2.1.

Vision of Private THz Chaotic Communications

Figure 1 shows the conceptual scenario of private THz chaotic communications involving multiple wireless end users who exchange broadband THz communication signals. In this example, a person-to-person communication system is depicted, where the legitimate transmitter, Alice, sends an encrypted THz signal to the legitimate receiver, Bob. The narrow beam of THz waves makes eavesdropping within the beam range challenging. However, due to the scattering and diffraction properties of THz waves, an eavesdropper, Eve, can intercept the signal by placing a passive object in the transmission path to scatter radiation toward him.

Fig. 1

Conceptual scenario of private person-to-person THz chaotic communication.

In this system, Alice encrypts the messages using physical chaos encryption, introducing intentional physical disturbances that can be reconstructed with shared hardware-based or software-based parameters as a private key. Bob, possessing the preshared key of parameters for the designed NNs from Alice, can decrypt the signal by reconstructing and eliminating the chaos, which Eve cannot do. It is important to note that this communication link is private rather than necessarily secure. Being secure implies that the system is entirely invulnerable, while being private indicates it provides a sufficient level of encryption to deter a less-determined eavesdropper.³³ This private wireless person-to-person communication system can potentially be expanded to other communication systems, such as wireless backhaul networks or satellite communication systems.

2.2.

Basic Architecture of THz Chaotic Communication Transmitter and Receiver

Figure 2 shows the basic architecture of THz chaotic communication transmitter and receiver. As for optical chaos generation, a nonlinear cavity is required to excite the chaotic oscillation, where an electro-optic modulator (EOM) is used as an instance of the key optoelectronic device for nonlinear electro-optic conversion in the external cavity. The amplifier in the cavity is essential to generate optical chaos by driving the EOM to fully utilize the nonlinearity. The chaotic oscillation is reflected on the intensity or phase of the output signal.

Fig. 2

Schematic of photonic heterodyne generation of encrypted THz signal with physical chaos.

As for THz chaos cipher generation, the output of chaos cavity is used for masking the optical plaintext in terms of intensity or phase. As shown in Fig. 2, the intensity chaos masking is, as an instance, where the plaintext-modulated optical signal with low amplitude is embedded within large intensity chaos fluctuations. In this case, messages with small amplitudes are controlled by a variable optical attenuator (VOA) and can be effectively concealed, thereby ensuring they do not interfere with larger chaotic disturbances and remain undetected by potential eavesdroppers. Concurrently, chaos introduces an expanded spectrum that overlaps with that of messages, thus making it more difficult for eavesdroppers to perform spectrum analysis. After chaos masking, optical message signal is coupled with the optical LO with the frequency of $f_2$ and injected to a broadband photodiode (PD) for photonic heterodyne detection. Encrypted signals with chaos masking in the THz domain can be generated in a photodiode, such as unitraveling-carrier photodiode (UTC-PD), and the frequency $f_{\mathrm{THz}}$ equals the difference of $f_2$ and $f_1$. To minimize the influence of phase noise induced by optical source, a coherent optical source can be applied to enhance the stability of the generated THz encrypted signal.

At the THz chaos receiver, the THz chaos-encrypted signal is received by a THz mixer and captured for further processing. After NN-based chaos synchronization and decryption, the plaintext can be obtained, which effectively simplifies the system complexity of the THz chaos receiver.

The chaotic encryption is achieved by chaos masking in terms of intensity with an optical coupler (OC) or phase with a phase modulator (PM). The general process of encryption can be expressed as

For a specific Mach–Zehnder modulator (MZM)-based chaos cavity, the numerical model is expressed by the Ikeda equation,³⁷ shown as

The left items of the equation refer to the output of an equivalent bandpass filter, which is characterized by the bandwidth $\mathrm{\Delta }\omega$ and the central frequency ${\omega }_0$. Especially, $c(t)$ in the equation is a dimensionless voltage, given as

2.3.

Neural Network Training and THz Chaos Synchronization

Figure 3 shows the principle of NN training and THz chaos synchronization for private THz chaotic communication. At the transmitter side (Alice), digital samples captured by the analog digital converter (ADC) from two PDs are used to train the NN, where the encrypted signal is used as the input matrix $\mathbf{X}$ and the generated chaotic signal is used as the desired output vector $\mathbf{Y}$. A digital lowpass filter and a least mean square filter are applied for PD and ADC channel prematching. The well-trained NN is considered as the private key that is preshared between Alice and Bob. At the legal receiver side (Bob), the encrypted THz signal needs to be downconverted to the intermediate frequency (IF) to be captured by the ADC. The received IF digital samples are then processed offline and the baseband samples are input to the NN to reconstruct the chaotic time series. The NN is applied to synchronize the chaos, which means NN reconstructs the chaotic signal $c(t)$ using the received signal $c(t)+m(t)$. The process of decryption is subtraction, where $m(t)$ can be obtained from the known $c(t)+m(t)$ and reconstructed $c(t)$.

Fig. 3

Principle of NN training and THz chaos synchronization.

3.1.

Experimental Setup of Photonic THz Chaotic Communication

Figure 4 shows the experimental setup of the private THz chaotic communication system. Here, in the experiment, the coherent optical source is realized by generating an optical frequency comb (OFC), which is excited by a 1550-mm continuous-wave (CW) laser. A PM is employed as the core device driven by a 30 GHz single-tone RF signal in the OFC generator. The upper inset of Fig. 4 depicts the optical spectrum of the OFC, where the comb space equals the driving frequency of 30 GHz. With a programmable wavelength selective switch, only two tones of the OFC remain, and the frequency difference is set to match the desired frequency of the desired THz carrier. The optical carrier at 193.461 THz is regarded as the optical LO, whereas the other with the frequency of 193.341 THz is amplified by an erbium-doped fiber amplifier (EDFA) and used for chaos cavity excitation. In the cavity, an EDFA and an RF amplifier (RFA) are used for optical and electrical amplification to obtain enough gain driving the MZM, respectively. The MZM is the key device for nonlinear electro-optical conversion to excite chaotic oscillations in the external cavity, whose DC bias point is affected by the experimental conditions and has a great impact on the dynamic characteristics and nonlinearities of the cavity.³⁸ In our experiment, the stability of the chaotic oscillation operating point is practically enough for several minutes of transmission and data capture after the DC bias is set. If longer stability is desired, active control of the operating point is needed.^39,40 There is an extra branch leading out of the cavity, where a PD is applied to capture the chaos as vector $\mathbf{X}$ only for training an NN.

Fig. 4

Experimental setup of private THz chaotic communication.

As for message loading, the optical carrier is modulated by a 5 Gbaud NRZ signal, and the frequency of the optical carrier needs to be the same as the central frequency of the generated optical chaos. A VOA is set after the MZM to adjust the mixing ratio of chaos to messages to be masked. After coupling, the encrypted signal needs to be captured as vector $\mathbf{Y}$ only for NN training outside the cavity. To achieve the photonic heterodyne detection, the encrypted signal needs to be coupled with the optical LO. The coupled optical spectrum of the chaos-masked signal and the optical LO is shown in the lower inset of Fig. 4; the frequency difference between two signals is 120 GHz. To optimize the photo-mixing efficiency of the UTC-PD, the optical power of the chaos-masked signal is adjusted to make it balanced with that of the optical LO. After being amplified by the EDFA, the coupled signal is adjusted in polarization and injected into the UTC-PD to generate the THz signal at 120 GHz. A polarization-maintaining VOA is employed to regulate the input optical power, consequently controlling the photocurrent of the UTC-PD. It should be mentioned that the heterodyne detection scheme benefits the flexible THz chaos-encrypted signal generation, which will facilitate the practical deployment of THz chaotic communication systems at a proper THz carrier frequency.

Influences induced by atmospheric factors^34,41–45 need to be considered for THz communications. In the experiment, the transmission distance is set as 0.5 m, which will be extended longer for practical deployments in the future. In this case, influences induced by atmospheric factors can be ignored in the experiment. After the wireless transmission, the electrical coherent manner is applied to receive the THz encrypted signal with a THz subharmonic mixer. The RF source offers a single-tone signal with the frequency of 10.833 GHz and drives the mixer with an electrical LO at 130 GHz after 12 times up-conversion. The IF signal with the frequency of 10 GHz is then sampled by a DSO (digital storage oscilloscope, i.e., ADC) with a sampling rate of $80\mathrm{GSa}/\mathrm{s}$, and the digital samples are stored for offline processing.

3.2.

Synchronization Performance for Photonic THz Chaotic Communication

Modeling of the optical wireless transmission link is essential to achieve high-quality chaotic synchronization. In the experiment, a full-connected NN is trained to reconstruct the nonlinear chaos cavity. The digital samples are divided into two parts for model establishment, where 80% of samples were used as the training subset, and the remaining 20% were used as the validation subset. The input dimension is set to 5000, and the NN features eight hidden layers, each with 770 neurons and an output layer with a single neuron. The network is trained using backpropagation (BP) and the Adam optimizer with a learning rate of 0.008. Rectified linear unit (ReLU) [max(0, $x$)] acts as the activation function. Mean squared error loss between the original chaos and the NN output is used for parameter updates, with training conducted over 60 epochs. Hyperparameters of the NN may need minor adjustments for optimal performance in the experiment.

Figure 5(a) shows the NN training performance in terms of C.C of up to 97.3% at Alice’s side, which indicates the nonlinear chaos cavity is excellently mathematically modeled. As is shown in Fig. 5(b), the temporal fluctuation of NN output behaves similarly to the original chaos. The well-trained NN is first self-tested using the encrypted signal at Alice’s side and the C.C results (red triangles) with seven different mask coefficients ($\alpha$) are shown in Fig. 5(c), where $\alpha$ is defined as, $V_{\text{ppMessage}}/V_{\text{ppSignal}}$, and the $V_{\mathrm{pp}}$ denotes the PtP value of the optical message to optical chaos. All C.Cs are over 93%, and the mean value reaches 94.6%, showing the excellent potential of trained NN for chaos synchronization. For the synchronization performance of Bob, the C.C deteriorates and the mean value decreases to 90.6%, which is shown by the blue pentagrams in Fig. 5(c). In general, a C.C above 90% indicates the chaos decryption can be achieved to support the chaotic communication. Device mismatch mainly causes the C.C’s degradation induced by different PDs and different channels of DSO. In the proposed scheme, the NN is trained on a specific nonlinear chaos cavity designed for a particular THz private communication system.^22,46–49 The nonlinearity inherent in the system acts as a hidden private physical key, enhancing the privacy of THz communication with chaos encryption.

Fig. 5

Photonic THz chaos synchronization performance. (a) NN training performance between physical chaos and chaos from NN with 45-deg line; (b) temporal waveform comparison between physical chaos and chaos from NN; (c) C.C results at Alice’s and Bob’s sides.

3.3.

Photonic THz Chaotic Communication Performance

Figure 6 shows the BER performance with different $\alpha$ in four different decryption situations. Considering the device mismatch at Alice’s side, there exists an optimal BER limit in the experiment, shown by the red line with a square. The optimal BER is tested in back-to-back transmission; the C.C is assumed to be 100%. It means that all experimental BERs will never be smaller than the optimal BER when $\alpha$ is fixed. The yellow line with a triangle is the BER performance of Alice, where the well-trained NN is self-tested for chaos synchronization. The result is obtained in back-to-back transmission, but unlike the test for an optimal BER line, the chaos synchronization is realized with a trained NN and is definitely not 100%, resulting in a BER penalty between Alice’s result and optimal BER limit.

Fig. 6

Transmission performance in terms of BER and eye diagram with different mask coefficient ($\alpha$).

The blue line with a hexagon is the BER performance of Bob, where Bob can decrypt the chaos-masked signal with the preshared NN with clear eyes. The BER of Bob can reach under the SD-FEC⁵⁰ limit @ $2.0\times {10}^{-2}$ when $\alpha$ is larger than 0.75 and under HD-FEC⁵¹ limit @ $3.8\times {10}^{-3}$ when $\alpha$ is larger than 0.90. As for Eve without the shared NN, the BER performance deteriorates with indistinguishable eyes by direct cracking of the encrypted signal, as shown by the gray line with a solid circle. The obvious BER penalty between Bob and Eve suggests chaos encryption has brought enough privacy to legal receivers.

The photocurrent of the UTC-PD at the transmitter is maximized to achieve the highest possible THz signal output power. The BER performance of the THz communication system without encryption is evaluated under these same conditions. The BER results were obtained below $3.6\times {10}^{-7}$. The comparison reveals a penalty for encrypted transmission compared to unencrypted transmission, primarily due to errors in chaotic synchronization on the receiver side. However, the enhanced privacy provided by chaos will offer greater benefits to the communication link than the trade-off between encrypted and unencrypted transmission.

The masking ratio between optical message and chaos significantly impacts both the privacy and reliability of the THz chaotic communication system.^35,46,49 In general, a larger $\alpha$ results in weaker privacy because the chaotic signals, being a smaller fraction of the combined signal, may be perceived as random noise that disrupts THz communications. Conversely, a smaller $\alpha$ enhances privacy but complicates decryption for legitimate receivers, as the plaintext signal is perceived as low-amplitude noise relative to chaotic signals. Since chaotic synchronization in legitimate receivers is always imperfect, errors in synchronization affect the accuracy of decrypting low-amplitude plaintext signals. Thus, there is an inherent trade-off between privacy and transmission performance for a given $\alpha$, which is experimentally determined for each specific THz chaos system. In the experiment, when $\alpha$ is adjusted to smaller than 0.75, optimal BER limit is larger than the SD-FEC limit with the deteriorated performance of legal decryption at both transmitter side (Alice) and legal receiver side (Bob). High interference introduced by chaos degrades the BER performance of Alice and Bob, even though Eve’s BER deteriorates more severely. Conversely, setting $\alpha$ to greater than 1 introduces significant security risks, with Eve’s BER dropping below 0.1, even though Alice and Bob achieve optimal BER performance, which is not acceptable. Therefore, proper masking coefficient ($0.8\le \alpha \le 0.95$) for THz chaotic system enables information to be effectively hidden and recovered by legal receivers and resist eavesdropping.

3.4.

Privacy Analysis of Photonic THz Chaotic Communication

To further quantitatively evaluate the transmission performance and privacy enhancement brought by chaotic encryption for THz communications, the data capacities of the legitimate channel and illegitimate channel are first calculated, which are defined as $C_{\text{Bob}}$ and $C_{\mathrm{Eve}}$, respectively.⁵² The data capacity means the maximum average mutual information in the transmission channel and the value of $C_x$ closer to 1 refers to a better transmission performance in this channel:

The $C_{\text{Bob}}$ and $C_{\mathrm{Eve}}$ versus the mask coefficient are shown in Fig. 7, which characterize both transmission and privacy performance. Here, Eve conducts brute-force attacks to the THz chaotic communication system, which means Eve does not obtain the chaotic time series but demodulates the signal directly. It should be mentioned that larger $C_{\mathrm{Eve}}$ means lower privacy. It is clearly shown that the transmission performance makes obvious progress characterized by $C_{\text{Bob}}$ with the increasing $\alpha$, while $C_{\mathrm{Eve}}$ also has an increasing trend with $\alpha$ increasing, indicating the deteriorating privacy performance. Moreover, the penalty between Bob and Eve in terms of $C_{\text{Bob}}-C_{\mathrm{Eve}}$ can be used for comprehensive assessment of the privacy and transmission performance, where larger value indicates a better comprehensive performance of privacy and transmission. As is shown in the inset of Fig. 7, the penalty has a stable and slight decrease when $\alpha$ is set at a proper interval from 0.75 to 1, while it declines dramatically when $\alpha$ is larger than 1. It also shows when $\alpha$ is set to smaller than 0.75, the penalty declines, indicating the start of deteriorating performance.

Fig. 7

Data capacities $C_{\text{Bob}}$ (blue line with hexagon) and $C_{\mathrm{Eve}}$ (red line with dot) versus the mask coefficient ($\alpha$).

Brute force attack to the NN also needs to be considered for privacy analysis. In a full-connected NN, weight, bias, and activation function of each neuron layer present an infinite array of possibilities. The system with the parameter space larger than $2^{100}$ can exhibit enough sensitivity to prevent brute force attacks.^22,53 In the experiment, the NN contain more than 11,000 parameters. Assuming each parameter can take $k$ discrete values, parameter combinations exceed more than $k^{11000}$, making it difficult to brute force the network by parameter enumeration, thus demonstrating the privacy of NN.

In addition, assuming that Eve has the ability to fully reconstruct the same chaotic signal as Alice’s, the privacy of chaotic communication is analyzed by chaos mismatch, which is characterized by the mismatch time, with $\alpha$ is fixed at 0.85. The information interception probability $\eta$ of Eve can be used for the evaluation of privacy and is defined as the ratio of the illegitimate channel capacity in the source entropy:⁵²

The $\eta$ of Eve is shown in Fig. 8. The value of $\eta$ that is closer to 1 means worse privacy of the THz chaotic communication system. It can be clearly seen that only if the chaos of Eve matches Alice’s exactly can Eve decrypt the encrypted signal with clear eyes and $\eta$ of 0.993 and tiny time mismatch (only 1 symbol) makes it hard to crack for Eve, thus ensuring the high privacy of transmission between Alice and Bob.

Fig. 8

Information interception probability ($\eta$) of Eve with chaos mismatching time.

Considering the free ciphertext attack, it is assumed that Eve can reconstruct the chaotic time series without messages and can train an NN. The attack performance of Eve with the trained NN in terms of BER is stable with above 0.1 as $\alpha$ changes, which is shown in Fig. 9. The result demonstrates that optical messages play a crucial role in nonlinear opto-electro conversion within the chaos cavity. Even if Eve successfully reconstructs the chaotic time series with a C.C of up to 96.0%, Eve cannot retrieve the messages.

Fig. 9

BER performance of Eve with the free ciphertext attack.