Doppler Delay-Time Frequency Cross-Domain Joint High Security Transmission Scheme Based on Orthogonal Time Frequency Space

A doppler delay-time frequency (DD-TF) cross-domain high-security transmission scheme based on orthogonal time frequency space (OTFS) is proposed, which is compatible with high-speed and large-capacity optical transmission systems. In this scheme, 36 sets of chaotic sequences are mapped out through a two-dimensional chaotic system with variable control parameters and variable initial values for optical physical layer encryption. The data modulated based on OTFS is encrypted by multiple sets of chaotic sequences, achieving cross-domain joint encryption in the DD domains and TF domains. This method improves security performance of the optical transmission system. An experiment is conducted, in which the encrypted 16- quadrature amplitude modulation (QAM) OTFS signals over 25 km standard single-mode fiber (SSMF) is achieved, and the key space is 101140 to effectively prevent malicious attacks from illegal Optical Network Units (ONUs). This key space is the reported maximum key space of the physical layer of two-dimensional chaotic systems. Compared with orthogonal frequency division multiplexing (OFDM) encryption scheme, this scheme obtains better transmission performance. It is shown that OTFS system with cyclic prefix (CP) of 1/16 obtains 2.1 dB gain than the OTFS system with CP of 1/32 in terms of receiver sensitivity when the bit error rate (BER) is 10-3. Due to its comprehensive advantages in error rate and security performance, this chaotic encryption based on OTFS has a high application prospect in low-cost and reliable optical access system.

the cornerstone of global data transmission, the optical fiber communication system is facing enormous pressure on the optical fiber access system [2]. At present, the fiber-terahertz seamless fusion architecture has been proposed [3], [4]. This technology is considered to be one of the important potential technologies of 6G [5]. Terahertz communication can realize ultra-high-speed wireless communication due to its extremely rich spectrum resources [6]. With the help of photon-assisted terahertz communication, ultra-high-speed and large-capacity 6G wireless communication can be realized. As the fronthaul part of the core system, fiber channel is particularly important for its signal transmission quality. The terahertz wireless channel characteristics vary with time and space, and the multi-path response is also time-varying, thus the received signal will go through serious fading, which affects the reliability of the communication system [7]. Therefore, the quality of the fiber fronthaul signal is particularly important. At the same time, the transmission rate of information continues to increase, the number of user terminals continues to increase, and the information security between users cannot be ignored [8]. Due to its complex characteristics such as sensitivity to initial values and parameters, topological transitivity, aperiodicity, and high dynamic complexity, chaotic systems are widely used in secure communications [9]. Because the properties of chaotic map meet the requirements of cryptography, it is very suitable for the encryption of optical communication physical layer.
In recent years, many scholars have adopted multi-carrier technology in the digital domain to improve the spectral efficiency and transmission rate of optical communication systems, thereby further increasing the capacity of optical transmission systems [10], [11]. At present, the most commonly used multicarrier technology is orthogonal frequency division multiplexing (OFDM) technology [12], [13], [14], [15], [16]. The introduction of this technology has brought about a qualitative change in spectral efficiency. However, in the OFDM system, a cyclic prefix (CP) is introduced to avoid crosstalk between information. The introduction of the CP improves the quality of information transmission, but introduces an overhead of transmission capacity. In order to further shorten the CP, improving spectral efficiency and alleviating crosstalk caused by multipath effects, orthogonal time frequency space (OTFS) technology was proposed [17], [18]. OTFS is a two-dimensional modulation scheme that This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ introduces a doppler delay (DD) domain on which information modulation is performed. In the DD domain, information can be distributed in both time and frequency, and roughly orthogonality can be maintained [19]. OTFS is more applicable in high-speed communication scenarios, and it can better resist inter-carrier interference (ICI) and inter-symbol interference (ISI). OTFS can convert the fast time-varying channel into a slow-varying channel in the DD domain, and obtain nearly constant channel gain [20]. OTFS is also a new DD representation, which can also be called Heisenberg lattice representation, and its essence is a multiplexed quadrature amplitude modulation (QAM) information symbol modulation scheme [21]. Compared with OFDM, OTFS can effectively reduce the proportion of CP in the data stream, improve spectral efficiency, and effectively combat ISI and ICI caused by delay spread. The article analyzed the performance of OTFS and OFDM in high-speed environment, and the experimental results show that in the case of high-speed communication, OTFS outperforms OFDM [22]. Some scholars realized NOMA transmission and multi-user transmission under the OTFS transmission framework [23], [24].
With the number of users increasing and data traffic exploding, the security issues of data transmission between these terminals cannot be ignored, as it plays an important role in user data and national information security [25]. At present, the chaotic encryption method in the digital domain is widely used. It has high flexibility and can be compatible with the existing high-speed optical network communication system [26], [27]. Compared with the traditional upper-layer algorithm encryption scheme, the chaotic physical layer encryption scheme in the digital domain effectively avoids the risk of data header leakage, and is compatible with a variety of modulation methods. The chaotic system has highly nonlinear dynamic behavior and quasi-randomness [28], which is mainly manifested in: 1) extreme sensitivity of initial value, 2) boundedness, 3) ergodicity, 4) quasi-randomness, 5) fractal dimensionality. So many scholars have conducted extensive research and proposed many cryptographic algorithms based on chaotic system mapping. Chen uses a seven-dimensional chaotic system to encrypt data, but the complexity of the seven-dimensional chaotic system is greatly improved [29]. In terms of combining with other multicarrier technologies [30], [31], security encryption technologies of UFMC and FBMC have been proposed. In addition, with the continuous development of neural network technology, the scheme of combining chaotic physical layer encryption with neural network has also been reported [32]. The above schemes realize effective protection of data, but they all realize large key space through high-dimensional chaotic system, which increases the complexity of the system. In the fronthaul part of the 6G fiber-terahertz architecture, the combination of multi-carrier technology and chaotic encryption technology needs to be further optimized [33].
In this article, a joint cross-domain encryption based on doppler delay-time frequency (DD-TF) is proposed in the optical communication system. As the potential of 6G in the optical fiber-terahertz wireless fusion technology solution, the signal quality of its fronthaul part is particularly important. The use of OTFS in the fiber fronthaul part can effectively alleviate the degradation of information transmission. Compared with OFDM, in the case of CP = 1/16, OTFS has obtained improvement in receiving sensitivity. At the same time, in order to protect the data security in transmission, this article adopts DD-TF cross-domain joint encryption, and generates 72 chaotic sequences through the two-dimensional chaotic system to encrypt the data in the OTFS modulation process, among which the chaotic sequence x is used for DD domain encryption, the chaotic sequence of y group is used for TF domain encryption. This article verifies the feasibility of the scheme through experiments. In this article, encrypted signal transmission is realized on 25 km standard single-mode fiber (SSMF), the key space reaches 10 1140 , and the sensitivity of the encryption system reaches E-15. This key space is the reported maximum key space of the physical layer of two-dimensional chaotic systems.

II. PRINCIPLES
The principle of DD-TF cross-domain encryption based on OTFS proposed in this article is shown in the Fig. 1. The OTFS modulation is mainly divided into a transmitter and a receiver. At the transmitter, the symbols are generated in the DD domain, and then the information is transformed into the TF domain after the inverse symplectic finite fourier transform (ISFFT) transformation. Such changes make the signal the TF domain signal. The processed TF domain signal becomes a time domain signal after Heisenberg transformation, and the signal can reduce ICI and ISI after adding a guard interval to the signal. The signal can be sent out through the transmitter. At the receiver, the demodulation of OTFS can be completed through the reverse process. The time domain signal at the receiver is converted into a TF signal after Wigner transformation, then the DD domain signal is obtained after symplectic finite fourier transform (SFFT) transformation. In the encryption part, the multi sets masking factors generated by the key and the chaotic system are carried out in the DD domain and the TF domain respectively. The 36 x chaotic sequences are generated a set every 12 interleaving, and generate x 1 , x 2 , x 3 in total. These three sets of chaotic sequences are used for encryption on the DD domain. The y sequence is also processed in the same way to generate y 1 , y 2 , and y 3 in total. These three sets of chaotic sequences are used for encryption on the TF domain.

A. The Principle of OTFS Modulation
OTFS modulates symbols in the DD domain to preserve the orthogonality between the modulated symbols over time and frequency selective channels. This means that OTFS can avoid ICI and ISI caused by multipath and doppler effects. OTFS is improved on the basis of OFDM. The ISFFT is added at the transmitter, and the SFFT is added at the receiver. The symbols can be modulated into the DD domain by two-stage two-dimensional transformation. Compared with the OFDM system, OTFS is equivalent to adding a pre-coded module in the OFDM system, and such an operation can be fully compatible with the current communication system. In the transmission process of OTFS, this article sets the number of carriers as M and the number of symbols as N. In the transmitting part, the randomly generated 01-bit information is first modulated to obtain the symbols of the MN, and then these symbols are modulated into the DD domain, and the signal can be expressed as x [k, l]. After x [k, l] is transformed by ISFFT, the signal in the DD domain is transformed into a signal in the TF domain. The mathematical expression of ISFFT is: Afterwards, the signal in the TF domain is transformed into a time domain signal after undergoing Heisenberg transformation, and the Heisenberg transformation can be expressed as: where g t () is the shaping filter. T d is the symbol length of an OTFS • Δf d is the subcarrier interval. It is worth noting that the Heisenberg transform is the same as the modulation at the OFDM transmitter. The process at the receiver can be regarded as the inverse process at the transmitter. The time domain signal passing through the channel becomes a TF signal after being changed by Wiener, and the TF signal can be expressed as: where g * r () is the matched filter function at the receiver, and then the signal is converted into a DD domain signal through SFFT transformation. r(t) is the received signal. The change of SFFT can be expressed mathematically as: It can be seen from the above process that OTFS modulates the signal to the DD domain by adding two-dimensional changes to both ends of OFDM.

B. The Principle of Encryption
In the encryption part, this article uses the two-dimensional chaotic system to generate 72 chaotic sequences for data encryption in the OTFS modulation process. The specific twodimensional chaotic system can be expressed as [34]: where f (x n ) = −0.05ax n + a(π − ax n )x 2 n /(1 + x 2 n ), the n th output of the (5) is denoted by x n , y n , and the dynamical evolution is modified by parameters a and b.
In order to demonstrate the complex dynamics of the twodimensional chaotic map, the numerical analysis method is used to study the dynamic behavior related to its control parameters. The symmetric 2D Maximal Lyapunov exponents of the (5) are plotted in Fig. 2, just like the conservative systems in continuous chaotic systems, indicating that extreme multistability is extracted. It can not only display symmetrical LEs, but also generate an infinite number of attractors with different amplitudes and different shapes coexisting.
In this article, 36 sets of chaotic sequences are obtained by changing the control parameter a and the initial value y 0 , that is, 72 chaotic sequences, which are 36 x chaotic sequences and 36 y chaotic sequences, and their phase track diagrams are shown in Fig. 3. The values of y 0 in Fig. 3(a) is 1, 2, 3 , 4, 5, 6, 8, 9, and 11. The control parameter a of Fig. 3(a) is -0.5, the control parameter a of Fig. 3(b) is 0.1, the control parameter   Fig. 3(c) is 1, and the control parameter a of Fig. 3(d) is −0.8. The complex dynamic behavior of this two-dimensional map strongly depends on the initial state y 0 . It must be pointed out that when the parameter a is given different values, the twodimensional chaotic map has completely different bifurcation behavior as the initial condition y 0 changes, which further proves that there are infinitely many attractors in the map. In addition, this article also analyzed the system through sample entropy, as shown in the Fig. 4. The lower the sample entropy is, the higher the self-similarity of the sequence is. The higher the value of sample entropy, the more complex the sequence. It can be seen from the results in the figure that the system is highly complex and the sequence autocorrelation is small.
In this article, we group and recombine 36 sets of chaotic sequences for data encryption, that is, recombine 36 x sequences and 36 y sequences. Taking the first 12 x chaotic sequence as an example, this article recombines these 12 chaotic sequences, as shown in Fig. 5. The meaning of x ij is the j th sequence of x. Recombine 12 x sequences alternate weaving into a new x sequence. A total of 36 x sequences are recombined into 3 chaotic sequences, namely x 1 , x 2 , and x 3 . The y sequence is processed in the same way to generate y 1 , y 2 , y 3 . In this article, a total of 36 x chaotic sequences are generated. This article takes 12 sequences as a group and divides 36 x sequences into 3 groups in total. The first group is shown in the Fig. 5. The 12 chaotic sequences are rearranged to obtain a new chaotic sequence for subsequent encryption. In this article, we use 36 sets of chaotic sequences, that is, 72 masking factors to encrypt the data in the OTFS modulation process. In terms of encryption, this article mainly realizes the joint encryption of DD-TF domain. In terms of DD domain, this article uses x sequence to encrypt doppler domain and delay domain respectively. In terms of TF domain, this article uses y sequence to encrypt time and frequency respectively. In the case of double-domain encryption, the data in the transmission process is effectively protected.
In DD domain encryption, this part of the data is encrypted, which can be understood as the conversion from global encryption to small local fine encryption in Fig. 6. Such an optimization algorithm avoids the high complexity of direct micro local refinement. The encryption algorithm first encrypts the data on doppler domain, and then performs replacement encryption on delay domain. So far, global encryption has been completed.
In terms of small-scale fine encryption, on the basis of global encryption, the data is cut into 2 * 2. The small scale fine encryption is completed within the 2 * 2 data range, and the 2 * 2 data is finely scrambled. It is worth mentioning that the chaotic sequence x encrypts the DD domain, but the chaotic sequence cannot be directly used for the encryption of the DD domain. In this article, x 1 sequence is used to encrypt on doppler domain. It needs to be preprocessed. The specific processing process is as follows: where · is the process of transpose, Mat(·) means that the non-zero elements of the matrix are represented as zero. P is the permutation matrix. The x 1 chaotic sequence is preprocessed to generate a masking factor to mask data on doppler domain. The encryption of this part is shown in Fig. 6(b). It can be seen in Fig. 6(b) that the data is replaced vertically as a whole, which can realize the masking of the data. The specific processing process can be expressed as: ⎡ ⎢ ⎢ ⎢ ⎣ Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply. I i is the encrypted signal of frequency masking, and I i is the original signal in the frequency domain.
In addition to completing global encryption on doppler domain, this article also implements global encryption on delay domain. The specific processing process is similar to data encryption on doppler domain. The first is to process the chaotic sequence x 2 , and the specific processing process is as follows: where · is the process of transpose, Mat(·) means that the non-zero elements of the matrix are represented as zero. Q is the permutation matrix. The x 2 chaotic sequence is preprocessed to generate masking factors to mask data on delay domain. The encryption of this part is shown in Fig. 6(c). It can be seen in the figure that the data is replaced horizontally as a whole, which can realize the masking of the data. The specific processing process can be expressed as: ⎡ where G i is the encrypted signal of frequency masking.
In terms of small scale fine encryption, this article selects a 2 * 2 data range for permutation encryption, and the specific process is shown in Fig. 6(d). In the range of 2 * 2, this article masks the positions of these four data through the chaotic x 3 sequence. The specific encryption process is as follows: Z = G 1 F wis even Z = (G 1 ) T F w is odd (11) In this article, the chaotic sequence y is used to encrypt the TF domain in Fig. 7. The encryption method on the TF domain is generally similar to the encryption scheme on the DD domain. It is mainly divided into global encryption and small local fine encryption. In terms of global encryption, it mainly implements vertical overall displacement encryption on time domain, and completes horizontal overall data masking encryption on frequency domain. In terms of overall encryption, it can see the results in Fig. 7(b) and (c), which mainly implement encryption in the horizontal and vertical directions of data. In terms of small scale fine encryption, as shown in Fig. 7(d). The encryption method of the TF domain and the DD domain are the same. The overall data is cut into 2 * 2 small scale data blocks, and refined encryption is realized in the small scale data blocks. Compared with a single encryption scheme in TF domain, the scheme proposed in this article can realize the joint encryption of DD domain and TF domain, and improve the security of data. The encryption scheme in this article uses the combination of global encryption and small scale encryption to further optimize the encryption algorithm. The key space of 10 1140 is designed by using two-dimensional chaotic system.

III. EXPERIMENTAL SETUP AND RESULTS
In order to verify security performance of the proposed chaotic encrypted system, we test it on intensity modulation/direct detection (IM/DD) system based on 25 km SSMF. The diagram of the experimental device is shown in Fig. 8, and the whole is mainly divided into a transmitter, a link and a receiver. At the transmitter, we generate encrypted data through offline digital signal processing (DSP). In DSP processing, the encrypted signal is transmitted using 512 subcarriers, and the IFFT/FFT points number is 2048. The symbol number of each subcarrier is 48, and 16QAM-OTFS technology is used for modulation. The generated encrypted data is input into the arbitrary waveform generator (AWG, TekAWG70002A), which has a sampling rate of 10 GSa/s. In the AWG, the data has completed digitalto-analog conversion, and the output signal of the AWG is injected into the electrical amplifier (EA), and then input into the MachZehnder modulator (MZM) for photoelectric modulation. The light is generated by a continuous wave (CW) laser that the wavelengths and power are set at 1550 nm and 14.5 dBm, respectively. In the experiment, in order to obtain better signal quality,  the experiment also increased the erbium-doped fiber amplifier (EDFA). In the link, the experiment uses a 25km SSMF. At the receiver, this article uses variable optical attenuator (VOA) to achieve different power reception, and then uses photodiode (PD) to perform photoelectric conversion on the signal. The mixed signal oscilloscope (MSO, TekMSO73304DX) with a sampling rate of 50 GSa/s realizes analog-to-digital conversion, and finally the signal is recovered by off-line DSP. Fig. 9 shows the bit error rate (BER) curve of encrypted signal in five different situations. First, this article tests the BER performance of encrypted data after passing through 25km fiber. It can be seen from the results in Fig. 9 that as the received optical power decreases continuously, the BER also increases continuously. In addition, this article also carried out backto-back encrypted data transmission and unencrypted signal transmission via 25km fiber, and the trend is roughly the same. The performance of the unencrypted signal is slightly better than that of the encrypted signal, but the cost is acceptable, and the security of the signal is guaranteed. In terms of illegal ONUs and illegal keys, this article also carried out tests, and none of them can be decrypted normally at the receiving end, and the error rate is above 0.4. Fig. 10 shows the BER curves under different CP situations. This part is mainly for testing between OTFS and OFDM. The values selected by CP are 1/16, 1/32 and 1/64. When 1/16 is selected for CP, the BER performance is far better than that of the other two groups. When the BER is 10 -3 , the receiver sensitivity of the OTFS modulation signal with CP of 1/16 is 2.1 dB higher than that of the OTFS modulation signal with CP of 1/32. When the received optical power is large, the performance of OTFS transmission is slightly better than that of OFDM under the same CP. When the CP becomes small, the performance gap between OTFS and OFDM is not large. In addition, this article also conducts tests on unencrypted data. The BER performance trends of OTFS and OFDM are basically the same in Fig. 11, and the transmission performance is optimal when the CP is less than 1/16. The transmission performance of unencrypted data is better than that of encrypted data compared to the transmission performance of encrypted data. This is due to the introduction of encryption algorithms, but the total cost is acceptable, and the data in transmission is guaranteed. On the whole, the performance of OTFS is slightly better than OFDM, which is only the data transmission performance as a part of the pretransmission of photon terahertz transmission, which is conducive to the data security and transmission performance improvement of terahertz wireless channels.
In addition, this article also analyzes the complexity of OTFS and OFDM. This article mainly analyzes the addition and multiplication times of OTFS and OFDM in IFFT and FFT calculation. In both OTFS system and OFDM system, the IFFT/FFT points number are all 2048 in experiments. In the OTFS system, a total of 45056 additions and 22528 multiplications are performed. In the OFDM system, a total of 22528 additions and 11264 multiplications are performed. In general, OTFS is more complex than OFDM.
Finally, the article analyzes the sensitivity of the encryption system. The system sensitivity test results are shown in Fig. 12. It can be seen from that when a, b, x, and y change, the BER will rise sharply. The sensitivity value of a and b is E-12, and the sensitivity value of x and y is E-15. The meaning of E-12 is that when the variation range of a is greater than E-12, the signal cannot be parsed normally at the receiver, and the BER is above 0.4. Specifically, when a = −0.5, the receiving end can recover the signal normally. When a = −0.5+0.000000000001, the receiver cannot recover the signal. The key space of the system has a great relationship with the sensitive value. The key space of this scheme reaches 10 1140 , which is the largest key space of the physical layer encryption of the two-dimensional chaotic system that has been reported.

IV. CONCLUSION
This article proposes the DD-TF cross-domain joint optical communication physical layer encryption scheme based on OTFS, which uses a two-dimensional chaotic system to generate 72 chaotic sequences to encrypt data during the OTFS encryption process. At the same time, 72 chaotic sequences are interleaved and recombined in this article, and 3 sets of x sequences are generated to encrypt the DD domain, and 3 sets of y sequences are respectively used to encrypt the TF domain. Joint encryption realizes the conversion from global encryption to small local fine encryption. Experiments have verified the transmission of encrypted 16QAM-OTFS signals on 25−km SSMF. In terms of key space, this article uses a two-dimensional chaotic system to realize a key space of 10 1140 . Even if sufficient optical power is received, the illegal end cannot decrypt the original data because the BER rate is above 0.4.