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Abstract
A 4-dimensional (4-D) constellation construction and encryption scheme of dimension dissecting reorganization are proposed in this paper. In this scheme, the high-dimensional constellation is constructed by gradually decomposing and superimposing the low-dimensional constellation, and the mapping dimension, phase, and arrangement order of signals are scrambled to realize the encryption. This scheme uses the evolution from low dimension to high dimension to reduce the difficulty of constructing a high-dimensional constellation, and the confusion between dimensions facilitates the encryption of high-dimensional information. To verify the performance, an experiment to demonstrate the transmission of 46.7 Gb/s 4-D constellation mapping the intensity modulation/direct detection carrierless amplitude and phase on 2 km 7-core optical fiber has been successfully carried out.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, the continuous progress of information technology, such as meta-universe, virtual reality, the Internet of Things, 4 K video, and so on, has promoted the continuous development of communication technology in high-speed and large-capacity demand. Multiplexing and high dimensional constellation modulation are effective methods to improve information rate, capacity, or spectrum efficiency. The space division multiplexing (SDM) technology is often realized by multicore fiber, which can realize the independent transmission of each fiber core due to the ultra-low crosstalk between the fiber cores [1]. At the same time, combined with large-scale or high-dimensional modulation, the transmission rate of information can be better improved [2–5].
One of the main forms of modulation is the constellation diagram. The constellation diagram shows the projection of different information points on different orthogonal bases, which can directly reflect the relationship between information and channels. Constellation charts use the distance between constellation points to represent the difference between information points. Moreover, the constellation structure also reflects the mapping dimension, average power, and anti-interference ability of information to a certain extent. The increase of the mapping dimension of information, namely the dimension of the constellation, can greatly improve the diversity of information representation. In a 1-D constellation, the length of the line segment represents the change of constellation points, which is obvious in the ASK constellation. In a 2D constellation, the plane figures show the difference in constellation points. There are many examples in PSK, QAM [6,7], and geometric shaping [8]. In the 3-D constellation diagram, the superposition of 3-D figures indicates the difference between decision space and constellation points, such as the square 16-point constellation and the square 64-point stellation [9], etc. Some will use the compact structure of a regular quadrangle to reduce the overall average power of the signal as much as possible, such as stacked 16-point [10], cell-intensive 16-point constellation [11,12], and so on. Of course, there are other structures of the constellation diagram, regular polyhedron class [13], wrapped structure class, and so on [14]. But 4-D constellations [15,16], even 5-D and 6-D, and other high-dimensional constellations are less involved, owing to the difficulty of designing and planning high-dimensional constellation diagrams intuitively.
In this paper, we further update and optimize the 3D constellation and propose a simplified version of the 4D constellation structure, which is the construction and encryption scheme of dimension dissecting reorganization. In order to verify the performance, an experiment to demonstrate the transmission of 46.7 Gb/s 4D constellation mapping the intensity modulation/direct detection (IM/DD) carrierless amplitude and phase (CAP) on 2 km 7-core optical fiber has been successfully carried out.
2. Principle
2.1 General description of dimension dissecting reorganization scheme
The encryption system used in this paper is the construction and encryption scheme of dimension dissecting reorganization for the 4-dimensional (4-D) constellation, and the general process is shown in Fig. 1. After the bit serial-parallel conversion, the dimensions of parallel details are selected and cut, and different dimensions are divided. After the signal is recombined in the dimension, the whole phrase is scrambled to realize the signal encryption. Because the signal is finally reflected in the 4-D constellation, but the high-dimensional constellation is challenging to construct, it can also be decomposed step by step from the low-dimensional anatomy. These methods are called the construction and encryption scheme of the 4-D constellation dimension dissecting reorganization. The specific classification is described below.
2.2 Constellation construction scheme of dimension dissecting
The first problem of the 4-D constellation is to find the orthogonal basis function, and the emergence of multi-carrier, multi-phase, multi-band, and other multiplexing methods solves the problem of orthogonal basis. We can transmit signals in multiplexing branches with different dimensions. Because the degree and number of multiplexing are large enough, independent constellation mapping or joint constellation mapping can be performed in multiple dimensions, such as 4-D, 5-D, 6-D, or even higher. Here we take the distinguishing method with band or frequency as the orthogonal basis to illustrate the search and use of the orthogonal basis function.
Based on the principle of reuse and the related theoretical basis of the triangular orthogonal system, a 4-D orthogonal basis function is proposed.
Due to the influence of the sampling points of orthogonal functions, the order of magnitude obtained by the inner product of functions will be under-converged. Therefore, the selection of orthogonal functions can be carried out by cyclic iteration with different values of n until a satisfactory and qualified orthogonal basis function is obtained. The orthogonal basis function can be expanded, the ontology is a trigonometric function system, and the expanded term is the root raised cosine function, so that the symmetry of the root raised cosine function will not affect the orthogonality, and the functions such as signal carrier loading and filter correction can be realized. In essence, these orthogonal basis functions still belong to the category of wavelength division multiplexing, and pairwise orthogonality between functions is achieved by orthogonal isolation between different carriers.
However, it is difficult to design and imagine high-dimensional constellations such as four dimensions in an intuitive manner. To visually represent the constellation, we use color to represent the fourth-dimensional constellation points, symbolizing the density of the signal at a specific position in a certain 3-D space. The hyper-dimensional space of the 4-D constellation is difficult to design by the space stacking method, but it can be deduced from low dimension to high dimension, approaching a better design structure. For this reason, we propose a method of constructing high-dimensional constellations by dimension dissecting reorganization. After judging the dimension of the high-dimensional constellation, the dimension is cut and divided into the superposition of several dimensions, such as (2,2), (1,3), (2,1,1), and so on, which means the dimensional constellation is divided into the dimension of ${x_1}$, ${x_2}$, ${x_i}$, until to $n - \sum\limits_i {{x_i}}$. After the step of dimensions division, the optimal constellation structure can be obtained in each dimension, and finally, the overall constellation structure design can be completed by overall superposition.
The superposition of dimensions, which is always combined by simply adding 1 or -1 as a distinction, will lead to an increase in the energy value of the entire constellation. The specific addition quantity is based on the number of lower dimensions in the divided dimensions. Therefore, generally speaking, it is the best choice to divide into n-1 and 1. From the point of view of 3-D constellation design, a 4-D constellation only adds one more dimension to 3-D, which can optimize 3-D. The constellation design can be completed only by superimposing the fourth dimension, which can reduce the number of points in the original constellation design by half. At the same time, combined with the 3-D geometric structure with equal distance between the vertex of the regular tetrahedron structure in 3-D composition, the average energy value of each vertex is the smallest under the same Euclidean distance, which can achieve a better constellation structure in the 4-D space. Let's take the 16-point constellation as an example to explain it in detail. The related content is presented in Fig. 2.
Due to the dimension reduction and simplification, the 4-D constellation with 16-point is halved, and only an 8-point constellation needs to be designed. Starting with the regular tetrahedron (with the regular tetrahedron and four related constellation points), from the point of view of simplifying the overall energy value, it is better to expand from the surface than from the edge. It is only necessary to stack the unit cells based on the regular tetrahedron with each surface as the reference, to obtain the structure composed of five regular tetrahedrons with eight vertices. This structure has a relatively low energy-volume ratio, that is, a good constellation gain index. Combined with the geometric structure of the constellation, the coordinate information of the constellation shown in Table 1 can be obtained. Also, the structure of the 4-D constellation is depicted in Fig. 3.
Table 1. Coordinates of each point of the 16-point 4-D constellation
Fig. 3. The structure of the 4-D constellation: (a) is a top view; (b) is a front view; (c) is a 3-D view, and (d) is a 4-D view.
2.3 Chaos function of the encryption scheme
The physical layer encryption of the information mainly focuses on quantum noise encryption, noise shielding, and chaotic sequence encryption. This paper adopts the physical layer encryption scheme based on the chaotic sequence. This paper adopts the Lorentz chaotic system model in Ref. [17], and its mapping function is
The Lyapunov coefficient of the system is as follows,
Among them, the coefficients of Lyapunov coefficients in 3-D space are all greater than zero, which indicates that the system exists in the hyperchaotic state. However, when the overall coefficient is less than zero, the system will appear chaotic dynamic degradation. When the initial value is met as follows:
The system is in the hyperchaotic state, and the obtained projection and autocorrelation diagrams of different dimensions are shown in Fig. 4 below.
Fig. 4. Chaotic model phase diagram: (a) is about the variable of x and y; (b) is about the variable of x and z; (c) is about the variable of x, y, and z; (d) is about the variable of y, z, and k.
2.4 General description of the encryption scheme
We adopt the encryption scheme of dimension splitting rearrangement, and use the chaotic sequence of the chaotic system as the dimension selection and constellation reorganization of the constellation, to complete the joint encryption of information in another dimension. The union of different dimensions can be realized by splitting, such as (2, 2), (1, 3), (3, 1), (2, 1, 1), etc. Because there is a strict mapping order between messages, the change in the order will lead to a change in the mapping, and the order of encrypted messages will be disordered. Finally, through phase re-selection, the overall scrambling of the signal phase is realized. The detailed scheme is shown in Fig. 5.
Firstly, by rearranging the arrangement and mapping order of information in various dimensions, the arrangement order of information in a section is completely disrupted, the scrambling degree of information is improved, the first encryption of the information is completed, and the information mapping scheme is determined. Then, the chaotic sequence is used to select the degree of signal dimension segmentation. According to the number of signal dimensions, divide the signal into different dimensions, such as dividing a 4-D signal into (1, 3) or (2, 2), and dividing a 3-D signal into (2, 1), etc. Different dimensions lead to different symbol information after mapping, which increases information confusion and the difficulty of password cracking. All dimensions are rotated by the constellation rotation scheme mentioned above, which further scrambles the signal as a whole and increases the difficulty of signal decryption.
2.5 Information rearrangement of the encryption scheme
By selecting an appropriate number, the data is divided into segments, and the information in each segment is scrambled and rearranged, which is shown in Fig. 6. The rearrangement order is given by a chaotic sequence. The initial ranking can be given by the formula:
where w represents the position within each information segment. However, the way of confusion will lead to duplicate values in a piece of information, which will make it difficult to recover the information from duplicate values. Therefore, we sort the information of w, and take the address of the information as the position of the last message to avoid duplicate values. The formula is as follows where $sort( x)$ represents a sort function, ${f_t}$ denotes the ordering of the array x, ${w_t}$ returns the index sequence of ${f_t}$, which represents the correspondence between the elements in ${f_t}$ and the elements in the array x. Since there will be no repetition of ${w_t}$, the condition of random scrambling of positions is met.2.6 Dimension dissecting of the encryption scheme
Take the 4-D space as an example, and decide the number of split dimensions, such as 1, 2, and 3 split dimensions, without considering any extension of the dimensions alone. There are 6 kinds of 4-D space dissecting methods, and the specific process is detailed in Fig. 7.
As shown in the figure, the 4-D signal can be divided into seven modes after being cut: (1, 1, 1, 1), (2, 1, 1), (1, 1, 2), (3, 1), (1, 3) and itself. The chaotic sequence is used to select the mode, and the formula is as follows.
where ${n_c}$ represents the number of mode choices, which is represented by 0 ∼ 6, corresponding to the dimension shown in the above figure, so that the dimension change of each data can be realized. There is no need to pay attention to the problem of excessive repetition here, just select the mode in the map.2.7 Phase disturbance of the encryption scheme
Using the complexity of the 4-D constellation, data encryption can be carried out by scrambling the phase and position of the 4-D constellation, which can be seen in Fig. 8. Based on the chaotic system, this scheme generates four chaotic sequences, and scrambles the phase and position of each dimension of the signal. Each chaotic sequence is converted into the corresponding phase rotation value by the formula as follows.
The obtained phase parameters are rotated and transformed by the rotation formula below.
In this way, the constellation map can be scrambled in the whole constellation space, the key space can be further strengthened, the information leakage caused by excessive system operation times can be prevented, and the cracking difficulty can be reduced.
3. Experiments and results
The experimental setup of the intensity modulation/direct detection (IM/DD) CAP transmission system on 7-core optical fiber using the structure of dimension dissecting reorganization, is shown in Fig. 9. The original data is generated by MATLAB program derived from the pseudo-random binary sequence (PRBS) and then mapped into 16 QAM formats. Then the mapped signals are up-sampled by a factor of 6. Overlapping signals are loaded into an arbitrary waveform generator (AWG, TekAWG70002A), and digital-to-analog conversion (DAC) is performed at the sampling rate of 10 GSa/s. Following the linear amplification by an electrical amplifier (EA) with 20 dB gain, the analog signal is utilized to drive the Mach-Zehnder modulator (MZM) to implement intensity modulation. A laser with a line width of less than 100 kHz generates an optical carrier of 1550 nm for optical signal modulation. After being amplified by an optical coupler (OC), they are transmitted into the 7-core weak coupling MCF of 2 km through the fan-in device. After transmission, the 7-core channels are demultiplexed into single-mode optical fibers by a fan-out device. The resultant net bitrate is 46.7 Gb/s for 16QAM. At the receiver, a variable optical attenuator (VOA) is equipped to adjust the received optical power. Afte666r the analog-to-digital conversion (ADC) by a mixed signal oscilloscope (MSO, TekMSO73304DX) with a sampling rate of 50 GSa/s, offline DSP extracts the secret keys and recovers the original data.
Fig. 9. Experimental system device diagram. (AWG :arbitrary waveform generator; EA: electrical amplifier; MZM: Mach–Zehnder modulator; EDFA: erbium-doped fiber amplifier; OC: optical coupler; VOA: variable optical attenuator; PD: photodiode; MSO: mixed signal oscilloscope)
The employed 7-core fiber is a commercially procured weakly coupled MCF with a single core diameter of 8 µm and core pitch of 41.5 µm. The coating diameter and cladding diameters are 245 µm and 150 µm. Due to the influence of control parameters in the manufacturing process of optical fiber, it is difficult to guarantee the consistent performance of each core. To express and measure the transmission performance of the whole optical fiber, the transmission state of each core is shown in Fig. 10. As can be seen from the figure, the transmission bit error rate curve among the core is relatively stable, and the trend of the whole curve is fitted together. In the range of bit error rate $({{{10}^{ - 2.75}},\textrm{ }{{10}^{ - 1.5}}} )$, the difference of received optical power at the same bit error rate is about 0.5 dB. When the error rate is higher than ${10^{ - 1.5}}$ and lower than ${10^{ - 3}}$, the divergence of the curve starts to widen. The optical power difference received by most fiber cores is about 1.5 dB. The reason for this phenomenon is that in the 4-D signal transmission scheme in the experiment, a single symbol carries a high amount of information, and the correct discrimination of symbols makes a high contribution to the bit error rate. The figure also shows the bit error rate curve received by the illegal receiver, roughly distributed between 0.4 and 0.5, demonstrating the error rate receiving performance under conventional interception.
In order to show the advantages of the proposed 4-D constellation dimension dissecting reorganization construction scheme of CAP 1, two 4-D constellations are introduced as a comparison. One of them is the square 16-point 4-D constellation diagram, which is CAP 2. The other is the square abduction (3,1) dimensional reconstruction constellation, which is CAP 3. As can be seen from Fig. 11, after dissecting the reorganization construction scheme, the proposed 4-D optimization scheme has better transmission performance. When the bit error rate is ${10^{ - 3}}$, compared with CAP 2 and CAP 3, the received optical power is increased by 1 dB and about 1.4 dB respectively. Due to the dimension dissecting reorganization construction scheme, the gap between constellation points is narrowed, the usability is improved, the average energy of the signal is reduced, and the signal transmission performance is improved. Compared with the signal CAP 3, the received optical power of the signal CAP 2 is increased for the same reason.
4. Conclusion
We proposed a simplified 4D constellation construction and encryption scheme of dimension dissecting reorganization, which scrambles the signals and constructs constellations through the selection, derivation, and recombination of the dimensions. This scheme can reduce the difficulty of constructing high-dimensional constellations, and optimize the matching difficulty of information encryption through internal scrambling. Also, an experiment demonstrating 46.7 Gb/s 4D constellation mapping the IM/DD CAP transmission over 2 km 7-core optical fiber is successfully carried out to validate the performance.
Funding
National Key Research and Development Program of China (2022YFB2903104); National Natural Science Foundation of China (61720106015, 61727817, 61835005, 61875248, 61935005, 61935011, 61975084, 62035018, 62171227, U2001601); Jiangsu Provincial Key Research and Development Program (BE2022055-2, BE2022079); The Natural Science Foundation of the Jiangsu Higher Education Institutions of China (22KJB510031); Jiangsu team of innovation and entrepreneurship; 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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