Physical layer encryption for coherent PDM system based on polarization perturbations using a digital optical polarization scrambler

Xianfeng Tang; Zeyu Xu; Chuanwei Gao; Yang Xiao; Le Liu; Xiaoguang Zhang; Lixia Xi; Hengying Xu; Chenglin Bai

doi:10.1364/OE.497301

1. Introduction

With the deployment and maturity of the fifth-generation mobile network, the global data traffic has been experiencing explosive growth due to the rapid development of various mobile internet businesses. Additionally, the surge in demand for services such as the Internet of things, virtual reality, and cloud computing has led to an increased need for connections between data centers [1–5]. The optical transmission system is facing enormous challenges due to the massive data transmission demand. As a result, it is of great importance to establish high-rate, large-capacity, and high-spectral-efficiency optical transmission systems in optical network [6–8]. To address this challenge, diverse technologies have been developed, among which coherent optical polarization division multiplexing (PDM) systems are widely used in backbone networks and trending towards application in short-distance connection systems due to their high capacity and spectral efficiency [9–12].

As a multiplexing method, PDM technology utilizes the polarization property of lightwave to realize the simultaneous transmission of two independent data sequences through two orthogonal polarization states of light in the same wavelength channel [13–17]. The introduction of PDM technology can effectively double the total capacity of the system and improve the system's spectrum efficiency. Furthermore, PDM technology exhibits good compatibility with digital signal processing (DSP) technology and advanced systems such as orthogonal frequency division multiplexing (OFDM) [18,19]. However, most of the research on PDM has focused on channel capacity and spectrum efficiency while neglecting the security of the system. In optical transmission systems, the transmitted data can be easily eavesdropped upon, and physical disruptions can result in security degradation [20,21]. Therefore, in recent years, many researchers have proposed encryption schemes to overcome this problem.

Currently, encryption techniques in different Open System Interconnection (OSI) layers are proposed to ensure secure signal transmission in optical fiber communication systems. Among these techniques, optical physical layer encryption is widely studied due to its advantages of transparent protocol, full encryption, and low complexity [22–26]. Most of these encryption methods utilize technologies to spread or scramble signal in amplitude, frequency, and phase domains [27–32]. Moreover, multiple encryption schemes have been developed to exploit joint domains to further increase disturbance among diverse domains [33–37]. However, the polarization domain in PDM systems is seldom explored to enhance disturbance.

In this paper, we introduce polarization perturbations to the transmitted signal by digitally rotating the state of polarization at the transmitter through DSP, which explore the polarization property to realize encryption in PDM system. Rotation of state of polarization (RSOP) is a common phenomenon in the PDM system, resulting from the random birefringence of optical fiber, leading to the unpredictable variation of the polarization state of the signal transmitted in the channel and the incorrect separation of the dual-polarization signal at the receiver. In extreme scenarios such as lightning, ultra-fast RSOP up to Mrad/s can occur in optical fiber links, resulting in a significant number of errors in the communication system [38–42]. The distortion caused by RSOP is regarded as a challenge which inspires a lot of studies on RSOP model establishment and equalization technologies [43–48]. However, this study artificially introduces a key-controlled high-speed RSOP which is beyond the receiver's compensation ability to prevent eavesdropping from illegal users. In order to increase the flexibility of applying RSOP, the concept of digital optical polarization scrambler (DOPS) is adopted and a series of digital polarization device matrices are operated on the input signal in the DSP at the transmitter, which results in the equivalent effect of optical polarization perturbations.

DOPS can be implemented using matrices for digital polarization devices, including digital polarization rotator (DPRO) and digital phase retarder (DPRE), similar to conventional optical polarization devices. Four encryption modes are designed by using different combinations of DPRO and DPRE and by adjusting the key parameters in the digital polarization matrix. To expand the key space, a 5-D chaotic system is introduced to accommodate the five critical parameters involved in the encryption process. By incorporating the inherent randomness and unpredictability of the chaotic system, the encryption scheme becomes more resistant to potential attacks and increases the difficulty of unauthorized decryption. To verify the feasibility and security of the encryption algorithm, an experimental PDM-QPSK transmission system with a data rate of approximately 32 Gbps is employed. The experimental results demonstrate the effectiveness and safety of the encryption scheme, providing confidence in its practical application.

2. Principles

Figure 1 illustrates the schematic diagram of a physical layer encryption scheme based on polarization perturbations of transmitted signal using digital optical polarization scrambler (DOPS) to introduce a chaotic key-controlled high-speed RSOP in a coherent optical PDM system. In the electrical domain through DSP, two pairs of I and Q signals are digitally scrambled using digital polarization device matrices. This process results in the equivalent effect of RSOP perturbations applied to the optical signals after PDM modulation and delivers extensive flexibility for designing various encryption modes.

Fig. 1. Schematic diagram of the proposed encryption scheme based on polarization perturbations of transmitted signal using DOPS in coherent PDM system. DOPS: digital optical polarization scrambler.

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The encryption mode and the parameters of the DOPS can be selected as the keys such as polarization rotation angle and phase delay angle, which can be fully exploited to determine RSOP speed and polarization perturbation effect. Such RSOP perturbation introduced is far beyond the ability of the state-of-the-art algorithms at the coherent receiver, making it impossible for eavesdroppers to eliminate RSOP even with the most advanced adaptive DSP approaches without keys. Additionally, to further increase the resistance against brute force attack, a five-dimensional (5-D) chaotic system is implemented to generate key sequences with high randomness and substantially expand the key space of the encryption method. After processing in the DSP, the electrical encrypted signals are modulated to two mutually orthogonal polarized optical carriers via conventional dual-polarization IQ modulator and thus the optical polarization perturbed signal is generated. Subsequently, the encrypted signals are transmitted to the receiver over the optical fiber. The encryption algorithm is reversible, allowing legitimate receivers to obtain the correct information by implementing the corresponding decryption steps in the reverse process.

2.1 Encryption based on the digital optical polarization scrambler

Conventionally, polarization adjustment is accomplished using optical polarization devices such as polarization rotator and retarder. However, in PDM systems, the signal is modulated onto two orthogonal polarization states, which enables polarization adjustment in the electrical domain by pre-polarizing the signal using DSP technology at the transmitter. By appropriately manipulating the I and Q signals on the X and Y polarizations, polarization changes can be introduced in a manner equivalent to those achieved optically. This implementation approach offers superior flexibility and control in polarization management, enabling encryption based on polarization perturbations.

Figure 2 depicts the principle of transitioning from utilizing DOPS in electrical domain to inducing polarization perturbations to optical signal. In high-speed PDM coherent optical communication systems, a dual-polarization IQ modulator is employed to modulate the electrical signal onto the optical carrier. The transfer function of the modulator for each polarization branch can be expressed as follows:

(1)$$\begin{aligned} \quad \Psi &= \cos \left( {\frac{\pi }{{2{V_\pi }}}{V_1}(t)} \right) + \cos \left( {\frac{\pi }{{2{V_\pi }}}{V_2}(t)} \right){e^{j\frac{\pi }{2}}}\\ &= \cos \left( {\frac{\pi }{{2{V_\pi }}}({v_{RF1}}(t) + {V_{bias1}})} \right) + j\cos \left( {\frac{\pi }{{2{V_\pi }}}({v_{RF2}}(t) + {V_{bias2}})} \right) \end{aligned}$$

where V_π is half wave voltage; v_RF1(t), v_RF2(t) respectively refer to the signal voltages to be input by I, Q arms; V_bias1, V_bias2 are the bias voltages of I, Q arms respectively. In conventional PDM high-order modulation, the bias voltages are often set at the null point where V_bias1= V_bias2=-V_π. So formula 1 can be written as:

(2)$$\Psi = \sin \left( {\frac{\pi }{{2{V_\pi }}}{V_{RF1}}(t)} \right) + j\sin \left( {\frac{\pi }{{2{V_\pi }}}{V_{RF2}}(t)} \right)$$

Fig. 2. The principle of transitioning from utilizing DOPS in electrical domain to inducing polarization perturbations to optical signal. DOPS: Digital optical polarization scrambler; DPRO: Digital polarization rotator; DPRE: Digital polarization retarder.

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The input electrical signal can be represented as a vector:

(3)$$A = \left( \begin{array}{l} {I_X} + j{Q_X}\\ {I_Y} + j{Q_Y} \end{array} \right)$$

where I_X, Q_X, I_Y, Q_Y correspond to the in-phase and quadrature components that modulate the two orthogonal polarization states of the optical carrier in a PDM system. The state of polarization (SOP) of the modulated optical signal can be written as:

(4)$$|E \rangle = {\Psi _x}{\widehat e_x} + {\Psi _y}{\widehat e_y}$$

where ${\mathrm{\Psi }_x} = \textrm{sin}\left( {\frac{\pi }{{2{V_\pi }}} \cdot {I_X}} \right) + j\textrm{sin}\left( {\frac{\pi }{{2{V_\pi }}} \cdot {Q_X}} \right)$;${\mathrm{\Psi }_y} = \textrm{sin}\left( {\frac{\pi }{{2{V_\pi }}} \cdot {I_Y}} \right) + j\textrm{sin}\left( {\frac{\pi }{{2{V_\pi }}} \cdot {Q_Y}} \right)$;${\hat{e}_x} = \left( {\begin{array}{{c}} 1\\ 0 \end{array}} \right)$;${\hat{e}_y} = \left( {\begin{array}{{c}} 0\\ 1 \end{array}} \right)$. For the typical modulation formats in PDM systems, both amplitude and phase modulation are used to modulate the two orthogonal polarization states of the optical carrier in the electro-optical modulation process. Consequently, the amplitude ratio and phase difference in the Jones vector of the modulated signal are determined by the electrical signal vector A as:

(5)$$|E \rangle \propto A = \left( \begin{array}{l} {I_X} + j{Q_X}\\ {I_Y} + j{Q_Y} \end{array} \right)$$

The left panel of Fig. 2 illustrates the complex plane constellation of the dual-polarization electrical signal for the PDM-QPSK modulation, and its corresponding relationship with the Stokes vectors on the Poincaré sphere in the optical domain after modulation. It can be inferred that the polarization state of the modulated signal is strongly correlated with the distribution of the applied electrical signal A.

Thus, utilizing DOPS to disturb the electrical signal vector A, perturbations can be introduced to the corresponding SOP of the modulated optical signal on the Poincaré sphere, enabling optical polarization-perturbation-based encryption.

The digital optical polarization scrambler can be represented mathematically using a matrix:

(6)$$U = \left( \begin{array}{l} {U_{11}}\textrm{ }{U_{12}}\\ {U_{21}}\textrm{ }{U_{22}} \end{array} \right)$$

DOPS can be implemented using matrices for digital polarization devices, including DPRO and DPRE, similar to conventional optical polarization devices. The right panel of Fig. 2 shows the complex plane constellation and the corresponding distribution of Stokes vectors on the Poincaré sphere in the optical domain after modulation for polarization scrambling utilizing a digital polarization rotator. The matrix for the digital polarization rotator can be expressed as follows:

(7)$${U_1} = \left( \begin{array}{c} \cos \theta \textrm{ } - \sin \theta \\ \sin \theta \textrm{ }\cos \theta \end{array} \right)$$

where θ is the rotation angle in the range of [0, π]. As seen from the figure, the PDM-QPSK signal's constellation on the complex plane is perturbed, and the corresponding distribution of the optical signal's polarization state is scattered around the equator and poles of the Poincaré sphere. Additionally, the digital phase retarder, as another fundamental polarization device, can be utilized to further scramble the output polarization. The matrix for the zero azimuth digital phase retarder can be expressed as follows:

(8)$${U_2} = \left( \begin{aligned} \exp (\frac{{i\Delta }}{2}) \qquad \qquad0\\ 0 \qquad \quad\exp ( - \frac{{i\Delta }}{2}) \end{aligned} \right)$$

where Δ is the phase retardation angle in the range of [0, 2π]. For the phase retarder with an azimuth angle of µ, the corresponding matrix can be expressed as:

(9)$${U_3} = \left( \begin{array}{c} \cos \mu \textrm{ }\sin \mu \\ - \sin \mu \textrm{ }\cos \mu \end{array} \right)\left( \begin{array}{l} \exp (\frac{{i\Delta }}{2}) \qquad 0\\ \textrm{ }0 \qquad \quad \exp ( - \frac{{i\Delta }}{2}) \end{array} \right)\left( \begin{array}{l} \cos \mu \textrm{ } - \sin \mu \\ \sin \mu \textrm{ }\cos \mu \end{array} \right)$$

where µ is in the range of [0, π].

It can be observed that U₃ can be obtained by multiplying U₁ and U₂. Thus, by utilizing a certain number of digital polarization rotator and phase retarder matrices, DOPS has the ability to scramble SOP of the optical signal to any desired state, achieving encryption with a high degree of randomness.

2.2 Design of different encryption modes

Using the principle mentioned above, various encryption modes can be implemented by combining DPRO and DPRE, resulting in distinct polarization state trajectories of the encrypted signal in Stokes space. Four encryption modes are designed by using different combinations of DPRO and DPRE and by adjusting the key parameters in the digital polarization matrix.

(1) DPRO-based encryption mode

By utilizing a polarization rotation matrix with a changing rotation angle, the polarization states corresponding to the transmitted optical signal can be rotated and scrambled, which is equivalent to applying a fast RSOP to the transmitted signal.

The encrypted signal vector can be expressed as:

(10)$$A^{\prime} = \left( \begin{array}{l} {I_X}^\prime + j{Q_X}^\prime \\ {I_Y}^\prime + j{Q_Y}^\prime \end{array} \right) = \left( \begin{array}{c} \cos \theta \textrm{ } - \sin \theta \\ \sin \theta \textrm{ }\cos \theta \end{array} \right)\left( \begin{array}{l} {I_X} + j{Q_X}\\ {I_Y} + j{Q_Y} \end{array} \right)$$

where θ increases linearly in the range of [0, π], the initial value and increasing step serve as the encryption keys. Figure 3(e) illustrates the perturbed polarization state of the encrypted signal under the rotation angle depicted in Fig. 3(a). As can be observed, the distribution of the optical signal's polarization state is scattered around the equator and poles of the Poincaré sphere after DPRO-based encryption.

(2) DPRE-based encryption mode

Fig. 3. Changes of the rotation angle, retardation angle and azimuth angle in different encryption modes (a-d), and corresponding perturbed polarization states for DPRO mode (e), DPRE mode (f) and DPRO-DPRE modes (g-h)

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If a fast-changing azimuth angle polarization retarder is used as the encryption matrix, the encrypted signal vector can be represented as follows:

(11)$$\begin{aligned} A^{\prime} &= \left( \begin{array}{l} {I_X}^\prime + j{Q_X}^\prime \\ {I_Y}^\prime + j{Q_Y}^\prime \end{array} \right)\\& = \left( \begin{array}{c} \cos \mu \textrm{ }\sin \mu \\ - \sin \mu \textrm{ }\cos \mu \end{array} \right)\left( \begin{array}{l} \exp (\frac{{i\Delta }}{2})\textrm{ }0\\ \textrm{ }0\exp ( - \frac{{i\Delta }}{2}) \end{array} \right)\left( \begin{array}{l} \cos \mu \textrm{ } - \sin \mu \\ \sin \mu \textrm{ }\cos \mu \end{array} \right)\left( \begin{array}{l} {I_X} + j{Q_X}\\ {I_Y} + j{Q_Y} \end{array} \right) \end{aligned}$$

where azimuth angle µ changes in steps between 0 and π, and the value of phase retardation angle Δ changes periodically between 0 and 2π. Figure 3(b) depicts the changes in the two angles in this mode, and the corresponding perturbed polarization state is shown in Fig. 3(f). As can be observed, the distribution of the optical signal’s polarization state is symmetrically scattered on the Poincaré sphere after DPRE-based encryption. This DPRE-based encryption mode can also be referred to as the “eyeball mode”.

(3) DPRO-DPRE based encryption mode

Advanced encryption modes can be designed by combining DPRO and DPRE to achieve increased randomness in encryption. By cascading matrix U₁ and matrix U₂, the encrypted signal vector can be obtained in the following form:

(12)$$A^{\prime} = \left( \begin{array}{l} {I_X}^\prime + j{Q_X}^\prime \\ {I_Y}^\prime + j{Q_Y}^\prime \end{array} \right) = \left( \begin{array}{l} \exp (\frac{{i\Delta }}{2}) \qquad 0\\ \textrm{ }0 \qquad \quad \exp ( - \frac{{i\Delta }}{2}) \end{array} \right)\left( \begin{array}{c} \cos \theta \textrm{ } - \sin \theta \\ \sin \theta \textrm{ }\cos \theta \end{array} \right)\left( \begin{array}{l} {I_X} + j{Q_X}\\ {I_Y} + j{Q_Y} \end{array} \right)$$

Different encryption modes can be further achieved by varying the methods of setting rotation angle θ and retardation angle Δ. After encryption, the distribution of the optical signal's polarization state becomes more randomized and widely distributed on the Poincaré sphere.

In one mode, θ changes periodically between [0, π], and Δ changes in steps between [0, π]. In the other mode, θ changes in steps between [0, π], while Δ first changes periodically between [0, π] and then between [-π, 0]. Figure 3(c-d) shows the changes in the two angles in these two modes, and the corresponding perturbed polarization state are shown in Fig. 3(g-h).

The modes mentioned above represent common application approaches of polarization devices in optical systems. However, since DOPS is implemented in the digital domain, it allows for more diverse options in matrix design and connection techniques. By making adjustments to various design factors, such as selecting different polarization device matrices, varying their quantities, employing different cascading methods, and optimizing key parameters, it becomes feasible to broaden the range of available encryption modes and improve the flexibility and versatility of the system.

2.3 Encryption process of different modes and the key space

For each frame of data, the signal vectors are processed with specific matrices using angles that are pre-determined based on the selected encryption mode, RSOP speed, and initial values, according to the principle explained above. Figure 4 provides a detailed illustration of the encryption process for the DPRO-DPRE-1 mode. The encryption process begins by selecting the desired encryption mode. Next, the rotation speed of RSOP is set to the desired value. After that, key parameters for encryption are determined, which may include initial values, step sizes, or specific ranges for angles. Based on these parameters, the angles required for the polarization matrices are calculated. With the calculated angles, the corresponding encryption matrices are derived. Finally, these encryption matrices are applied to the data vectors to perform the encryption operation, ensuring the desired level of data security and confidentiality.

Fig. 4. Detailed encryption process for the DPRO-DPRE-1 mode.

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To prevent eavesdropping through compensation of the RSOP introduced in the encryption by the transmitter, it is important for the applied RSOP speed to exceed the upper limit of compensation achievable by existing methods such as constant modulus algorithm (CMA) and Kalman filter [44]. The RSOP speed is determined by considering the angle interval and time interval between two adjacent output polarization states. The specific values for these two parameters would depend on the encryption mode, system parameters, and desired level of security. These values need to be carefully chosen to ensure a sufficiently high RSOP speed that surpasses the compensation capabilities of potential eavesdroppers.

For the dual-polarization QPSK signal, where the bit rate at the transmitting end is denoted as r, the symbol rate R for each polarization branch can be calculated as:

(13)$$R = \frac{r}{4}$$

Assuming the mode cycle, representing the number of DOPS matrices, is denoted as N, and n represents the number of signal vectors that an encryption matrix operates on. Additionally, the number of periods of the polarization rotator in one mode cycle is denoted as m.

Supposing that the angle interval between two adjacent polarization rotation angles is uniform, we can derive the values for the angle interval a₁ and time interval Δt as follows:

(14)$${a_1} = \frac{{\pi m}}{N}$$

(15)$$\Delta t = \frac{n}{R}$$

Then the RSOP speed is derived as:

(16)$${V_1} = {a_1} \times \frac{R}{n} = \frac{{\pi m}}{N} \times \frac{R}{n}$$

In the designed modes, the encryption result is determined by the specific sequences of θ, Δ or µ angles. Therefore, the independent parameters mentioned above can be selected as the encryption keys. These parameters include the number of DOPS matrices N, the number of signal vectors n, the number of periods of the polarization rotator in one mode cycle m, the initial value θ₁, the initial value Δ₁, the initial value µ_1. The key space is determined by the hardware limitations and the requirement for RSOP speed.

Let’s assume that the minimum RSOP speed required for encryption is denoted as V_L, representing the maximum RSOP speed that can be compensated by the DSP at the receiver. In the DPRO-DPRE-1 encryption mode, the applied RSOP speed should satisfy the following condition:

(17)$${V_1} = {a_1} \times \frac{R}{n} = \frac{{\pi m}}{N} \times \frac{R}{n} > {V_L}$$

Therefore, to ensure that the minimum RSOP speed requirement is met, the value of the mode cycle N should satisfy:

(18)$$N < \frac{{\pi mR}}{{n{V_L}}}$$

For the DPRO matrix, the rotation angles within one period should be integers ranging from 2 to floor(2πR/(nV_L)), where the floor function rounds down the value. Let h denote the number of possible initial values for θ₁ and Δ₁. The number of periods of the polarization rotator in one mode cycle, denoted as m, is an independent parameter with a value range of integers from 2 to h/2. Therefore, the key space of the DPRO-DPRE-1 mode can be expressed as follows:

(19)$${P_1} = {h^2}(\frac{h}{2} - 1)(floor(\frac{{2\pi R}}{{n{V_L}}}) - 1)$$

Similarly, the key spaces of the other three modes can also be derived.

Taking a dual-polarization QPSK system with a transmission rate of 100 Gbps as an example, assuming that the DSP at the receiver can compensate for a maximum RSOP speed of 3 Mrad/s. Due to hardware limitations, the maximum number of possible initial values for the θ angle in the electrical domain is on the order of 10⁴. By substituting these values into the above equation, the key space for the DPRO-DPRE-1 mode in one encryption cycle is estimated to be on the order of 10¹⁶.

2.4 Extension of key space using 5-D chaotic sequences

Based on the discussion in Section 2.3, it has been recognized that the key space of encryption using DOPS within a single mode cycle is comparatively limited and insufficient to withstand brute-force attacks from eavesdroppers. To address this issue, a 5-D chaotic system is introduced to accommodate the five critical parameters involved in the encryption process. By incorporating the inherent randomness and unpredictability of the chaotic system, the key space is significantly expanded, making it more resistant to potential attacks and increasing the difficulty of unauthorized decryption.

The Lorenz chaotic system is a widely recognized nonlinear dynamic system renowned for its chaotic behavior. It is characterized by a distinct butterfly-shaped attractor in three-dimensional space. Operating in continuous time, the Lorenz system allows for smooth and continuous evolution of its variables. It offers several advantages, including a wide range of possible trajectories, making it suitable for applications requiring real-time or continuous data encryption. Additionally, it exhibits sensitive dependence on initial conditions and lacks long-term predictability. Here, a 5-D Lorenz chaotic system is designed as an extension of the classic Lorenz system. By incorporating five variables instead of three, this enhanced system introduces increased complexity and dynamical behavior. As a result, it exhibits more intricate and unpredictable chaotic dynamics compared to the original 3-D Lorenz system. The additional dimensions provide greater freedom for chaotic trajectories to explore in a higher-dimensional space.

The equations of the extended 5-D Lorenz chaotic system are as follows:

(20)$$\begin{array}{l} \mathrm{\dot{x}\ =\ y\ +\ m\ +\ z(}{\textrm{n}^\textrm{2}}\textrm{ - 3)}\\ \mathrm{\dot{y}\ =\ -\ x\ +\ zm\ +\ m}\\ \mathrm{\dot{z}\ =\ -\ ym\ +\ }{\textrm{n}^\textrm{2}}\textrm{ - 3}\\ \mathrm{\dot{m}\ =\ -\ x\ -\ y\ +\ }{\textrm{n}^\textrm{2}}\textrm{ - 3}\\ \mathrm{\dot{n}\ =\ -\ xz\ -\ z\ -\ m} \end{array}$$

These equations describe dynamics of the variables x, y, z, m and n as they evolve with time. The 5-D chaos systems are employed to derive the values of N, m, n, and the initial angles, which function as the encryption keys for the DOPS method, as outlined in the following expressions:

(21)$$\begin{array}{c} \textrm{X = mod(x} \times \textrm{1}{\textrm{0}^{\textrm{15}}}\textrm{,10000)}\\ \textrm{Y = mod(y} \times \textrm{1}{\textrm{0}^{\textrm{15}}}\textrm{,10000)}\\ \textrm{Z = mod(z} \times \textrm{1}{\textrm{0}^{\textrm{15}}}\textrm{,10000)}\\ \textrm{M = mod(m} \times \textrm{1}{\textrm{0}^{\textrm{15}}}\mathrm{,2\pi )}\\ \textrm{N = mod(n} \times \textrm{1}{\textrm{0}^{\textrm{15}}}\mathrm{,\pi )} \end{array}$$

Unpredictability is a critical characteristic of a chaotic system, where the values in the output sequences exhibit random variations with time. Figure 5(a) depicts the state variables generated by the 5-D chaotic system, demonstrating the substantial changes in state values as the number of iterations increases. The autocorrelations of the x and X sequences are illustrated in Fig. 5(b), while the cross-correlations between the x and n, X and N sequences are shown in Fig. 5(c), providing evidence of the exceptional autocorrelation and cross-correlation performance of the sequences derived from the 5-D chaotic system. Consequently, the high level of randomness in the output sequences effectively guarantees the unpredictability of the 5-D chaotic system, thereby enhancing its overall security.

Fig. 5. (a) Change of state values of 5-D chaos system with time. (b) Auto-correlations of sequence x and X. (c) Cross-correlations between sequences x and n, X and N.

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The distribution characteristic of the chaotic sequence is an important criterion for assessing its randomness. According to Shannon's theorem, if the distribution of a random sequence is non-uniform, the amount of information decreases, and the decryption difficulty decreases accordingly. In Fig. 6(a), the distribution of the five normalized chaotic variables is presented. Here, x’ to n’ represent the sequences after being normalized to the range of 0 to 1. Figure 6(b) displays the distributions of the derived key sequences after applying a modulus operation. The distribution of the key sequences appears to be more uniform due to the incorporation of multiplication and division operations. This enhanced uniformity in the distribution of sequences contributes to the overall randomness of the 5-D chaotic system, thereby reinforcing its encryption capability.

Fig. 6. Distributions of the normalized 5-D chaotic sequences (a) and derived key sequences (b).

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Furthermore, the designed 5-D chaotic system demonstrates a notable sensitivity to initial values, indicating that even a slight alteration in the initial value can lead to a completely different system state after a short time. Figure 7(a) illustrates the presence of positive Lyapunov exponents (LE), signifying the system's transition to chaotic motion. Moreover, Fig. 7(b) shows that a slight perturbation of 10⁻¹² in the initial value of the chaotic sequences results in a completely distinct sequence. This characteristic of high sensitivity to initial values endows the designed 5-D chaotic system with the potential to achieve a large key space, enhancing its security and resistance against cryptographic attacks. Moreover, by increasing the reading interval of the chaotic values, it is possible to further enhance the sensitivity to initial values, thereby increasing the key space.

Fig. 7. Lyapunov exponents (LE) of the 5-D chaotic sequences (a) and correlation coefficients of the derived key sequences with the initial disturbance of x (b).

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By incorporating the 5-D chaotic system, the total key space of the proposed system is expanded to 10⁶⁰. This substantial increase in the key space ensures a higher level of security and significantly enhances the system's resistance against brute-force attacks and other cryptographic vulnerabilities.

3. Experimental setup and results

The experimental setup for the proposed PDM system with enhanced security using polarization perturbation through DOPS is depicted in the Fig. 8. The data is first encrypted in the electrical domain at the transmitter DSP. Thereafter, the encrypted signal is utilized to modulate the laser using a PDM-IQ modulator. At the receiver, the received optical signal is coherent-detected and then converted into an electrical signal. Subsequently, decryption and DSP equalization are conducted to recover the original data.

Fig. 8. Experimental diagram of the security enhanced PDM-QPSK system based on polarization perturbation using DOPS. ECL: External Cavity Laser; DAC: Digital-to-Analogue Converter; EA: Electrical Amplifier; PDM-IQM: Polarization Division Multiplexing I/Q Modulator; TOF: Tunable Optical Filter; VOA: Variable Optical Attenuator; ICR: Integrated Coherent Receiver; DSO: Digital Storage Oscilloscope.

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In the experiment, QPSK is utilized as the modulation format in the PDM system. In the off-line DSP, the PDM-QPSK signals are encrypted using the aforementioned encryption modes. Parameters in the DOPS are set as follows: the mode cycle N is 4000, the initial values of angles in the encryption matrices are 0, the number of periods of the polarization rotator in one mode cycle m is 20, and the number of signal vectors that an encryption matrix operates on n is 1. Subsequently, the four encrypted IQ signals are converted from digital to analog using an arbitrary waveform generator (AWG) and then amplified using electrical amplifiers (EA) before being loaded onto the optical carrier using a PDM-IQ modulator. An external cavity laser (ECL) laser with a center wavelength of 1550 nm is used as the light source. The symbol rate for each polarization branch is configured to be 8 Gbaud, leading to a corresponding data bit rate of 32 Gbps. At the receiver, the optical signal with a power of -8.9 dBm undergoes coherent detection using an integrated coherent receiver (ICR). The received optical signal is then converted into an electrical signal and digitized using a digital storage oscilloscope (DSO). For the authorized users, decryption is implemented using the inverse DOPS matrices in the decryption module. Equalizations are performed in the receiver DSP to compensate for impairments such as random RSOP and phase noise.

Figure 9 displays the constellation diagrams of the polarization perturbed signal and the received signal with and without decryption in the X polarization for the four encryption modes. All the applied RSOP speeds for the four encryption modes exceed the upper limit of compensation provided by the classical CMA algorithm, which is typically around 2-3 Mrad/s. This means that the RSOP speeds used in the system are beyond the compensation capability of the CMA algorithm for all four encryption modes. For the DPRO and DPRO-DPRE-1 modes, where the number of angle values per period of the polarization rotator is P = 200, the calculation for RSOP is as follows:

{V_{DPRO}} = {V_{DPRO - DPRE - 1}} = \frac{\pi }{P} \times \frac{R}{n} = \frac{\pi }{{200}} \times \frac{{8 \times {{10}^9}}}{1} \approx 126\textrm{Mrad/s}

Fig. 9. Constellation diagrams of the polarization perturbed signal and the received signal with and without decryption in the X polarization for the four encryption modes.

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And for the DPRO and DPRO-DPRE-1 modes, the number of angle values per period of the polarization rotator is P = 4000, and the calculation for RSOP is as follows:

{V_{DPRO}} = {V_{DPRO - DPRE - 1}} = \frac{\pi }{P} \times \frac{R}{n} = \frac{\pi }{{4000}} \times \frac{{8 \times {{10}^9}}}{1} \approx 6.3\textrm{Mrad/s}

As observed in the constellations in the first row, the constellation points, represented by different colors, are substantially scrambled and scattered after encryption. The second row displays the constellations that are intercepted by unauthorized users, while the third row illustrates the constellations after successful decryption has been applied. Comparison between these constellations reveals that authorized users are able to successfully decrypt the received signal, while unauthorized eavesdroppers are unable to obtain useful information, resulting in a maintained bit error rate (BER) of approximately 0.5. This indicates that the received signal can be accurately recovered, and the proposed encryption scheme effectively enhances security in high-speed PDM systems. It is important to note that at the receiving end, the utilization of the CMA algorithm is still necessary after decryption to handle the randomly changing RSOP introduced by the channel during the transmission process.

4. Conclusion

This paper proposes a physical layer encryption scheme for coherent PDM system based on polarization perturbations, where a 5-D chaotic system is utilized to generate encryption keys. The proposed scheme applies a key-controlled high-speed RSOP to the transmitted signal, leveraging the concept of digital optical polarization scrambler. Four encryption modes are designed by using different combinations of digital polarization device matrices and adjusting their key parameters. The experimental setup demonstrates successful transmission of a 32 Gb/s encrypted PDM-QPSK signal. The results indicate that the proposed scheme achieves secure transmission in the optical physical layer, protecting against unauthorized eavesdropping. By harnessing the inherent randomness and substantial entropy of the 5-D chaotic system, the proposed scheme offers robust protection against chosen-text attacks, with an expanded key space of 10⁶⁰. Consequently, the proposed physical layer encryption scheme enhances the security of coherent PDM transmission.

Funding

National Natural Science Foundation of China (62001040); State Key Laboratory of Information Photonics and Optical Communications (IPOC2021ZT12); Science and Technology Foundation of State Grid Corporation of China (5700-202119188A-0-0-00); Research Innovation Fund for College Students of Beijing University of Posts and Telecommunications.

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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Physical layer encryption for coherent PDM system based on polarization perturbations using a digital optical polarization scrambler

Abstract

1. Introduction

2. Principles

2.1 Encryption based on the digital optical polarization scrambler

2.2 Design of different encryption modes

2.3 Encryption process of different modes and the key space

2.4 Extension of key space using 5-D chaotic sequences

3. Experimental setup and results

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (23)

Optics Express