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Abstract
A mode division multiplexing (MDM) chaotic encryption scheme based on key intertwining and accompanying transmission is proposed in this paper. Based on the weakly coupled few-mode fiber (FMF), data and time-varying keys can be accompanied by transmission in two modes, LP01 and LP11, respectively. In order to generate a new key, the current key is XORed with all of the keys from all the preceding moments, one by one. To implement chaotic masking in the digital domain, the three chaotic sequences corresponding to the new key are adopted to encrypt the data at the constellation phase, data symbol block, and subcarrier levels. An 8.89 Gb/s encrypted 16QAM-OFDM signal transmission over 1 km weakly-coupled FMF is experimentally demonstrated. The receiver with the correct key can recover the data normally, while the BER of the illegal receiver remains around 0.5. In the case of the key transmission bit rate of 1 Gb/s, the cracking efficiency threshold of the time-varying key encryption scheme is 5.21 × 106 times that of the time-invariant key encryption scheme, which suggests that the proposed work is a promising candidate for future physical layer security.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Introducing new services like the Internet of Things, the metaverse, and digital twin technologies have ushered in a new era of interconnection. The optical fiber transmission infrastructure, which serves as the primary conduit for Internet traffic, must contend with greater demands for high capacity and information security issues. Data security protocols and digital certificates are examples of upper-layer security encryption schemes. Quantum encryption, optical chaotic encryption, and digital chaotic encryption are examples of physical layer security encryption schemes emerging one after another to ensure information security. Eavesdroppers can perform brute force breaking for the upper-layer encryption techniques by copying a huge amount of data and processing them using parallel signal processing computing units. It is simple to actualize such data cracking with the rapid development of computing power, such as quantum computing. However, compared with the former, the physical layer security schemes encrypt the optical signal. The security of the data can essentially be guaranteed even if an eavesdropper manages to intercept a partial power of the optical signal in the optical fiber transmission link since it cannot understand the genuine plaintext.
The physical layer encryption schemes have a trade-off between high-security performance and strong communication capability. Quantum key distribution (QKD) generates and shares verifiably secure keys between two users via quantum mechanical properties [1–5], which can not only distribute keys but also realize the perception of eavesdroppers. Currently, it is considered one of the safest encryption schemes. However, compared with the transmission rate of ciphertext, the rate of keys generation is too low to support the simultaneous transmission of time-varying keys and ciphertext. Additionally, optical chaotic encryption is a security method for encryption in the optical domain that either applies broadband chaos as an optical carrier to transform the transmitted data into a noise-like signal [6–9] or utilizes the chaotic signal as a driving signal to perform optical phase disturbance [10–12]. Based on the chaotic synchronization mechanism of a semiconductor laser, the parameters of a slave semiconductor laser in the transmitter and the receiver are matched, realizing the synchronization of encryption and decryption. However, in the actual chaotic optical system, the increase in transmission distance will introduce the accumulation of optical fiber dispersion, which makes it challenging to maintain the chaotic synchronization of high-speed optical signals and restricts the transmission distance of physical layer secure optical communication. Besides, the above schemes are incompatible with the existing optical transmission architecture. Before they are applied in the high-speed optical transmission, it is necessary to modify the communication system equipment.
Digital chaotic encryption based on digital signal processing (DSP) has attracted extensive attention from researchers in recent years [13–21]. This scheme realizes the chaotic masking of data by perturbing the process of bit coding, constellation mapping, and multi-carrier modulation in DSP, which is compatible with the existing optical transmission architecture so that the strong communication capability of a high-speed optical transmission system can be maintained. An offset quadrature amplitude modulation filter bank multi-carrier (OQAM-FBMC) encryption scheme based on 7-dimensional (7D) Cellular Neural Network (CNN) was proposed in [22], in which the key space of 7D CNN can reach 101575, and the scrambling degree can be maintained at 100%. An encryption scheme based on manifold learning-assisted generative adversarial networks (MFGANs) was proposed in [23], in which the complex structures from various chaotic models were learned to achieve the sizeable key space. A novel probabilistic shaping (PS) based constellation encryption scheme was proposed in [24], where two bit-level encryption operations were firstly performed according to chaotic sequences and hash values, and the proposed PS-based encryption scheme can obtain approximately 2.4 dB gain in terms of receiver sensitivity. Chaotic compressive sensing (CS) encryption algorithm for orthogonal frequency division multiplexing passive optical network (OFDM-PON) was proposed in [25], which aimed at compressing the transmitted data and enhancing the security of data transmission. Recently, researchers have reported a secure key distribution and synchronization method by embedding the key into the synchronization header in OFDM-PON [26]. In addition, we also proposed a noise-masking secure key distribution by utilizing power dimension in our previous work [27]. These schemes have striking advantages including low implementation complexity and high system compatibility. But they use the fixed key in a chaotic system, that is, they do not support time-varying chaotic encryption and key distribution. Even if the chaotic system has a large enough key space, it still offers eavesdroppers the possibility to decipher the information.
In this paper, a mode division multiplexing (MDM) chaotic encryption scheme based on key intertwining and accompanying transmission is proposed, in which LP01 and LP11 modes are respectively used for the simultaneous transmission of ciphertext and time-varying keys in weakly coupled few-mode fiber (FMF). Each key has a connection to all the keys that came before it. Given the circumstances, it is still impossible to discern genuine key information even if the eavesdroppers simultaneously steal both the key and the ciphertext. In addition, the three chaotic sequences generated corresponding to the actual key after intertwining are used to encrypt the data at three levels of constellation phase, data symbol block, and subcarrier to realize the chaotic masking in the digital domain. An 8.89 Gb/s encrypted 16QAM-OFDM signal transmission over 1 km weakly-coupled FMF is experimentally demonstrated to verify the feasibility of the proposed scheme.
2. Principle
2.1 MDM chaotic encryption architecture
The proposed MDM chaotic encryption scheme based on key intertwining and accompanying transmission is shown in Fig. 1, which employs two nearby optical modes, LP01 and LP11. As the fundamental mode, LP01 is relatively less crosstalk introduced by other modes, making it more suitable to be used as the data bearing pipeline. Compared with the number of bits in the data part, the number of bits to be transmitted in the key part is much less. This paper uses LP11, a higher-order mode, to transmit the key. Note that the proposed scheme can be extended to multimode multiplexing systems, since the bits of transmitted key are fewer, we can still adopt one of the high order modes to transmit keys with low transmission rate, and adopt other modes to transmit the synchronous encrypted signals. The bitstream frame structures of key and data are shown in Fig. 1(b). The time consumption of transmission of the two parts is the same, and each frame corresponds one to one to realize the chaotic data encryption with a time-varying key. It is worth noting that since the number of bits of the key is less, a lower order modulation format can be used to reduce the net bit rate of the key channel utilizing interpolation or circular key sending. This can keep the frame length of the KEY consistent with that of the data and ensure the reliability of key transmission.
Fig. 1. The architecture of MDM chaotic encryption based on key intertwining and accompanying transmission.
2.2 MDM chaotic encryption architecture
To be more explicit, the chaotic encryption is performed with a classical three-dimensional Chua’s circuit map, which can be formulated as [28]:
The MDM chaotic encryption scheme proposed in this paper puts a forward higher requirement on the initial value sensitivity of the chaotic model. Figure 3 shows the phase diagrams for two cases where the initial values $\{{{x_0},{y_0},{z_0}} \}$ are $\{{ - 1.0,0.2,1.1} \}$ (corresponding to the yellow line) and $\{{ - 1.0 + {{10}^{ - 15}},0.2 + {{10}^{ - 15}},1.1 + {{10}^{ - 15}}} \}$ (corresponding to the blue line). It can be seen from the partially enlarged figure that when the initial value has a tiny change of 10−15, the phase trajectories of a chaotic model under these two conditions are quite different. This is also the significance of the time-varying key encryption scheme, that is, the constant change of the key over time brings about great changes in the chaotic sequence, effectively enhancing the difficulty of eavesdropping.
Fig. 3. Phase diagrams with initial values of {−1.0, 0.2, 1.1} and {−1.0 + 10−15, 0.2 + 10−15, 1.1 + 10−15}.
In this paper, OFDM-based multicarrier modulation is adopted for data transmission. So, according to the three chaotic sequences x, y, z generated by the chaotic model, the data are perturbed at three levels of constellation phase, data symbol block, and subcarrier, respectively. The specific data perturbation process includes the following two steps:
Firstly, the obtained chaotic sequences x, y, and z are processed as follows:
Secondly, an M-order square matrix Q is constructed, where the value of the column represented by q is set to 1 row by row. And the N-order square matrix W is constructed, where the value of the row represented by w is set to 1 column by column. The data encryption of the OFDM signal is realized according to the constellation phase disturbance factor and the constructed two disturbance matrices Q and W:
The encryption processes are shown in Fig. 4. After a phase disturbance in a constellation, there will be a shift in where the points make up the constellation based on how severe the disturbance was. QAM symbol positions on the time-frequency block in OFDM are entirely jumbled after disruption of OFDM symbols and subcarriers. Note that, because the digital domain encryption doesn’t affect the key accompanying transmission scheme, other digital chaotic encryption methods can also be added or replaced to the encryption process.
Fig. 4. Schematic diagram of (a) constellation phase perturbation; (b) OFDM time-frequency scrambling.
2.3 Key intertwining and accompanying transmission
In this paper, the initial values $\{{{x_0},{y_0},{z_0}} \}$ are used as the key. The decimal key is converted into a binary bit stream with a fixed length to enable the accompanying transmission of key and data. Since x, y, z in Chua's circuit map can support even a tiny change ∼10−15, it is determined that each initial value retains 16 significant digits, including the one integer digit and the 15 decimal digits. And the decimal to binary conversion is carried out according to the order of the key bits. Among them, the value of the integer digit is less than 8, which can be represented by 3 bits. Each digit on the decimal part is an integer in the range of [0, 9], which is expressed with 4 bits following the 8421BCD code, namely {0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001}. In addition, an extra bit is used with the key as a sign bit, where ‘0’ is negative and ‘1’ is positive. Therefore, the key starts with the sign bit. The decimal point is eliminated during the transformation. This way, the initial decimal values {x0, y0, z0} are converted into three sets of 64-bit keys, where the 1st bit represents the sign, the 2nd to 4th bits represent the integer part, and the 5th to 64th bits represent the decimal part.
The time-varying key is transmitted with data in the proposed chaotic encryption scheme. To prevent eavesdroppers from intercepting key and data simultaneously during transmission and performing ciphertext cracking, the 5th to 64th bits (the decimal part of the initial key value) are intertwined by the stepwise XOR operation. Specifically, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} _{5\sim 64B}}^{(i)} = {({x_0}^{(i)},{y_0}^{(i)},{z_0}^{(i)})_B}$ represents the 5th to 64th bits of the ith key transmitted in LP11 mode, which is the key loaded at the transmitter. When the perturbation is performed on data, the key is processed in two steps:
Firstly, the 1st to the 4th bits in the key remains unchanged, and the 5th to the 64th bits are XORed step by step with the corresponding bits in the previous key, establishing the intertwined association between the ith key and the previous 1st to the (i-1)th key:
Secondly, since the decimal to the binary conversion of the decimal part of keys is based on 8421BCD code, six invalid cases (1010,1011,1100,1101,1110,1111) will overflow after the above XOR operation. Therefore, we convert the obtained ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} _{5\sim 64B}}{^{(i)^\prime}} = ({x_0}^{(i)^\prime},{y_0}^{(i)^\prime},{z_0}^{(i)^\prime})_B$ to a decimal number by the following procedure:
The above operation ensures that each number in the decimal position is an integer in the range of 0 to 9. By doing so, the current key is linked to the keys from all previous times. Since the key was not available when the ciphertext was generated, it cannot be cracked even if eavesdroppers suddenly eavesdrop on the intermediate link and obtain both the key and data information simultaneously. Figure 5 shows the key $\{{{x_0},{y_0},{z_0}} \}$ transmitted in LP11 mode and the key used after key intertwining association $\{{{x_{0real}},{y_{0real}},{z_{0real}}} \}$, as well as their difference. It can be seen that the key intercepted by the eavesdropper is completely different from the actual key.
For the time-varying key chaotic encryption scheme, the bit length and update frequency of the keys determine the anti-cracking ability. In this paper, a parameter called cracking efficiency threshold (CET) is defined to measure the security performance of this scheme quantitatively:
where n is the bit length of a single set of the keys and 2n is the key space with a time-invariant key, R is the key transmission bit rate, and $R/n$ represents the update times of the key per unit time, that is the update frequency of the key. The unit of CET is s−1, representing the number of calculations per second to decipher the key successfully. Figure 6 shows the curve of CET as a function of R and n. It can be seen that a higher key transmission rate and a longer key bit length mean a higher cracking efficiency threshold. The values of R and n are adjustable according to the actual system requirements. One can choose larger values of n and R if the system requires higher security performance. While one can choose smaller values of n and R if the system requires higher transmission performance.3. Experiment and results
An experiment is conducted, as shown in Fig. 7(a), in which the key and encrypted data are transferred based on intensity modulation direct detection to test the effectiveness of the proposed MDM chaotic encryption system based on key intertwining and accompanying transmission. Specifically, at the transmitter, the key and encrypted data generated by offline DSP are fed into an arbitrary waveform generator (AWG, Tek AWG70002A) with a maximum sampling rate of 25 GSa/s for digital-to-analog conversion. Then the electrical signals are amplified by the electrical amplifiers (EAs), and injected into the Mach-Zehnder modulator (MZM) for intensity modulation. Two continuous-wave (CW) lasers with a linewidth of 100 kHz are used as the light sources, whose wavelength is 1550 nm, and optical power is 10 dBm. A mode multiplexer multiplexes two optical signals together. the applied mode multiplexer is a spatial mode multiplexer and demultiplexer based on multi-plane light conversion (MPLC) with a size of 64 × 64 × 46 mm. It can support four modes including LP01, LP11a, LP11b, and LP21. The working wavelength range of the mode multiplexer is 1530 nm to 1565 nm. The insertion loss is less than 6 dB and the average mode isolation is greater than 10 dB at the wavelength of 1550 nm. The key is loaded onto the LP11 mode, and the encrypted data is loaded onto the LP01 mode. In the transmission link, a two-mode fiber with a length of 1 km is used, which can support the transmission of LP01 and LP11 modes. At the receiver, the same MPLC-based mode demultiplexer as the transmitter is applied to demultiplex the key and encrypted data optical signals. Two variable optical attenuators (VOAs) are placed to adjust the received optical power (ROP) for subsequent optical signal detection via photodetector (PD). And a mixed signal oscilloscope (MSO, TekMSO73304DX) with a single channel maximum sampling rate of 100 GSa/s is used for analog-to-digital conversion of the detected electrical signals. The offline DSP performs data decryption and recovery reversed to the transmitter.
Fig. 7. Experimental setup (AWG: arbitrary waveform generator; EA: electrical amplifier; MZM: Mach-Zehnder modulator; VOA: variable optical attenuator; PD: photodiode; MSO: mixed signal oscilloscope).
The detailed DSP processes of key and data at the transmitter and receiver are shown in Fig. 7(b-e). Both parts of the information are modulated with OFDM-based multicarrier modulation. A total of 256 subcarriers are adapted to carry the key or data, while another 256 subcarriers are used to carry the corresponding complex conjugates to fulfill the Hermitian symmetry to ensure that the output signal is a real value. The IFFT size is 1024, and 1/8 of the OFDM symbol length is added as the cyclic prefix (CP). The key adopts BPSK modulation, while the data adopts QPSK or 16QAM.
As described in principle, the bit rates of key and data are not the same during the transmission. The higher the key transmission rate, the higher the update frequency of the key, and the larger ROP required for a key signal transmitted on LP11 mode. Therefore, we fixed the sampling rate of AWG as 10 GSa/s, and the data bit rates based on QPSK-OFDM and 16QAM-OFDM are 4.44 Gb/s and 8.89 Gb/s, respectively. The minimum ROP (namely ROP threshold) required by the proposed encryption scheme is analyzed by changing the bit rate ratio of data to key. As the experimental results shown in Fig. 8, it can be seen that the key parts are the same regardless of the modulation format, so the ROP thresholds at the same key bit rate in Fig. 8(a) and (b) are consistent. As the bit rate of the key decreases, the bit rate ratio of data to key gradually increases, and the ROP threshold decreases. However, since the key bit rate is gradually reduced to below 1 Gb/s, under such a low transmission rate, the change of bit rate ratios of data to key has little effect on the ROP threshold. This also guides the selection of the key rate; that is, a larger ratio can be selected under the same ROP condition to increase the update frequency of the key and improve the system’s security.
Fig. 8. ROP thresholds for (a) QPSK-OFDM and (b) 16QAM-OFDM at various bit rate ratios of data to key.
Figure 9 shows the BER curves of QPSK-OFDM and 16QAM-OFDM before and after the encryption, as well as at the illegal receiver, respectively, in which the ROP of the key signal is fixed as −18 dBm. With the increase of ROP, the BER of the unencrypted signal and the legal receiver with the correct key decrease gradually, while the BER of the illegal receiver remains around 0.5, which shows the security of the proposed encryption scheme. And it is undeniable that the encryption process also comes with a slight power penalty in the experiment, where the receiver sensitivity penalty at the BER of 1 × 10−3 is 0.6 dB and 0.9 dB for QPSK-OFDM and 16QAM-OFDM, respectively.
Fig. 9. (a) BER curves of the encrypted QPSK-OFDM, normal QPSK-OFDM, and illegal receiver; (b) BER curves of the encrypted 16QAM-OFDM, normal 16QAM-OFDM, and illegal receiver.
Due to the use of key intertwining, each received set of key ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} ^{(i)}} = ({x_0}^{(i)},{y_0}^{(i)},{z_0}^{(i)})$ needs to be associated with the previous key ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} ^{(0)}}$, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} ^{(1)}}$,…, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} ^{(i - 1)}}$ before it can be substituted into the chaotic model as a correct key for decryption. To verify the security performance of this key intertwining scheme, we conduct security experiments on the following three conditions: (a) illegal receivers do not eavesdrop on any key information; (b) illegal receivers only eavesdrop ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} ^{(i)}}$; (c) illegal receivers eavesdrop on three sets of key ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} ^{(i)}}$, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} ^{(i + 1)}}$, ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over K} ^{(i + 2)}}$. The experimental results are shown in Fig. 10. It can be seen that the BER in the three cases maintains at about 0.5, which verifies that the proposed intertwining scheme exhibits good security performance. In addition, when the three initial values of the key x0, y0, z0 are changed by 10−15, respectively, the corresponding difference of the 15th digit after the decimal point of the chaotic sequence is shown in Fig. 11, which is randomly distributed within the integer range in [−9,9]. The combination of Chua’s circuit map with good initial value sensitivity and key intertwining scheme can effectively strengthen the security of digital chaotic encryption.
Fig. 11. When (a) x0, (b) y0, and (c) z0 is changed by 10−15, the corresponding difference of the 15th digit after the decimal point of the chaotic sequence.
Finally, we compare the CET of the time-varying and time-invariant key chaotic encryption schemes at various key bit rates. The results are shown in Fig. 12. Taking the total transmission of the 1s key as the calculation standard, the bit length of the key in this paper is 192 bits, and thus the key space of the time-invariant key chaotic encryption scheme is 2192. If cracking time is limited within 1s, the CET is 2192 = 6.28 × 1057 s−1. For the time-varying key encryption scheme, according to the key transmission rate of 1 Gb/s, its CET is 109/192 × 2192 = 3.27 × 1064 s−1, which is 5.21 × 106 times of the former. With the increase of the key transmission bit rate, the CET of the time-invariant key encryption scheme remains constant, while the CET of the time-varying key encryption scheme increases linearly.
Fig. 12. The CET of time-varying and time-invariant key chaotic encryption schemes at various key bit rates.
4. Conclusion
We have proposed an MDM chaotic encryption scheme based on key intertwining and accompanying transmission, where LP01 and LP11 modes are respectively adopted for the accompanying transmission of ciphertext and time-varying keys in weakly coupled FMF. The key at each moment is intertwined with the keys of all previous moments, which brings a great challenge to the illegal receivers who want to decipher the information by intercepting the key and data simultaneously. Moreover, the CET of the time-varying key encryption scheme is 5.21 × 106 times that of the time- invariant key encryption scheme in the case of the key transmission bit rate of 1 Gb/s. The feasibility of the proposed scheme is verified by an 8.89 Gb/s encrypted 16QAM-OFDM signal transmission over 1 km weakly-coupled FMF, proving that it could be an effective solution for future physical layer security.
Funding
National Key Research and Development Program of China (2018YFB1800901); National Natural Science Foundation of China (61935011, 61835005, 62171227, 61727817, U2001601, 62035018, 61875248, 61935005, 61720106015, 61975084); Jiangsu Provincial Key Research and Development Program (BE2022079, BE2022055-2); The Natural Science Foundation of the Jiangsu Higher Education Institutions of China (22KJB510031); Jiangsu team of innovation and entrepreneurship; The Startup Foundation for Introducing Talent of NUIST.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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