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Abstract
This paper proposes a 3-dimensional (3D) carrier-less amplitude and phase modulation (CAP) based on a pyramid constellation design encryption scheme for 7-core fiber in passive optical network (PON). The chaos sequences generated by the 4D hyperchaotic system are applied to produce the masking factor, and the pyramid Rubik's cube rotation rules are used to flip and rotate the constellation points. To verify the performance of the proposed 3D CAP-PON system, 25.5Gb/s encrypted Pyraminx-3D-CAP-16 signal transmission over 2 km 7-core fiber is experimentally demonstrated. Compared with the traditional 3D-CAP-16 signal, the proposed Pyraminx-3D-CAP-16 signal achieves a sensitivity gain of 0.5 dB under the limit of hard decision forward error correction (HD-FEC) 3.8 × 10−3. Compared with the Pyraminx-3D-CAP-16 signal, the encrypted Pyraminx-3D-CAP-16 signal has little difference from that before encryption, which has a great application prospect in the physical layer security enhancement of the 3D-CAP-PON system.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Demand for digital communications broadband services has skyrocketed alongside the rise of the Industrial Internet, 6 G, and data centers. This trend will continue to rise at an uncertain rate in the future. As the capacity of optical networks grows and their reach expands, however, the current short-range infrastructure will need to expand rapidly to keep up. The intensity modulation direct detection (IMDD) system [1–3] has been widely employed in short-range communication [4] due to its inexpensive cost, low power consumption, and small footprint in comparison to the current popular coherent approach. With its high bandwidth, low cost, and low power consumption, a passive optical network (PON) [5] is quickly becoming the market-dominating, future-oriented network architecture [6]. Modulation in the form of carrier-less amplitude and phase (CAP) uses a digital filtering method to create multichannel signals [7]. CAP is an excellent and well-liked modulation format in the IMDD-based short-reach situation since it does not need complicated frequency conversion sources and radio frequency (RF) for down-conversion and is expected to allow various levels and modulations [8]. The demand for flexible communication systems is increasing as optical communication systems evolve and optical fiber quality improves. CAP has expanded to 3-dimensional, 4-dimensional, and even higher dimensions in order to assign distinct users a different service. There aren't many studies on 3D constellation encryption at the moment. However, the 3D constellation can increase the minimum Euclidean distance (MED) between constellation points while using the same amount of power [9], and the position change of constellation points is more selective, making the optical communication system more adaptable and enabling it to perform better in terms of bit errors. Due to the advantages of high transmission rate, low complexity, flexible bandwidth, and large capacity, CAP-PON [10] is considered one of the ideal architectures for the next-generation optical access network [11]. However, the remarkable architecture of the PON system brings some security issues. Although the PON system expands the access range, it broadcasts the downlink data to the optical network unit (ONU) terminal. All ONUs connected to the optical line terminal (OLT) can receive downlink data frames. Illegal ONUs can often pretend to be legitimate ONUs for eavesdropping and various attacks. Before the downlink transmission through the PON system, it is imperative to encrypt information [12].
Data encryption is generally implemented in upper or physical layers to improve the system's security [13]. However, due to encryption the existence of naturally occurred defects in the data cannot be fully protected in the optical network, therefore, encryption in the upper layer alone is not enough to ensure the safety of the whole PON system. While physical layer encryption can enhance the network's overall security, it is more conducive to enhancing the system's security performance [14]. Compared with optical network upper-layer encryption, physical layer encryption can make use of mature digital signal processing (DSP) [15] to process data efficiently, flexibly, and compatible so that that information can be transmitted to the user more safely [16]. A secure optical 3D probabilistic shaping CAP based on spherical constellation masking is proposed in [17]. The Reference [18] ensures the security of users through a physical layer encryption method of chaotic in CAP-PON system based on floating probability disturbance. A high-security 3D CAP modulation is used for short-distance multi-core fiber communication in [19]. Most current encryption techniques, however, advocate adding phase noise at the transmitter (TX) to disrupt the constellation and make the signal pseudo-noisy, which in turn increases the bit error rate (BER) and decreases the transmission performance of the system during decryption at the receiver (RX). An immediate issue is a need to boost the system's transmission performance without compromising security.
This paper proposes a new 3D encryption scheme based on a pyramid constellation design for CAP-PON in 7-core fiber. The regular tetrahedron is used as the base element for 3D constellation geometric shaping, and the constellation points are clustered inward as much as possible to optimize the geometric distribution of the constellation points and maximize the constellation figure of merit (CFM) of the 3D constellation in the situation of unchanged constellation power. Using four-dimensional hyperchaotic mapping to generate four masking factors, the high-security encryption of multiple dimensions can be realized with only one chaotic model. The pyramid constellation masking chaotic encryption scheme can always keep the constellation point on the vertex of the regular tetrahedron during the disturbance, and the overall position of the constellation point has not changed before and after encryption, which can give the illegal receiver the illusion of disguising the constellation diagram. Effectively it improves the security performance of 3D-CAP-PON, minimizes the complexity, and improves the anti-interference ability and error code performance of the system. In order to verify the performance of our encryption scheme, the encrypted 3D-CAP-16 signal transmission is experimentally demonstrated over 2-km 7-core fiber. The results show that the scheme has high-security performance, and the key space reaches to 10126, which can significantly improve bit error performance.
2. Principle
The proposed CAP-PON pyramid constellation masking chaotic encryption scheme is shown in Fig. 1, describing the detailed procedure occurred with the data signal from TX to RX, where the corresponding operations between two sides are labeled with the same color. Similar to the principle of 2D-CAP, the original data is converted into multiple multi-level symbols after 3D constellation mapping, and the initial key is generated into a chaotic sequence by 4D hyperchaotic mapping. According to the chaotic sequence, the constellation flip and the constellation rotation vectors are respectively generated. After two vector encryption, the positions of constellation points are exchanged. By processing these three signals, 3D-CAP combines them into a single signal for transmission. The data stream is recovered at the receiver after 3D-CAP demodulation, and the constellation point is restored to its original position using the key. Finally, the original data is obtained after 3D mapping. The signal processing is mainly carried out in the off-line DSP at the transmitter and receiver, which has no impact on the transmission system and can be compatible with the existing optical network. The system designed in this manuscript only adds the process of encryption on the transmitter and receiver, and only needs to add the corresponding module in its DSP chip to be compatible with the existing optical network.
Fig. 1. Schematic illustration of the proposed pyramid constellation masking chaos encryption for CAP-PON.
In the traditional 2D-CAP, the impulse responses of two filters using sine and cosine waveforms can form a Hilbert pair [20] to make them orthogonal. Still, this two-dimensional filter cannot be directly extended to three dimensions. Reference [21] gives the optimization algorithm to find the 3D orthogonal FIR filter $\{ f0,f1,f2\}$. The optimization expression can be formulated as follows:
The optimization expression is stated in Eq. (1) to determine three sequences $\{ f0,f1,f2\}$ with the given frequency magnitude response. Assuming that our linear contract condition is $F_{TX}F_{RX} = {Z^{ - n}}I$. The perfect reconstruction (PR) condition can be satisfied in this condition, and the data streams can be separated without crosstalk. The waveform of the transmitter FIR filter can be calculated by the above method [22]. To better maintain the orthogonality among the three groups of filters, we set the tap number of filters to 11 and the up-sampling factor to 11, which can be used to avoid inter-symbol interference (ISI).
Figure 2(a) describes a traditional 3D-CAP-16 constellation composed of two concentric cubes, so its constellation diagram is also the simplest. The 16 constellation points are distributed vertically on all inner and outer cubes. Figure 2(b) is a three-layer constellation design named Pyraminx-3D-CAP-16. The minimum Euclidean distance between adjacent constellation points is set to 1. The first layer is a regular hexagon composed of six equilateral triangles, and the second and third layers are two equilateral triangles consisting of four and one equilateral triangle, respectively. By connecting the points with a distance of 1 between different layers in the constellation diagram, we can get Fig. 2(c). It is a pyramid cube with four small regular tetrahedrons removed. We can get the pyramid cube constellation diagram in Fig. 2(d) by adding up the four missing small regular tetrahedrons.
Fig. 2. (a) traditional 3D-CAP-16 constellation; (b) Pyraminx-3D-CAP-16 three-layer constellation; (c) Pyraminx-3D-CAP-16 constellation; (d) extended pyramid Rubik’s cube constellation.
As the minimum Euclidean distance is larger, the anti-noise performance of the constellation is better. When the minimum Euclidean distance is fixed, the average power of the constellation is smaller, the sensitivity of the constellation to noise is lower, and the signal-to-noise ratio efficiency of the modulation method is higher. Assuming that the minimum Euclidean distance of the constellation is ${d_{\min }}$ and the average power of the constellation is P, CFM [23] is defined as the ratio of the square of the minimum Euclidean distance to the average power of the constellation. The mathematical expression of CFM is expressed as:
Compared with the typical 2D constellation, 3D constellation can further expand the Euclidean distance between constellation points, so that geometry of constellation points can be optimized to provide more space for distribution. On the condition that the minimum Euclidean distance is fixed, the constellation points are gathered inward as much as possible, and the CFM of the constellation is increased. On the condition that the minimum Euclidean distance MED is set to 1, the CFM of the traditional 3D-CAP-16 constellation is 0.4726 by calculation, and the CFM of our proposed Pyraminx-3D-CAP-16 constellation is 0.8571. Detailed mapping rules of the two constellations are shown in Table 1.
Table 1. Parameters of 16-ary 3D constellations
The 4D hyperchaotic map generates set of chaotic sequences, the corresponding constellation flip factor, and the constellation rotation factors. Compared with other systems such as Lorentz chaotic system, the four-dimensional hyperchaotic system, with higher dimensions, is selected to generate four independent chaotic sequences at once. Although the combination of multiple low-dimensional chaotic systems can obtain multiple chaotic sequences, it is not as flexible as the direct use of four-dimensional chaotic systems. The 4D hyperchaotic map can be represent as:
In Eq. (3), ($a$, b, c, d, $r$)is a parameter and ($\alpha$, $\beta$, $\chi$, $\delta$) is a variable, so our initial key is ($a$, b, c, d, r, ${\alpha _0}$, ${\beta _0}$, ${\chi _0}$, ${\delta _0}$), which is set as (35, 3, 12, 7, 0.5, 3, -1, 4, 2) in this paper. As can be seen from Fig. 3, when a driver key is given, there is a complex bifurcation with complex chaotic characteristics, which is exactly what secure optical communication needs. The trailing line dragged out of the phase diagram is due to the different initial values. Different initial values will make the initial position of the phase diagram different, but eventually it will be within the scope of chaotic system after iterations.
We propose a Rubik's Cube encryption method for the Pyraminx-3D-CAP-16 constellation based on constructing the pyramid Rubik's Cube. Since the four vertices of the pyramid only fall on four vertices when rotating according to Rubik's Cube rules, we place 16 constellation points on 16 points except for the vertices. When the pyramid cube rotates, the rule of rotating around the Z axis is simple, but turning around the other three axes is very complicated, so we set the rotation rule as revolving around the Z axis. Still, we can constantly change the plane and coordinate system before each rotation, equivalent to turning each complex step into a simple one-step flip and one-step spin. In particular, the flip and rotation of each step are carried out based on the flip and rotation of the previous step, which further enhances security in the information transmission process. Suppose that the coordinates of three-dimensional constellation points are (${x_m}$, ${y_m}$, ${z_m}$) after encryption for m times.
The specific steps of scrambling are as follows:
Step 1: Concerning the factor ${\alpha _n}$ generated by the 4D hyperchaotic mapping in Eq. (3), we can flip the constellation through it. As shown in Fig. 4, the four vertices of the pyramid cube are O1, O2, O3, and O4, respectively. The O4 to O3 are defined as the observation directions.
When Eq. (4) holds, the constellation of our pyramid Rubik's Cube passes through Method 1 in Fig. 4, and the constellation remains unchanged.
When Eq. (5) holds, the constellation of our pyramid Rubik's Cube passes through Method 2 in Fig. 4, and the whole constellation turns $\pi - \arccos (1/3)$ counterclockwise around O3O4 edge. At this time, O1-O3-O4 plane is the bottom surface. As the axis of turning is not the coordinate axis, we reset the origin by translating along the X axis and along the Z axis respectively, as shown in Eq. (6). The last step is to rotate the whole constellation counterclockwise around the Z axis by 120 degrees, as shown in Eq. (7), so that the inverted constellation does not change in the overall position compared with the initial constellation, which is convenient for subsequent operation.
When Eq. (8) holds, the constellation of our pyramid Rubik's Cube passes through Method 3 in Fig. 4, The whole constellation turns $\pi - \arccos (1/3)$ anticlockwise around the O3O4 edge, and the O1-O3-O4 plane is the bottom. Similarly, we reset the origin by translating along the X-axis and the Z-axis, respectively, as shown in Eq. (6). The last step is to rotate the whole constellation clockwise around the Z-axis by 120 degrees, as shown in Eq. (9).
As for the constellation points of the second layer, we use the factor ${\chi _n}$ to rotate it. Similarly, the proportional amplification of ${\chi _n}$ generates the remainder, resulting in three rotation angles of 120, 240, and 360, which makes the second layer of the constellation rotate. The specific implementation is reflected in Eq. (11).
The constellation points of the first layer are rotated by the factor ${\delta _n}$, which is the same as before. The remainder is generated by the proportional amplification of ${\delta _n}$, resulting in three rotation angles of 120, 240, and 360, which makes the first layer of the constellation rotate. The specific implementation is reflected in Eq. (12).
Like our pyramid Rubik's Cube rotation rules, each encryption is carried out based on the previous two steps. An intact Rubik's Cube can be quickly restored with few rotations, but the more rotations, the more difficult it is to recover the reverse operation. Our constellation is scrambled on this basis. Although the overall position of the 3D constellation has not changed after m times of scrambling, the location of each constellation point has been chaotically jumbled. The benefit of this procedure is that it can achieve the effect of masking the constellation, making it impossible for an unauthorized receiver to intercept and steal the information, considerably enhancing the system's security and transmission capabilities.
3. Experiment and results
The 3D-CAP-16 secure communication system based on pyramid Rubik's Cube constellation masking has been established experimentally, shown in Fig. 6, where the 7-core fiber is deployed for the secure transmission. At the OLT side, the digitally encrypted Pyraminx-3D-CAP-16 signal is generated in the electrical domain by the offline digital signal processing (DSP). The arbitrary waveform generator (AWG, TekAWG70002A) with a sampling rate of 10 GSa/s is used for digital-to-analog conversion (DAC). The corresponding waveform is generated and amplified by an electric amplifier (EA), and then sent to a Mach-Zehnder modulator (MZM) for intensity modulation. In this experiment, the power and wavelength of continuous-wave laser (CW) are set to 12 dBm and 1550 nm, respectively, and the transmission length of 7-core fiber is set to 2 km. After amplifying and compensating for the power loss by an Erbium-doped fiber amplifier (EDFA), the signal is divided into 1: 8 by power splitter (PS) and decorrelated by delay line (DL). Seven groups of signals are coupled into the 7-core fiber, transmitted by fan-in equipment, and spatially demultiplexed into single-mode fiber by a fan-out device. The variable optical attenuator (VOA) in the ONU is used to adjust the received optical power. The received optical signal is converted into an electrical signal by a photodiode (PD), then passed through a mixed-signal oscilloscope (MSO, TekMSO73304DX) with a sampling rate of 50GSa/s, and subjected to analog-to-digital conversion for further off-line DSP operation. The DSP at the ONU side performs the process opposite to that at the OLT end, including matched filter, downsampling, decryption, demapping, and finally, restoring the original data.
Fig. 6. The experimental setup (OLT: optical line terminal; ONU: optical network unit; CW laser: continuous-wave laser; AWG: arbitrary waveform generator; EA: electrical amplifier; MZM: Mach-Zehnder modulator; EDFA: Erbium-doped fiber amplifier; DL: delay line; PS: power splitter; VOA: variable optical attenuator; PD: photodiode; MSO: mixed-signal oscilloscope)
The information entropy carried by each constellation point is 4 bits/symbol. We use 7-core fiber for transmission. Due to the 11 times up-sampling of DAC, the transmission rate of the signal is 10 GSa/s × 4bits/symbol × 7 /11 = 25.5Gb/s. Key space is a parameter to evaluate the robustness of the system quantitatively. The key consists of the initial value and control parameters of the 4D hyperchaotic system, and the encryption effectiveness and information security depend on the sensitivity to the initial value and control parameters. Figure 7 is a partial result of iteration under different parameters. In Fig. 7(a), we set the initial value a to 35.000000000000000001 and 35.00000000001 in turn, while other parameters remain unchanged. Even if the parameters are slightly different, the chaotic sequences after iteration have almost nothing in common, indicating that the 4D hyperchaotic system is susceptible to change the initial value. Likewise, in Fig. 7(b), 7(c), and 7(d), only a slight change of ${10^{ - 14}}$ is given to the initial values b, c, and d, respectively. Suppose our system is attacked and stolen by illegal ONU, even if they have ${10^{ - 14}}$-level differential keys, they can't get valid information.
Figure 8(a) gives the autocorrelation function of the sequence $\alpha$, which is approximately an impulse function, and Fig. 8(b) shows the cross-correlation function of two lines $\alpha$ obtained by changing the initial value of ${\alpha _0}$, and the correlation between the two sequences is close to 0. From Fig. 7 and Fig. 8, we can see that our encryption scheme has super high-security performance. The key in our system consists of initial values ($a$, b, c, d, r, ${\alpha _0}$, ${\beta _0}$, ${\chi _0}$, ${\delta _0}$), and each initial key can be changed, which can be calculated conservatively. The chaotic sequences after iteration are almost entirely different, which shows that our key parameters are very sensitive, and the key space is (1014)7 = 10126, which is enough to resist the exhaustive attack and theft of illegal ONU.
Figure 9 is a schematic diagram of the error rate curve of encrypted Pyraminx-3D-CAP-16 signal after 2 km transmission in 7-core optical fiber. It is easy to find that the error rate of information transmission on each core decreases with the increase of power at the legal ONU end, and at the same time, the error rate curves of each core almost coincide. With the HD-FEC BER of $3.8 \times {10^{ - 3}}$, the sensitivity difference between the best and worst core is about 0.5 dB, which proves that the 7-core fiber used in this experiment has excellent stability and uniformity. The illegal ONU can't get the right key, so the information can't be cracked. The constellation's position has shifted significantly, and it has been rolled into a sphere so that no useful information can be obtained. With the increased optical power, the BER keeps at about 0.5, and there is no noticeable change. This proves that the encryption scheme proposed in this paper can effectively improve the system's security.
In order to demonstrate the advantages of our pyramid constellation design, we evaluate how well Pyraminx-3D-CAP-16 performs compared to the more typical 3D-CAP-16. Figure 10 shows the BER curve of signals sent in core 4 using Pyraminx-3D-CAP-16 and standard 3D-CAP-16. Due to serious noise interference, there is an interesting phenomenon when the received optical power is lower than -22dBm. The BER of the Pyraminx-3D-CAP-16 signal is even higher than that of the traditional 3D-CAP-16 signal. On the condition of low received optical power, the signal-to-noise ratio has become very low, the constellation points are all very concentrated, and the traditional 3D-CAP-16 constellation points are more dispersed. The proposed scheme constellation is relatively concentrated, so the BER of the Pyraminx-3D-CAP-16 is higher than the traditional 3D-CAP-16. With the increase of optical power, the BER of Pyraminx-3D-CAP-16 signal and traditional 3D-CAP-16 signal are gradually reduced, the constellation points at the receiving end are not shifted, and the bit error performance of the system is improved. At the limit of HD-FEC BER of $3.8 \times {10^{ - 3}}$, the optical power of Pyraminx-3D-CAP-16 and traditional 3D-CAP-16 signals are -20.9 dBm and -20.4 dBm, respectively, achieving a sensitivity gain of 0.5 dB.
Figure 11 is the BER curve of the Pyraminx-3D-CAP-16 signal before and after encryption in core 4. As seen from the figure, with the increase of the received optical power, the BER performance of an unencrypted signal is almost the same as that of an encrypted signal. This is because our encryption scheme has not changed the overall position of constellation points before and after encryption and hardly brings phase noise caused by constellation disturbance to the system. Hence, their BER difference is minimal. It can be seen from the figure that the optical power of the Pyraminx-3D-CAP-16 signal is -20.9 dBm under the limit of HD-FEC BER. After encryption, the optical power of the Pyraminx-3D-CAP-16 signal is -20.5 dBm; that is to say, at the expense of 0.4 dB, we have improved the key space of 10126, which fully proves the high security and high gain performance of Pyraminx-3D-CAP-16 encryption scheme based on pyramid constellation proposed in this paper.
Besides, to verify the performance of our encryption scheme on images, we transmitted a picture of a giant panda lying on a seat in this system. The images and histograms before encryption are shown in Fig. 12(a) and Fig. 12(b). There is no error in image recovery at the legal ONU with the correct key. However, the images and histograms received at the illegal ONU are shown in Fig. 12(c) and Fig. 12(d) because there is no correct key. The images are blurred, and the gray histograms are evenly distributed, so there is no valuable information to take. This shows that our encryption scheme has good security performance, which can ensure that the transmitted data will not be stolen and attacked by illegal receivers.
Fig. 12. (a) the image before encryption; (b) Histogram before encryption; (c) the encrypted image; (d) Histogram before encryption.
In the end, we restore the original images of 20% of malicious attacks, which strongly verifies the robustness of the proposed encryption scheme against attacks. Figure 13 (a) is the original image, and Fig. 13 (b) is the restored image of the unencrypted image after a 20% malicious attack. It can be seen that after 20% of malicious attacks by the hijackers, the restored unencrypted image has lost essential parts, making it challenging to identify the information expressed by the image. Figure 13(c) is an encrypted image with a 20% malicious attack, and Fig. 13(d) is an image recovered after a 20% malicious attack. Even after 20% of malicious attacks by illegal attackers, the image quality is unaffected, and most information can be displayed. The result shows that our system has a strong anti-interference ability and even can transmit helpful information to legal ONU under the malicious attack of the illegal receiver.
4. Conclusion
This paper proposes a new encryption scheme of high-security 3D CAP-PON system based on a pyramid constellation design, and experiments verify it. By using the 4D hyperchaotic system, four chaotic sequences are generated. Then the constellation flip and constellation rotation factors are generated by these sequences, and the encryption of the pyramid cube is realized. To verify the feasibility of our proposed scheme, the transmission of 25.5-Gb/s encrypted Pyraminx-3D-CAP-16 signal on a 2 km 7-core optical fiber was demonstrated experimentally. The transmission performance difference between the best and worst cores is about 0.5 dB. Compared with the traditional 3D-CAP-16 signal, our Pyraminx-3D-CAP-16 signal achieves a sensitivity gain of 0.5 dB at the limit of HD-FEC BER of $3.8 \times {10^{ - 3}}$, and the encrypted Pyraminx-3D-CAP-16 signal sacrifices 0.4 dB. In terms of security performance, this scheme can obtain a substantial key space of 10126 with low complexity and can effectively resist all kinds of attacks and thefts of illegal ONU. This scheme can recover the original image well. When the attack range of encrypted data reaches 20%, the outline and general information of the decrypted image can still be observed, which has little impact on the vision. The experimental results further prove our proposed scheme's high sensitivity and security and show its great potential in physical layer security enhancement of the 3D-CAP-PON system.
Funding
The Startup Foundation for Introducing Talent of NUIST; The Natural Science Foundation of the Jiangsu Higher Education Institutions of China (22KJB510031); Jiangsu Provincial Key Research and Development Program (BE2022055-2, BE2022079); National Natural Science Foundation of China (61835005, 61935005, 61935011, 61975084), 62035018, 62171227, U2001601); National Key Research and Development Program of China (2021YFB2800904).
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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