A High-Security Probabilistic Constellation Shaping Transmission Scheme Based on Recurrent Neural Networks

: In this paper, a high-security probabilistic constellation shaping transmission scheme based on recurrent neural networks (RNNs) is proposed, in which the constellation point probabilistic distribution is generated based on recurrent neural network training. A 4D biplane fractional-order chaotic system is introduced to ensure the security performance of the system. The performance of the proposed scheme is veriﬁed in a 2 km seven-core optical transmission system. The RNN-trained probabilistic shaping scheme achieves a transmission gain of 1.23 dB compared to the standard 16QAM signal, 0.39 dB compared to the standard Maxwell-Boltzmann (M-B) distribution signal, and a higher net bit rate. The proposed encryption scheme has higher randomness and security than the conventional integer-order chaotic system, with a key space of 10,163. This scheme will have a promising future ﬁber optic transmission scheme because it combines the efﬁcient transmission and security of ﬁber optic transmission systems.


Introduction
With the development of modern technologies, such as 6G, artificial intelligence, and the Internet of Things, the demand for high network bandwidth is growing exponentially.More advanced coding and modulation techniques are being explored to meet the demand for efficient optical transmission.Constellation shaping has received much attention as a typical modulation optimization technique, generally including geometric constellation shaping (GCS) and probabilistic constellation shaping (PCS).At present, many scholars have conducted research on PCS techniques [1], significantly reducing the average constellation power by increasing the probability of the occurrence of inner constellation points and decreasing the likelihood of the occurrence of outer constellation points, thus obtaining a more efficient transmission gain [2,3].Various deep learning applications in system design, system identification, and network monitoring have recently attracted significant research interest in optical communications [4,5].
Deep learning technology has been applied to communication systems by approximating and optimizing the transmitter and receiver functions using neural networks, known as autoencoder.In the autoencoder, there are two neural networks, i.e., the encoder and decoder.For coded modulation systems, a better communication can be achieved by embedding the channel model into the autoencoder for training and then using the neural network to train the constellation [6].Compared to other neural networks, a recurrent neural network (RNN) naturally adapts to processing data with time series or sequence structures.It can flexibly handle input sequences of different lengths and capture dependencies within the sequences.RNN adopts a weight-sharing strategy when processing sequences, which means using the same weight at different time steps.This can reduce the number of model parameters and reduce the risk of overfitting.To date, most research based on autoencoders has focused on GCS.At present, comparatively mature PCS techniques are implemented using constant component distribution matchers (CCDM) [7], but CCDM modules have high computational complexity, and constant probability distributions cannot achieve optimal transmission gain.Therefore, it is exciting to understand how to optimize PCS techniques under different channel models.
The advantages of passive optical networks (PONs) over standard electrically connected networks include greater flexibility, lower power consumption, and higher bandwidth [8,9].PONs are also considered to meet the growing demand for bandwidth and higher capacity in optical access networks.However, since the PON network downlink uses a broadcast mechanism to broadcast information to each optical network unit (ONU) [10,11], the transmitted information is highly susceptible to interception and decryption by illegal ONUs, thus imposing higher requirements on the security performance of the transmission scheme [12][13][14].Physical layer encryption based on DSP technology can fundamentally protect the transmitted data from malicious attacks involving upper-layer encryption.It is widely used in secure communication because of its advantages, such as pseudo-randomness, high storage capacity, and sensitivity to initial values [15].The early chaotic encryption used for physical layer encryption consisted of simple low dimensional chaotic maps, such as logical maps, sine maps, and Lorentz maps [16,17].Although using high-dimensional chaotic models can improve the chaos of the mapping, sometimes it not only fails to improve the security performance of the system, but also makes the encryption system more complex and less practical.Therefore, integer-order chaotic systems still face the risk of illegal decryption.Compared to integer-order chaotic systems, fractional-order chaotic systems have a larger key space and more complex dynamic behavior in the same dimension, thus having a broader application potential [18].
This paper proposes a high-security probabilistic constellation shaping transmission scheme based on recurrent neural networks.Firstly, the constellation point probability distribution is trained using a RNN for higher transmission gain.Then, four sets of mask vectors are generated by a 4D biplane fractional-order chaos (BFC) system [19] to perturb the bit stream, constellation map, subcarrier, and symbols.The four-dimensional perturbation based on the 4D biplane fractional-order chaos system effectively increases the key space of the transmission scheme compared to the proposed scheme and achieves high receiver sensitivity gain and high-security performance in a 2 km seven-core fiber-optic transmission system, confirming the feasibility of the proposed scheme.
The paper is organized as follows: the parameter settings of the RNN neural network are first introduced; the encryption scheme settings based on the 4D BFC hyperchaotic system are introduced in Section 2; the specific experimental setup based on a 2 km sevencore fiber optic transmission system is illustrated in Section 3; the experimental results are analyzed and discussed in Section 4; and, finally, the conclusions are drawn in Section 5.

Principle
The specific framework of the recurrent-neural-network-based high security constellation probabilistic shaping transmission scheme is shown in Figure 1.First, a pseudo-random binary sequence (PRBS) is generated at the transmitter side as the original input data, using the chaotic sequence A generated by BFC to perform XOR encryption on PRBS and perform serial/parallel (S/P) transformation to generate a matrix.The generated matrix is constellation-mapped by a pre-trained probability distribution, followed by the rotation vector encryption of the constellation points using the chaotic sequence B to generate random rotation phase information, followed by OFDM modulation via an inverse fast Fourier-transform (IFFT) matrix, and frequency masking vectors to scramble the subcarriers and symbols via the chaotic sequences C and D. Finally, a cyclic prefix (CP) is added.The encrypted RNN-PS-OFDM signal is transmitted through the channel.The original bitstream can be retrieved by decrypting the secured information at the receiver side using the original key.
Photonics 2023, 10, x FOR PEER REVIEW 3 of 13 pseudo−random binary sequence (PRBS) is generated at the transmitter side as the original input data, using the chaotic sequence A generated by BFC to perform XOR encryption on PRBS and perform serial/parallel (S/P) transformation to generate a matrix.The generated matrix is constellation−mapped by a pre−trained probability distribution, followed by the rotation vector encryption of the constellation points using the chaotic sequence B to generate random rotation phase information, followed by OFDM modulation via an inverse fast Fourier−transform (IFFT) matrix, and frequency masking vectors to scramble the subcarriers and symbols via the chaotic sequences C and D. Finally, a cyclic prefix (CP) is added.The encrypted RNN−PS−OFDM signal is transmitted through the channel.The original bitstream can be retrieved by decrypting the secured information at the receiver side using the original key.

Training Framework Setup Based on RNN Probability Distribution
The probability distribution scheme used in this experiment was based on RNN training.RNN naturally adapts to processing data with time series or sequence structures.It can flexibly handle input sequences of different lengths and capture dependencies within the sequences.RNN adopts a weight−sharing strategy when processing sequences, which means using the same weight at different time steps.This can reduce the number of model parameters and reduce the risk of overfitting.The designed training framework is shown in Figure 2. Learning is performed by embedding the channel model in the autoencoder.The autoencoder consists of an encoder side, a channel side, and a decoder side, which is mathematically described as follows: where  (•) is the coding side,  (•) is the embedded channel model, and  (•) is the decoding end.The training objective is to solve the original input s at the decoding end r utilizing the original message F and the post−over−channel message G− The constellation of order  is trained with the so−called one−hot encoded vectors s ∈ S =  | = 1. . , where  is equal 1 at row i, or else 0. The decoding side ends with the SoftMax function [20], which produces a probability vector p ∈  ∈  | ∑  = 1 .During the training of the set,  =  ,  represent the training weights of the RNN as well as the biases.A minimization cross−entropy loss function () [20] is used to train .It is defined as: where  is the train batch size, generally a positive integer multiple of .The larger  is, the slower the convergence, but the better the training effect.(•) is defined as the

Training Framework Setup Based on RNN Probability Distribution
The probability distribution scheme used in this experiment was based on RNN training.RNN naturally adapts to processing data with time series or sequence structures.It can flexibly handle input sequences of different lengths and capture dependencies within the sequences.RNN adopts a weight-sharing strategy when processing sequences, which means using the same weight at different time steps.This can reduce the number of model parameters and reduce the risk of overfitting.The designed training framework is shown in Figure 2. Learning is performed by embedding the channel model in the autoencoder.The autoencoder consists of an encoder side, a channel side, and a decoder side, which is mathematically described as follows: where f θ f (•) is the coding side, C θ c (•) is the embedded channel model, and g θ g (•) is the decoding end.The training objective is to solve the original input s at the decoding end r utilizing the original message F and the post-over-channel message G-The constellation of order M is trained with the so-called one-hot encoded vectors s ∈ S = {e i |i = 1 . . .M}, where e i is equal 1 at row i, or else 0. The decoding side ends with the SoftMax function [20], which produces a probability vector p ∈ p ∈ R M + ∑ M i=1 p i = 1 .During the training of the set, θ = θ f , θ g represent the training weights of the RNN as well as the biases.A minimization cross-entropy loss function L(θ) [20] is used to train θ.It is defined as: where K is the train batch size, generally a positive integer multiple of M. The larger K is, the slower the convergence, but the better the training effect.I(•) is defined as the entropy value of the probability distribution after each iteration round.Due to the use of one-hot encoding, s only maintains a single non-zero value, so the internal summation on M only requires one evaluation.The average value of cross entropy on all samples ( 4) is calculated and used to estimate the gradient θ relative to the model parameters.Gradient θ is used to update the parameters to minimize losses.θ iterative updates based on the stochastic gradient descent (SGD) [21] are defined as: where η is the learning rate, j is the iteration time step, and ∇ θ L is the gradient estimation.We used the method of adding dropouts to the RNN structure to solve the potential overfitting problem during the training process.The addition of dropout randomly hides half of the hidden layer's neural units in each iteration of the RNN, which is not updated during that iteration.The hidden neural units in different iteration rounds are completely random.The specific parameters of the RNN training network are shown in Table 1.The definitions, parameter settings, and operations of RNN are shown in Algorithm 1.  4) is calculated and used to estimate the gradient relative to the model parameters.Gradient is used to update the parameters to minimize losses.iterative updates based on the stochastic gradient descent (SGD) [21] are defined as: where is the learning rate, is the iteration time step, and ∇ is the gradient estimation.We used the method of adding dropouts to the RNN structure to solve the potential overfitting problem during the training process.The addition of dropout randomly hides half of the hidden layer's neural units in each iteration of the RNN, which is not updated during that iteration.The hidden neural units in different iteration rounds are completely random.The specific parameters of the RNN training network are shown in Table 1.The definitions, parameter settings, and operations of RNN are shown in Algorithm 1.    Figure 3a shows the constellation's RNN-based trained probability distribution scheme when the training batch is 400, and the information entropy of the trained probability dis-tribution scheme was calculated as 3.9448 bits/symbol.The following equation calculates the information entropy: where p(x i ) is the probability of the occurrence of constellation points, and the informa- tion entropy of a uniformly distributed 16QAM was calculated to be 4 bits/symbol and 3.7864 bits/symbol based on the CCDM standard M-B distribution scheme.The net bit rate of the three schemes was calculated using the following formula: where AWG_sampling_rate is the sampling rate, set to 10 Gb/s in the experiment.
IFFT_bin_length is the data length, set to 2048 in the experiment.GI is the protection interval, set to 512 in the experiment.GIP is the cyclic suffix length, set to 256 in the experiment.Pre f ixratio is the protection interval ratio, set to 1/4 in the experiment.
until Achieve the optimization objective.return p.
Figure 3a shows the constellation's RNN−based trained probability distribution scheme when the training batch is 400, and the information entropy of the trained probability distribution scheme was calculated as 3.9448 bits/symbol.The following equation calculates the information entropy: where ( ) is the probability of the occurrence of constellation points, and the information entropy of a uniformly distributed 16QAM was calculated to be 4 bits/symbol and 3.7864 bits/symbol based on the CCDM standard M−B distribution scheme.The net bit rate of the three schemes was calculated using the following formula: where __ is the sampling rate, set to 10 Gb/s in the experiment.__ℎ is the data length, set to 2048 in the experiment. is the protection interval, set to 512 in the experiment. is the cyclic suffix length, set to 256 in the experiment. is the protection interval ratio, set to 1/4 in the experiment.Figure 3b shows the NGMI curves calculated under the corresponding signal−to−noise ratios of the three schemes, while Figure 3c shows the net bit rate curves corresponding to the sampling rates of the three schemes.It can be seen that, compared to the M−B distribution in the simulation experiments, the probability distribution based on RNN training loses some GMI performance, but has a higher information entropy and smaller transmission rate loss.Figure 3b shows the NGMI curves calculated under the corresponding signal-to-noise ratios of the three schemes, while Figure 3c shows the net bit rate curves corresponding to the sampling rates of the three schemes.It can be seen that, compared to the M-B distribution in the simulation experiments, the probability distribution based on RNN training loses some GMI performance, but has a higher information entropy and smaller transmission rate loss.

4D Biplane Fractional Order Chaotic Systems
As most of the current optical network architecture's downlinks adopt broadcast mechanisms, the transmitted information is highly vulnerable to eavesdropping by illegal ONUs.Therefore, a 4D biplane fractional-order chaotic system was introduced to obtain and encrypt chaotic sequences in four dimensions, bitstream, constellation points, subcarriers, and symbols, to ensure data security.Chaotic systems are sensitive to initial values and possess strong pseudo-randomness and ergodicity.The Lyapunov index is an important quantitative indicator for measuring the dynamic characteristics of a system.The hyperchaotic system has more positive Lyapunov exponents, which means that, even if the difference between the initial values of two orbits is small, over time, the differences will be separated exponentially, leading to the local instability and global stability of the system.Compared with traditional integer-order hyperchaotic systems, the proposed BFC system introduces fractional-order operators, making its dynamic behavior more complex and possessing stronger chaotic properties compared to integer-order chaotic systems.The BFC system employed can be represented as follows: where (a,b,c,d,e) are the parameters, D is the fractional operator, α is the fractional order, and the four chaotic sequences generated are x,y,z,w.The traditional fractional-order calculus has three definitions as the Caputo calculus definition [23], the Riemann-Liouvile (R-L) calculus definition [24], and the Grunwald-Letnikov (G-L) calculus definition [25].The 4D fractional-order system in this paper used the Caputo definition, which can be expressed as where Γ(•) is the Gamma function; t,u are the upper and lower limits of integration, respectively; and f (•) is the continuous quadratic function on the interval [u,t], where the fractional order α satisfies n − 1 < α < n.The fractional order of the discussion is controlled in the (0, 1) interval; so, we set n = 1.When α = 1, the system is equivalent to an integer-order chaotic system.Figure 4a shows the maximum Lyapunov exponent of the system for α when a = 20, b = 2, c = 20, d = 20, e = 0.2.It can be seen that, when α is between 0.800 and 1, the maximum Lyapunov exponents of the BFC system are all greater than 0, proving that the BFC system is in a hyperchaotic state at this time.The phase diagram of the BFC system is shown in Figure 4b, where α = 0.996 and (a, b, c, d, e, x 0 , y 0 , z 0 , w 0 ) were set to (20, 2, 20, 20, 0.2, 5, 1, 1, 1).It can be seen that the BFC system at this time presents a complex chaotic dynamic behavior and pseudo-randomness.

RNN−PS−OFDM Encryption Scheme
The BFC system can generate four chaotic sequences X, Y, Z, and W after M iterations at time t.The bit stream, constellation points, subcarriers, and symbols are perturbed.Firstly, the bitstream is heterogeneously encrypted, and the encryption matrix A is generated by the chaotic sequence X.The rules for generating A are as follows: where mod(•) is the remainder function and floor(•) is the downward integer function.Ten decimal places were selected for the operation to ensure the security of the encrypted sequence, and A eventually becomes a sequence of 0, 1 integers of the same length as the initial bit stream; the original bit stream is heterogeneous, with the following rules:

RNN-PS-OFDM Encryption Scheme
The BFC system can generate four chaotic sequences X, Y, Z, and W after M iterations at time t.The bit stream, constellation points, subcarriers, and symbols are perturbed.Firstly, the bitstream is heterogeneously encrypted, and the encryption matrix A is generated by the chaotic sequence X.The rules for generating A are as follows: A = f loor(mod(X•10 10 , 2)) (8) where mod(•) is the remainder function and floor(•) is the downward integer function.Ten decimal places were selected for the operation to ensure the security of the encrypted sequence, and A eventually becomes a sequence of 0, 1 integers of the same length as the initial bit stream; the original bit stream is heterogeneous, with the following rules: baseband_out_bits = bitxor(baseband_out, A) where baseband_out is the initial bitstream, bitxor(•) is the XOR operation, and baseband_out_bits is the heterodyne-encrypted bitstream.The encrypted bitstream is subsequently PS-mapped, and the constellation points are rotated and encrypted.The encryption matrix B is generated through the chaotic sequence Y.The generation rules are as follows: where round(•) is a rounding function and fix(•) is a function that rounds to 0. B eventually becomes a set of the [−π, π] sequence of radians for the phase rotation of the constellation points, and the rotation angle generation rules are as follows: where reshape(•) is an array reconstruction function that reconstructs the encryption matrix B into a set of m x n matrices' rotation angles, where m is the number of symbols per subcarrier and n is the number of subcarriers.This is added to each constellation point's phase angle to complete the constellation points' rotational encryption.Finally, the subcarrier frequencies and symbols are cryptographically scrambled, and the cryptographic matrices C and D are generated through chaotic sequences Z and W, respectively, with the following generation rules: The complex_carrier_matrix shows the final encryption. Figure 5 shows the schematic diagram after frequency and symbol perturbation.At the receiving end, if the receiver has the correct key and the corresponding decoder, it can decode the correct message.
where Tra(•) is the transpose function, sort (•) is the array−sorting function from most minor to most significant, and C and D are constant matrices that undergo multiple primary transformations.The subcarrier frequency perturbation is accomplished by multiplying the transpose transform matrix C with the constellation point matrix, and symbol perturbation is accomplished by multiplying the transpose transform matrix D with the IFFT matrix.The perturbation rules are as follows: :); complex carrier matrix complex carrier matrix C complex carrier matrix complex carrier matrix D The complex_carrier_matrix shows the final encryption. Figure 5 shows the schematic diagram after frequency and symbol perturbation.At the receiving end, if the receiver has the correct key and the corresponding decoder, it can decode the correct message.

Experimental Setup
To verify the performance of the proposed transmission scheme, an experimental system, as shown in Figure 6, was built in the laboratory to implement the RPS modulation of the raw data as well as the encryption of the BFC−based system via DSP.

Experimental Setup
To verify the performance of the proposed transmission scheme, an experimental system, as shown in Figure 6, was built in the laboratory to implement the RPS modulation of the raw data as well as the encryption of the BFC-based system via DSP.The signal was sent to an arbitrary waveform generator (Tektronix, AWG70002A) with a sampling rate of up to 25 GSa/s to generate the corresponding electrical waveform, which was then injected into a Mach-Zehnder modulator (MZM) for intensity modulation.It then passed through an erbium-doped laser amplifier (EDFA) for optical amplification.The seven-core fiber system used in the experiments had a consistent and low attenuation of 0.3 dB/km at 1550 nm wavelength, while the average insertion loss per core was approximately 1.5 dB.A continuous wave (CW) laser with a wavelength of 1550 nm and an emitted optical power of 14.5 dBm was used at the transmitter side.At the receiving end, the received optical power was adjusted by a variable optical attenuator (VOA), and finally, the optical conversion was performed using a photoelectric converter (PD).Finally, the electrical signal was received by a mixed signal oscilloscope (MSO, TekMSO73304DX), and the signal was demodulated and decrypted by an offline DSP.

Results and Discussion
After building the experimental setup shown in Figure 6, only one core was selected to transmit the standard 16QAM−OFDM signal, the M−B−PS−OFDM signal, and the trained RNN−PS−OFDM signal.To ensure the experimental fairness of 10 Gbps, the baud rates of the three modulated signals in the experiment were set to 2.5, 2.64, and 2.53 Gband.The test results are shown in Figure 7.When the BER is equal 1 × 10 −3 , the RNN− PS−OFDM signal after RNN training achieves an additional 1.23 dB of shaping gain compared to the standard 16QAM distribution.It achieves an additional 0.39 dB of shaping gain compared to the M−B distribution.A continuous wave (CW) laser with a wavelength of 1550 nm and an emitted optical power of 14.5 dBm was used at the transmitter side.At the receiving end, the received optical power was adjusted by a variable optical attenuator (VOA), and finally, the optical conversion was performed using a photoelectric converter (PD).Finally, the electrical signal was received by a mixed signal oscilloscope (MSO, TekMSO73304DX), and the signal was demodulated and decrypted by an offline DSP.

Results and Discussion
After building the experimental setup shown in Figure 6, only one core was selected to transmit the standard 16QAM-OFDM signal, the M-B-PS-OFDM signal, and the trained RNN-PS-OFDM signal.To ensure the experimental fairness of 10 Gbps, the baud rates of the three modulated signals in the experiment were set to 2.5, 2.64, and 2.53 Gband.The test results are shown in Figure 7.When the BER is equal 1 × 10 −3 , the RNN-PS-OFDM signal after RNN training achieves an additional 1.23 dB of shaping gain compared to the standard 16QAM distribution.It achieves an additional 0.39 dB of shaping gain compared to the M-B distribution.The designed transmission scheme was extended from a single−mode fiber to a seven−core fiber system to test the system stability of the proposed scheme.Figure 8 compares the BER performance of the different cores after 2 km of transmission in the seven−core fiber (where the AWG sampling rate was set to 10 Gsa/s and the MSO sampling rate was set to 10 Gsa/s).As shown in Figure 7, the measured BER curves for the seven cores almost overlap, proving that the experimental seven−core fiber transmission system is highly stable over a range of 2 km.When the BER of the system is 1 × 10 −3 , the difference in the received optical power between the best and worst cores is 0.34 dB, which proves that the experimental seven−core fiber optic transmission system has a good uniformity over a range of 2 km.In addition, the BER performance of all seven cores is within the forward error correcting (FEC) threshold when the received optical power is greater than −20 dBm.As the optical power increases, the BER performance of each core shows a significant improvement, which proves that the proposed transmission scheme still has a good transmission performance in a seven−core fiber optic transmission system.The designed transmission scheme was extended from a single-mode fiber to a sevencore fiber system to test the system stability of the proposed scheme.Figure 8 compares the BER performance of the different cores after 2 km of transmission in the seven-core fiber (where the AWG sampling rate was set to 10 Gsa/s and the MSO sampling rate was set to 10 Gsa/s).As shown in Figure 7, the measured BER curves for the seven cores almost overlap, proving that the experimental seven-core fiber transmission system is highly stable over a range of 2 km.When the BER of the system is 1 × 10 −3 , the difference in the received optical power between the best and worst cores is 0.34 dB, which proves that the experimental seven-core fiber optic transmission system has a good uniformity over a range of 2 km.In addition, the BER performance of all seven cores is within the forward error correcting (FEC) threshold when the received optical power is greater than −20 dBm.As the optical power increases, the BER performance of each core shows a significant improvement, which proves that the proposed transmission scheme still has a good transmission performance in a seven-core fiber optic transmission system.The designed transmission scheme was extended from a single−mode fiber to a seven−core fiber system to test the system stability of the proposed scheme.Figure 8 compares the BER performance of the different cores after 2 km of transmission in the seven−core fiber (where the AWG sampling rate was set to 10 Gsa/s and the MSO sampling rate was set to 10 Gsa/s).As shown in Figure 7, the measured BER curves for the seven cores almost overlap, proving that the experimental seven−core fiber transmission system is highly stable over a range of 2 km.When the BER of the system is 1 × 10 −3 , the difference in the received optical power between the best and worst cores is 0.34 dB, which proves that the experimental seven−core fiber optic transmission system has a good uniformity over a range of 2 km.In addition, the BER performance of all seven cores is within the forward error correcting (FEC) threshold when the received optical power is greater than −20 dBm.As the optical power increases, the BER performance of each core shows a significant improvement, which proves that the proposed transmission scheme still has a good transmission performance in a seven−core fiber optic transmission system.We tested the BER performance of RNN-PS-OFDM signals in the BTB mode.For comparison, we selected the best BER performance of the cores of RNN-PS-OFDM signals transmitted through a 2 km seven-core fiber transmission system.Finally, we tested the BER performance in the case of eavesdropping or brute force decryption by an illegal ONU at the receiving end without the correct private key at the receiving end performance.The experimental results are shown in Figure 9. Considering the experimental errors, it can be concluded that difference in BER performance between the RNN-PS-OFDM counterpart using the 2 km seven-core fiber system and the BTB mode transmission is small and within an acceptable range, demonstrating the feasibility of the proposed RNN-PS-OFDM transmission scheme based on a 4D biplane fractional-order chaotic system.In addition, the BER value received for forced decryption at the illegal ONU end is 0.49, confirming the security of the proposed encryption scheme.ONU at the receiving end without the correct private key at the receiving end performance.The experimental results are shown in Figure 9. Considering the experimental errors, it can be concluded that the difference in BER performance between the RNN−PS−OFDM counterpart using the 2 km seven−core fiber system and the BTB mode transmission is small and within an acceptable range, demonstrating the feasibility of the proposed RNN−PS−OFDM transmission scheme based on a 4D biplane fractional −order chaotic system.In addition, the BER value received for forced decryption at the illegal ONU end is 0.49, confirming the security of the proposed encryption scheme.Finally, the key space of the proposed RNN−PS−OFDM transmission scheme based on the 4D biplane fractional−order chaotic system was calculated.As shown in Figure 10, the key includes the initial value, control parameters, fractional order, and step size of the 4D biplane fractional−order chaotic system, i.e., {x,y,z,w,a,b,c,d,e,α,n}.Therefore, it was necessary to calculate the initial value sensitivity of each parameter.Taking variable x as an example, when the iteration step size of x is less than 10 −16 times, the BFC system loses its sensitivity to the initial value of x.Therefore, it was considered that the key space of variable x is 10 16 , and so on.For the calculation of the key space, due to the independent initial sensitivity of each parameter in the BFC hyperchaotic system used in this article, the corresponding key space is multiplied.If the step size is [1, 10 3 ], the key space is experimentally calculated as 10 15 × 10 16 × 10 15 × 10 15 × 10 17 × 10 17 × 10 16 × 10 16 × 10 16 × 10 17 × 10 3 = 10 163 , as the key space is so large that it takes a long time to find the correct key, thus effectively preventing the hijacker from obtaining the key.Finally, the key space of the proposed RNN-PS-OFDM transmission scheme based on the 4D biplane fractional-order chaotic system was calculated.As shown in Figure 10, the key includes the initial value, control parameters, fractional order, and step size of the 4D biplane fractional-order chaotic system, i.e., {x,y,z,w,a,b,c,d,e,α,n}.Therefore, it was necessary to calculate the initial value sensitivity of each parameter.Taking variable x as an example, when the iteration step size of x is less than 10 −16 times, the BFC system loses its sensitivity to the initial value of x.Therefore, it was considered that the key space of variable x is 10 16 , and so on.For the calculation of the key space, due to the independent initial sensitivity of each parameter in the BFC hyperchaotic system used in this article, the corresponding key space is multiplied.If the step size is [1, 10 3 ], the key space is experimentally calculated as 10 15 × 10 16 × 10 15 × 10 15 × 10 17 × 10 17 × 10 16 × 10 16 × 10 16 × 10 17 × 10 3 = 10 163 , as the key space is so large that it takes a long time to find the correct key, thus effectively preventing the hijacker from obtaining the key.
Photonics 2023, 10, x FOR PEER REVIEW 11 of 13 ONU at the receiving end without the correct private key at the receiving end performance.The experimental results are shown in Figure 9. Considering the experimental errors, it can be concluded that the difference in BER performance between the RNN−PS−OFDM counterpart using the 2 km seven−core fiber system and the BTB mode transmission is small and within an acceptable range, demonstrating the feasibility of the proposed RNN−PS−OFDM transmission scheme based on a 4D biplane fractional −order chaotic system.In addition, the BER value received for forced decryption at the illegal ONU end is 0.49, confirming the security of the proposed encryption scheme.Finally, the key space of the proposed RNN−PS−OFDM transmission scheme based on the 4D biplane fractional−order chaotic system was calculated.As shown in Figure 10, the key includes the initial value, control parameters, fractional order, and step size of the 4D biplane fractional−order chaotic system, i.e., {x,y,z,w,a,b,c,d,e,α,n}.Therefore, it was necessary to calculate the initial value sensitivity of each parameter.Taking variable x as an example, when the iteration step size of x is less than 10 −16 times, the BFC system loses its sensitivity to the initial value of x.Therefore, it was considered that the key space of variable x is 10 16 , and so on.For the calculation of the key space, due to the independent initial sensitivity of each parameter in the BFC hyperchaotic system used in this article, the corresponding key space is multiplied.If the step size is [1, 10 3 ], the key space is experimentally calculated as 10 15 × 10 16 × 10 15 × 10 15 × 10 17 × 10 17 × 10 16 × 10 16 × 10 16 × 10 17 × 10 3 = 10 163 , as the key space is so large that it takes a long time to find the correct key, thus effectively preventing the hijacker from obtaining the key.

Conclusions
In paper, a recurrent-neural-network-based probabilistic shaping transmission scheme for high-security constellations was proposed, and a 4D biplane fractional-order chaotic system was introduced to ensure the security performance of the system.Compared with integer-order chaotic systems, the introduction of fractional order makes the chaotic system have a more complex nonlinear dynamical behavior and pseudo-randomness, and the key space reaches 10 163 .The proposed scheme achieved a shaping gain of 0.39 dB compared to the standard M-B distribution and 1.23 dB compared to the standard 16QAM signal.The proposed scheme has a higher net bit rate than the standard M-B distribution scheme.When the AWG sampling rate was 10 Gb/s, the proposed scheme achieved a net bit rate of 48.82 Gb/s (6.97 Gb/s × 7 = 48.82Gbit/s) for the transmission of encrypted RNN-PS-OFDM signals.Therefore, the proposed transmission scheme balances the efficient transmission and security of the fiber optic transmission system, which is a promising future fiber-optic transmission scheme.

Figure 2 .
Figure 2. Training framework for the RNN−based probability distribution schemes.

Figure 2 .
Figure 2. Training framework for the RNN-based probability distribution schemes.

Figure 3 .
Figure 3. (a) Probability distribution at K = 400.(b) Simulated NGMI curves for the three signals.(c) Net bit rate curves for the three signals.

Figure 3 .
Figure 3. (a) Probability distribution at K = 400.(b) Simulated NGMI curves for the three signals.(c) Net bit rate curves for the three signals.

Figure 4 .
Figure 4. Maximum Lyapunov exponent (a) and phase diagram (b) of a 4D biplane fractional−order chaotic system.

Figure 4 .
Figure 4. Maximum Lyapunov exponent (a) and phase diagram (b) of a 4D biplane fractional-order chaotic system.

Figure 5 .
Figure 5. Schematic diagram of the frequency and symbol perturbation.

Figure 5 .
Figure 5. Schematic diagram of the frequency and symbol perturbation.

Figure 8 .
Figure 8. BER performance of the RNN−PS−OFDM signal after 2 km transmission in a seven−core fiber system.

Figure 8 .
Figure 8. BER performance of the RNN−PS−OFDM signal after 2 km transmission in a seven−core fiber system.

Figure 8 .
Figure 8. BER performance of the RNN-PS-OFDM signal after 2 km transmission in a seven-core fiber system.

Figure 9 .
Figure 9. BER curve for the RNN−PS−OFDM signal transmission in a seven−core fiber system for 2 km, BTB mode, and illegal ONU.

Figure 9 .
Figure 9. BER curve for the RNN-PS-OFDM signal transmission in a seven-core fiber system for 2 km, BTB mode, and illegal ONU.

Figure 9 .
Figure 9. BER curve for the RNN−PS−OFDM signal transmission in a seven−core fiber system for 2 km, BTB mode, and illegal ONU.

Figure 10 .
Figure 10.Key space of the proposed RNN-PS-OFDM scheme.
Algorithm 1:Training process of RNN Definition 1. Build the RNN as shown in Figure2;2.Obtain the coding−side data s; 3. Obtain the decoding−side data r; 4. Define the optimization objective as the optimal transmission performance;