to randomly swap the locations of the constellation points, and the other is to randomly compensate or rotate the phase of the constellation points. Bowen Zhu et al. used random interleaving to encrypt probabilistic shaping 64-QAM signals in the intensity modulation and direct detection (IM/DD) discrete multi-tone transmission system [12]. Chaotic neural network and nonlinear encryption were applied to the orthogonal frequency division multiplexing (OFDM) WDM-PON system. A key sensitivity of 10−15 and a key space of more than 10279 were achieved [13]. It can be seen that physical layer high-dimensional constellation encryption has the potential to become a research hotspot [14,15].
This paper proposes a multi-dimensional multiplexed high-security transmission scheme based on quaternion chaotic encryption. Three-dimensional (3D) 24-ary constellation mapping is designed by constellation compression shaping (CCS), and three-dimensional constellation points are encrypted by quaternion chaotic encryption, eliminating the singularity of rotation of 3D constellation points and reducing redundancy. Quaternion chaotic encryption is a unique encryption method that involves constellation perturbation and rotation. This method utilizes the chaotic sequences generated by the Chen’s chaos model to rotate the three-dimensional constellation points, resulting in the encryption of data. The encrypted data are formed into data to be sent through code division multiplexing and four-dimensional carrier-less amplitude phase (4D-CAP) joint modulation to implement orthogonal multiplexing and dimensionality reduction transmission. Incidentally, the orthogonal basis of code division multiplexing and 4D-CAP also has the characteristics of data encryption, which can be considered as secondary encryption. A multi-core optical transmission experimental system has successfully demonstrated the viability and security of multi-dimensional multiplexing transmission systems.
2. Principle
The multi-dimensional multiplexed high-security optical transmission scheme based on quaternion chaotic encryption is shown in a block diagram in Fig. 1. A pseudo-random binary sequence generates original data on the transmitting side. The original data $\{ {b_i}\}$ becomes a parallel data stream after series-to-parallel (S/P) conversion. The sequence of 3D constellations is obtained from a parallel data stream of 3D constellation compression shaping modulation. More specifically, Chen's chaotic circuit generates 3D chaotic sequences (X, Y, Z), which is transformed into a set of rotational angles ${\{ \phi ,\theta ,\varphi \} _i}$ during quaternion rotational encryption. Namely, quaternion encryption consists of rotating the 3D constellation points around the three axes x, y and z, respectively, and setting the angle as $\phi ,\theta ,\varphi$. Chen's chaotic circuit’s parameters and initial values are set as keys here. The training sequence is then added to the encrypted data to get the data stream $\{ {m_{ij}}\}$ generated from the CCS-16-8 constellation mapping table, which can be considered a reference standard for equalization compensation of received data. The data stream $\{ {m_{ij}}\}$ is multiplied by the orthogonal matrix H of the code division multiplexing to obtain a multiplexed parallel data stream. A 4D-CAP technique based on orthogonal basis is employed to stack multiplexed parallel data streams to implement the transmitted data $\{ {s_i}\}$. The transmitted data arrives at the receiver via a multi-core fiber transmission link. For the received data, 4D-CAP demodulation and code division demodulation are performed first, and then the training sequence is removed from the received data. To balance and compensate for the separated information at the receiver, the training sequence is required due to the presence of phase noise and power loss. Then the quaternion decryption, 3D constellation de-mapping, and decompression are performed on the compensated data successively to recover the binary data at the receiver. Finally, the bit error rate (BER) is calculated as the evaluation parameter of system transmission performance.
Fig. 1. Block diagram of multi-dimensional multiplexing high-security optical transmission scheme based on quaternion chaotic encryption.
2.1 Three-dimensional constellation compression shaping modulation and demodulation methods
The constellation compression shaping modulation is designed to construct a 3D non-2n constellation model. Figure 2(a), (b) show the flow diagram of 3D CCS modulation and demodulation. As shown in Fig. 2(a), the constellation compression modulation [19] is adopted to convert 32-ary modulation format into 16-ary and 16-ary into 8-ary, respectively. The resulting data are marked as CCS-32-16 and CCS-16-8. In addition, the encoding diagram of constellation compression modulation of CCS-32-16 is illustrated in Fig. 2(c). The squares represent a symbol, and the squares with different colors represent different symbols. After that, convert one or more $\{ {b_1}{b_2}{b_3}{b_4}{b_5}\}$ to $\{ b_1^{\prime}b_2^{\prime}b_3^{\prime}b_4^{\prime}\}$, and then label $\{ {m_1}{m_2}\}$ to get the final data $\{ b_1^{\prime}b_2^{\prime}b_3^{\prime}b_4^{\prime}{m_1}{m_2}\}$. The CCS modulation also has the effect of probabilistic shaping. The probability ratio of “0” and “1” of CCS data is about 0.64:0.36. Afterwards, CCS-32-16 and CCS-16-8 are mapped to 3D constellation points, respectively.
Fig. 2. The flow diagram of 3D constellation compression shaping: (a) modulation and (b) demodulation, (c) schematic diagram of CCS-32-16 encoding.
Additionally, Table 1 provides the mapping relationships between constellation point coordinates and binary data. The average energy of the 3D data stream after CCS data mapping is about 2.31. The demodulation flow diagram of 3D constellation compression shaping is shown in Fig. 2(b). After constellation de-mapping, data recovery employs constellation decompression.
Table 1. Mapping relations between binary data and coordinates of 3D constellation points
The 3D constellation map procedure proposed in this paper is shown in Fig. 3.
- 1) The CCS-16-8 symbols are mapped to the eight vertices of the cube centered on the origin in the three-dimensional constellation diagram, as shown in Fig. 3(a);
- 2) To create four vertices, take four CCS-32-16 symbols and transfer them to the cube’s four sides like a standard tetrahedron. Connecting lines have nicely finished off shown in Fig. 3(b);
- 3) The other twelve CCS-32-16 symbols are mapped respectively to the vertices of an equilateral triangle with the twelve sides of the cube as the base and an angle of 45 degrees to get the final three-dimensional constellation diagram like in Fig. 3(e).
Fig. 3. (a)-(e) The 3D constellation design flow chart, (f)-(h) different views of the 3D constellation.
If only the twelve constellation points are connected, Fig. 3(c) is finished. To observe the three-dimensional constellation map more directly, we connect all constellation points differently, and Fig. 3(d) and (e) are represented, respectively. Moreover, Fig. 3(f) and (g) are the front view and top view of (e), respectively. And the left view of (e) is the same as the front view (f). Figure 3(h) is a top view of (d).
2.2 Quaternion Chen's chaotic encryption method
To increase data transmission security, we propose a quaternion chaotic encryption method. The main application of quaternions is to rotate the 3D constellation coordinates. In contrast to a rotation matrix, rotation vector, and Euler angle, quaternions are compact, have no singularities, and do not lose freedom. At first, chaotic sequences (X, Y, Z) are generated by Chen's circuit, and its mathematical model [20] is as follows:
Fig. 4. Phase diagram of chaotic model: (a) about xyz axis, (b) about variables xy axis, and (c) about yz axis; constellation diagram: (d) before quaternion chaotic encryption, (e) after quaternion rotation encryption, (f) a quaternion rotation example.
The rotation encryption angle ${\{ \phi ,\theta ,\varphi \} _i}$ of three-dimensional constellation points is transformed from chaotic sequences ${\{ X,Y,Z\} _i}$:
The quaternion form of rotation angles ${\{ \phi ,\theta ,\varphi \} _i}$ are:
2.3 Code division multiplexing and 4D-CAP joint modulation method based on orthogonal basis
To reduce the symbolic crosstalk during 3D data transmission, we apply the code division multiplexing and 4D-CAP joint modulation based on orthogonal basis to modulate the data. First, an orthogonal transformation is applied to the data stream after quaternion encryption:
3. Experiments and results
We construct a seven-core optical fiber transmission system, shown in Fig. 6, to verify the feasibility of multi-dimensional multiplexing optical transmission based on quaternion chaotic encryption. At the transmitter, offline digital signal processing (DSP) performs constellation compression shaping modulation, chaotic encryption, adding training sequences, code division multiplexing, and 4D-CAP joint modulation on the original data to generate the encryption data for transmission. The transmitted digital data is converted to analog data by an arbitrary waveform generator (AWG, TekAWG70002A) with a sampling rate of 25 GSa/s. The radio frequency signal is amplified by an electric amplifier and fed into a Mach-Zehnder modulator (MZM) for intensity modulation. In this experiment system, an external cavity laser is employed to produce 193.4 THz, 1550 nm, 14.5 dBm output light as the optical input of MZM. After that, the output data is amplified by an erbium-doped fiber amplifier to ensure that the seven-core fiber has enough input optical power. The seven-core fiber adopted is a commercial weak-coupled multi-core fiber with single core, coating and cladding diameters of 8 µm, 245 µm and 150 µm, respectively, and core spacing of 41.5 µm. The amplified data is coupled by an optical coupler and then transmitted through a fan-in the device to a 2 km seven-core weakly coupled fiber for transmission. The fan-out device separates the seven cores of the data and transmits them separately through a single-core optical fiber. A variable optical attenuator is adopted to adjust the received optical power for measurement. The received data is converted into digital data by a photodiode, and then collected by a mixed signal oscilloscope (MSO, TekMSO73304DX) with a sampling rate of 50 GSa/s. Finally, the offline DSP is used to demodulate, decrypt, and recover the data in the receiving end.
Fig. 6. Experimental setups (DSP: digital signal processing, ECL: external cavity laser, AWG: arbitrary waveform generator, EA: electrical amplifier, MZM: Mach-Zehnder modulator, EDFA: erbium-doped fiber amplifier, OC: optical coupler, MCF: multi-core fiber, VOA: variable optical attenuator, PD: photodiode, MSO: mixed signal oscilloscope).
Figure 7 illustrates the BER of multidimensional multiplexed data with quaternion chaotic encryption after transmission through a seven-core fiber as a function of the received optical power. As can be seen in Fig. 7, the measured BER curves of seven fiber cores after the transmission follow the same trend, and the transmitted BER curves are consistent with the principles of information theory. When the BER is 3.8×10−3, the maximum difference of received optical power between fiber cores is about 1.14 dB. As the received optical power increases, the difference in the receiving sensitivity between the fiber cores increases. This phenomenon is because, in addition to a non-uniform core loss per core during the fabrication of a multi-core fiber, there is also a large amount of information carried by the 3D signal, resulting in a large contribution of symbol discrimination to the BER. Besides, Fig. 7 also shows that the BER received by the illegal receiver is around 0.5, and the illegal receiver cannot get the correct information under regular interception. In addition, when the received optical power is further increased to -6 dBm or even greater, the system always measures a BER of 0. This is because the initial information bit is around 104, hence the lowest achievable BER is only on the order of 10−4. By increasing the amount of initial information bit, we can achieve a smaller BER. Figure 8 gives the constellation obtained by illegal and legitimate reception of core 4 after transmission, when the received optical power is -8 dBm. Figure 8(a)-(c) shows the 3D constellation image received illegally and its projection on the x-y and x-z planes, while Fig. 8(d)-(f) shows the 3D constellation obtained normally and its projection on the x-y and x-z planes. This also indicates that our proposed quaternion chaotic encryption method is efficient and viable, and can make transmission more secure.
Fig. 8. Constellations of illegal and legal reception: (a) illegal constellation, (b) x-y plane projection of (a), (c) x-z plane projection of (a), (d) legal constellation, (e) x-y plane projection of (d), (f) x-z plane projection of (d).
We thoroughly investigate the sensitively of the initial values and circuit parameters to determine the proposed system’s encryption security level. After multi-core fiber transmission, when the received optical power is -7 dBm, the initial value (x0, y0, z0) of core 7 is set to (0.20, -0.10, 0.20). As shown in Fig. 9(a), when x0 changes by 10−17, or y0 changes by 10−18, or z0 changes by 10−16, the BER obtained is the same as that under the correct key, but once it is greater than this critical value, BER will be increased significantly, even close to 0.5, and this means that the information cannot be recovered correctly. The subfigure in Fig. 9(a) shows the difference between the chaotic sequence at the initial value of x0 and the chaotic sequence when x0 is varied by 10−16. The two lines are different; hence, it can be found that our proposed quaternion chaotic encryption method is hypersensitive to the initial values. It can be seen from Fig. 9(b) that the system is also highly sensitive to the circuit parameters (a, b, c) of Chen's chaotic model. When a changes to 10−16, or b changes to 10−17, or c changes to 10−16, there is no obvious change in the BER compared to the original correct parameter value. When the change of the circuit parameter a is larger than 10−15, the BER of system is dramatically increased and has no significant change when it continues to increase. Similar conclusions can be drawn for circuit parameters b and c.
We send a picture of a dog standing on grass into the proposed system in order to further investigate the efficacy of our encryption approach in information transfer. Figure 10(a), (d) present the image and its histogram before encryption. Following encryption, the illegally received image and histogram are shown in Fig. 10(b), (e), while the normally received image and histogram are displayed in Fig. 10(c), (f). It is evident that the illegal receiver fails to properly receive the image compared to the normal receiver. Additionally, the histograms of illegal decryption differ significantly from those of normal decryption, resulting in blurry images. This demonstrates the illegal receiver’s inability to decipher any meaningful information. Furthermore, Fig. 11 showcases the normal and illegal information bit stream received before and after encryption. It is also apparent that illegal reception fails to retrieve the correct information, underscoring the importance of accurate decryption operations.
Fig. 10. Images: (a) before encryption, (b) after illegal reception, (c) after normal reception, and histograms: (d) before encryption, (e) after illegal reception, (f) after normal reception.
Fig. 11. The information bit stream: (a) before encryption, (b) after illegal reception, and (c) after normal reception.
4. Conclusions
The paper proposes a high-security multi-core optical transmission method based on quaternion encryption. 3D coded modulation is used in conjunction with constellation compression shaping, constellation design recombination, and mapping to make high-dimensional constellation construction formation easier. The 3D constellation points are mixed up in quaternion chaotic encryption, which increases the information’s chaos and security. The 4D-CAP signal is successfully transmitted over a seven-core optical fiber. The experimental results demonstrated that under legal reception, when the BER is 3.8×10−3, the maximum difference of the received optical power between the fiber cores is 1.14 dB. Since the illegal receiver cannot get the correct information, the proposed scheme is feasible and has strong security. Multi-dimensional multiplexed coded modulation based on quaternion chaotic encryption has broad application prospects in multi-dimensional multi-core optical transmission in the future.
Funding
National Key Research and Development Program of China (2018YFB1800901); National Natural Science Foundation of China (61835005, 61935005, 62171227, 62205151, 62225503); Jiangsu Provincial Key Research and Development Program (BE2022055-2, BE2022079); The Natural Science Foundation of the Jiangsu Higher Education Institutions of China (22KJB510031); The Startup Foundation for Introducing Talent of NUIST.
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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