Enhanced chaotic communication with machine learning

In the era of big data, information encompasses various aspects of life, ranging from national security issues to individual privacy concerns. Therefore, achieving secure communication has become an urgent problem, where chaotic communication has gained prominence in recent years.^1–3 Chaotic communication is a communication technology that utilizes the dynamical characteristics of chaotic signals for information transmission and secrecy. A key feature of chaotic systems is their extreme sensitivity to perturbations, where small changes can lead to significant variations in outcomes, resulting in seemingly random and unpredictable behavior,⁴ which greatly ensures the security of the communication process. In addition, chaotic signals are also one of the natural candidates for spread-spectrum modulations owing to their inherent wideband characteristics.⁵ 

In the early stages of development, most chaotic communication processes relied on the synchronization or coherence between dynamical systems to achieve signal transmission. In Refs. 6 and 7, the strategy of Chaotic Masking (CM) was proposed, where the signal desired to be transmitted was superimposed on a chaotic signal, and decoding was accomplished using the self-synchronization of the chaotic system. However, this method is highly sensitive to noise. To enhance the robustness of communication processes against noise, the scheme of Chaotic Shift Keying (CSK) was proposed in Ref. 8, which was specifically designed for digital signals. This method encodes different bit values using different chaotic systems, but heavily relies on the selection of chaotic systems for confidentiality, and attractors with significant differences can be easily deciphered. Therefore, in Ref. 9, the authors proposed the method of parametric modulation, which achieved the encoding of original digital signals through parameter switching. In addition, there have been some other improvements^10–12 made to chaotic communication systems based on synchronization or coherence; however, there are still several common issues such as high time cost to achieve synchronization and complex synchronization circuit constructions. Moreover, noise is still an unsolved problem, which has a significant impact on the chaotic synchronization process. Even though the aforementioned developments have addressed this issue in a targeted manner and some noise filtering techniques, such as the Extended Kalman Filter (EKF), have been introduced,¹³ their effectiveness is still not satisfactory.

To overcome the problems brought by synchronization, another category of methods based on non-coherent schemes has been proposed, among which the most extensively studied is Differential Chaotic Shift Keying (DCSK). DCSK was first proposed in Ref. 14, which employed two signals of equal length to transmit the original bit values: one signal served as a reference, while the other hid the actual information, and they were sent with a delay to the receiver. At the receiving end, synchronization strategies were no longer adopted; instead, a correlator was used to decode the original signal. Although DCSK has shown significant improvements compared to synchronization or coherence-based communication designs, traditional DCSK still has some issues: it requires transmitting a reference signal, leading to low data rates and high energy consumption. In addition, other drawbacks including weakened information security and the use of wideband delay lines¹⁵ make it difficult to implement in the CMOS technology.^16,17 Subsequent efforts have been made to overcome the mentioned weaknesses of the DCSK scheme: Permutation Multiple Access Differential Chaotic Shift Keying (PMA-DCSK) enhances the security of communication, Noise Reduction Differential Chaotic Shift Keying (NR-DCSK) is proposed as a solution to reduce the noise variance present in the received signal to improve the performance,¹⁸ and Continuous Mobility Differential Chaotic Shift Keying (CM-DCSK) is designed for supporting continuous-mobility scenarios¹⁹ to address the problem of delay lines; although Code-Shifted Differential Chaotic Shift Keying (CS-DCSK)¹⁶ and High-data-rate Code-Shifted Differential Chaotic Shift Keying (HCS-DCSK)¹⁷ have made improvements to the aforementioned drawbacks, they affect the non-coherent nature of the traditional DCSK system due to the use of synchronization of chaotic or Walsh codes at the receiver.

On the other hand, over the past few years, Machine Learning (ML) has undergone rapid development and has already been applied in the secure communication field successfully.^20–23 Reservoir Computing (RC), as a special type of Recurrent Neural Network (RNN), has become a hot topic in the field of ML in recent years due to its simple structure and easy training.²⁴ It has shown great success in applications such as nonlinear system identification,^25,26 time series prediction,²⁷ speech recognition,²⁸ and hardware implementation in optical devices,²⁹ among which, RC has achieved outstanding performance in chaotic time series prediction and chaotic attractor reconstruction.³⁰ Moreover, Refs. 31–33 directly verify that RC can synchronize with chaotic dynamical systems, which lays the foundation for the application of RC in chaotic communication. In Ref. 34, the author has demonstrated the use of RC as a chaotic filter to replace the decoder at the receiving end. In this paper, we propose a new framework based on RC for the CSK scheme, which not only achieves synchronization more simply and quickly but also enhances the robustness of the entire communication system through noise training. Our design fully harnesses the predictive capability of RC, makes the practical application of the CSK scheme possible, and provides new avenues for the application of neural networks in other structures of chaotic communication.

This work is organized as follows: in Sec. II, we will briefly introduce the preliminary knowledge of synchronization or coherence-based chaotic communication as well as reservoir computing. In Sec. III, we will provide a detailed overview of the components of the transmitter and receiver in the chaotic communication framework, as well as the complete process of signal transmission and restoration. To validate the effectiveness of the proposed method, numerical simulations have been conducted in Sec. IV using the Lorenz system as encoder, and finally, we primarily discuss the impact of channel noise strength, time window length, and training noise strength on the accuracy of signal restoration in Sec. V. In addition, we also test our proposed model on M-ary digital signals.

Chaotic communication, an intriguing field within the broader domain of information theory, explores the utilization of chaotic systems for secure and robust information transmission.³⁵ Unlike traditional communication methods, chaotic communication leverages the inherent unpredictability and sensitivity to small perturbations found in chaotic systems, making it challenging for unintended recipients to decipher.³⁶ In the framework of chaotic communication, signals are often encoded and decoded using chaotic trajectories, which involves exploiting the unique properties of chaotic attractors, and the complexity and pseudo-randomness of chaotic trajectories provide a promising avenue for developing encryption techniques that are resistant to conventional cryptographic attacks.

For synchronized or coherent chaotic communication, frameworks have been categorized into two major conceptual paradigms in Ref. 1: the additive chaos-masking scheme and the chaotic shift keying scheme. The fundamental concept of CSK is to assign binary or M-ary data symbols to distinct non-periodic and wideband chaotic signals,³⁷ which is designed so that the transmitter alternates between different chaotic attractors by adjusting the parameters of the chaotic system and the receiver estimates whether the secret message corresponds to one of its states. However, regardless of how the idea of implementing chaotic communication evolves, due to the inherent nature of chaotic systems, almost all schemes are inevitably susceptible to noise interference.

Therefore, in this paper, we propose a CSK framework based on the reservoir computing technique, which can significantly simplify the process and enhance its robustness against noise deterioration. This approach extends beyond other uses of RC in chaotic communication, providing a more precise understanding and fully leveraging the RC’s potential in the predictive ability of time series.

Reservoir computing, as a variant of RNN, has garnered widespread attention and discussion in recent years, with its conceptual origins rooted in Echo State Networks (ESN)³⁸ and Liquid State Machines (LSM),³⁹ and recent developments have emerged in the theoretical understandings.^30,40,41 Its core idea is the replacement of the complex hidden layers in RNN with a reservoir, and RC is primarily composed of three components: the input layer, the reservoir, and the output layer.

Compared to the traditional RNN, RC owns a simple structure and convenient training, which eliminates the need for backpropagation and achieves its objectives through straightforward linear regression.⁴² 

During the training process, time series from different attractors are sequentially connected and accompanied by various indices, and the training results of RC_index are independent of the order of attractor arrangement and indices selection. The training of the output layer is the same as Eqs. (2) and (3).

The CSK framework is primarily designed for transmitting digital signals, whose structure consists of two main components, a transmitter and a receiver, connected through a transmission channel, as illustrated in Fig. 2(a). In the transmitter, multiple chaotic systems are employed to distinguish between different values of original digital signals. Typically, the same chaotic system is chosen but with different parameter configurations. During signal transmission, the transmitter switches between different chaotic attractors based on the original values, facilitating the effective transfer of the signal.

In the design of the receiver, it is necessary to pre-train an RC_index, the purpose of which is to make it memorize the chaotic systems used by the transmitter with the help of indices, and set up a corresponding number of duplicate reservoirs based on the number of chaotic systems. Obviously, these RCs are identical, and we name them $RCindexi$⁠. Therefore, when we receive the transmitted chaotic signal, we add the same index_i to the entire sequence and then input it into the corresponding $RCindexi$ to carry out a one-step prediction task, resulting in multiple error curves. Subsequently, we reconstruct the original digital signal based on these error curves. It is well known that the Echo State Property (ESP) is a crucial feature to ensure that RC functions effectively.^38,48 Simply put, the ESP guarantees that the state of neurons within the reservoir is uniquely determined by the input data, independent of initial values, which, in turn, ensures that the RC’s training result is a global optimum. If predictions are made when RC has not yet achieved the ESP, which means RC is still in its transient state, then the final prediction result will largely depend on the choice of initial values. In other words, even with the same input sequence, different initial values could yield different outcomes, which contradicts our original intent. Therefore, in this decoding process, to mitigate the impact of the RC’s initial lack of ESP on the prediction accuracy, we no longer employ a point-by-point restoration method; instead, we sum the errors from a certain part of each segment for comparison.

To gain a clearer understanding of the entire process of chaotic communication, we will illustrate the whole procedure using a 0–1 binary digital signal and the Lorenz system as an illustrative example in the following, and then provide a detailed explanation of the aforementioned process.

Figure 3(a) displays the shapes of the two chaotic attractors, both exhibiting the characteristic butterfly shape and demonstrating a substantial overlap in the x-z space, which imparts crucial security assurances for subsequent chaotic communication. Moreover, Fig. 3(b) shows, from top to bottom, the trajectories of two Lorenz systems in their respective three dimensions, providing a more intuitive representation of how each dimension of the two systems moves within the same range.

When generating the chaotic signal, to illustrate how to derive s(t) from m(t), let us consider a simple example with m(t) = [10010]. In Figs. 2(b)–2(d), bits “1” are decorated with a gray background, while bits “0” are marked with a white background, and the original signal m(t) is represented by a blue solid line in Fig. 2(d). At first, it is necessary to discard the transient state initially to ensure that the chaotic signals converge onto attractors. Once the transient state is over, based on the values of the original digital signal, we take the end of the previous segment as the initial value and drive the Lorenz system using corresponding parameters to produce the transmitted chaotic signal s(t). Finally, the signal travels through a transmission channel and arrives at the receiving end as v(t) = s(t)+w(t), whose x-component is shown in Fig. 2(b). Here, each bit in the original digital signal corresponds to a segment of a chaotic signal with 50 discrete steps.

In the receiving end, we need to pre-train an RC_index with noise training, where the training data correspond to the Lorenz system (4) driven by two parameter sets, and the training process is as described in Sec. II B and Ref. 43. Here, we use indices {1, 2} to differentiate between attractors corresponding to “0” and “1,” and replicate two identical RCs according to the above pre-trained RC_index, named as $RCindex1$ and $RCindex2$⁠, respectively.

To further validate the effectiveness of the proposed framework, we conduct a series of numerical simulations, with a particular focus here on showcasing the results of chaotic encoding of binary digital signals using the Lorenz system. In the experiments, noise in the transmission channel is assumed to be additive white Gaussian noise (AWGN).

During the data acquisition process, we employ the fourth-order Runge–Kutta method for numerical discretization, with a time interval Δt set to 0.01, and generate a chaotic signal segment with a switching speed of 20 discrete steps. Subsequently, we concatenate them in chronological order to form a complete chaotic signal, which is illustrated in Fig. 4(a).

When pre-training RC_index in the receiver, the Gaussian white noise with a standard deviation of 0.1 is added into the normalized training data, and the remaining parameters are as shown in Table I. We carry out the numerical simulations according to Eqs. (5)–(7) in Sec. III and obtain the decoding results of the original signal, as shown in Fig. 4(d). Compared to the noisy signal in Fig. 4(a), the curves predicted by the RC_index with noise training, which are presented in Fig. 4(b), are notably smoother than the received one. In addition, from the locally magnified plot in Fig. 4(b), it can be discerned that the prediction results of the two are distinctly opposite, and the errors in Fig. 4(c) exhibit a “complementary” pattern.

We set the number of bits for the original digital signal to be 100 and switching speeds to be 20 and 50 discrete steps, respectively. Under these assumed conditions, their simulated BER values are 0.08 and 0. From Fig. 5(b), it can be observed that increasing the time window length results in more time needed to transmit the same original signal, but the accuracy of signal restoration at the receiving end improves. Due to its simple structure and minimal computational overhead, the RC does not significantly increase the time required in practical applications even when predicting longer chaotic time series. Therefore, we believe that sacrificing some degree of transmission efficiency is worthwhile.

As shown in Fig. 5(c), with the increase in the noise strength, the BER gradually rises. If the time window T is set to the previously discussed value of 0.2, experimental results indicate a high BER of 0.2335 when the standard deviation of the Gaussian white noise is 2. However, as the time window increases, there is a noticeable reduction in the BER values. When the time window T = 2, our RC-based model performs exceptionally well under the range of noise strength that we have set. However, as for Fig. 5(d), the performance of the traditional AST method is far inferior to the RC, which is because for AST, the time window is too short to allow it to identify the correct parameters effectively, and the addition of noise further exacerbates its performance. These two comparative experiments demonstrate the strong robustness of our proposed RC-based model, and it can effectively handle strong noise with simple parameter adjustments.

To address strong noise, we also can adjust the noise intensity added during the pre-training of RC while keeping the time window length unchanged, which enables the RC to handle a wider range of noise intensities as well. In Ref. 43, the authors have already confirmed that the reservoir achieves optimal performance when equal-intensity noise is added to both training and testing data. Therefore, in response to varying levels of channel noise, we employ a strategy of adjusting different training noise intensities to observe the resulting changes in decoding performance. During the pre-training of RC, we normalize the raw data to have a mean of 0 and a variance of 1. However, the range of the chaotic signals transmitted during numerical simulations far exceeds that of the training data, and as a result, the magnitude of the training noise is significantly smaller than the actual channel noise.

Following the same experimental procedure as before, we transmit a random 500-bit binary signal, and for each set of training noise and channel noise, we repeat the process 20 times and calculate the final BER, whose results are shown in Fig. 6. It can be observed that the entire communication process is highly sensitive to noise when no noise is added or only very small noise is added during the pre-training of RCs at the receiver, and even extremely minimal noise disturbances can lead to significant adverse effects on the results. Furthermore, if the noise added during pre-training is too large, although there is decent performance when the channel noise is relatively small, as the strength of channel noise increases, there is a rapid and pronounced increase in the BER, which is because excessive noise can interfere with the pre-training process, causing the behavior of adding noise to become counterproductive. When the Gaussian white noise with a standard deviation of approximately 0.1 is added during pre-training, the entire chaotic communication process can withstand a wide range of channel noise.

We conduct a similar traversal simulation as described above: we initially generate 1000-bit binary digital signals, which can be sequentially integrated into 500 meaningful digital messages, and then according to the encoding principle, generate the corresponding chaotic signals. We repeat this process 20 times, equivalent to transmitting a total of 10 000-bit meaningful digital signals. The results of the RC-based model are shown in Fig. 7(a), showing a trend similar to the binary digital signals: as the time window increases, the tolerable noise intensity also increases, and synchronization can be achieved in a short time window as well. However, the performance of AST shown in Fig. 7(b), while following an analogous trend, lags far behind RC in terms of overall decoding capability. These two numerical simulations perfectly validate the superiority of our proposed RC-based model, which not only achieves simpler and faster synchronization but also exhibits robustness against noise, and, therefore, provides a potential pathway for the CSK scheme to transition from theory to practice.

In the preceding text, we mention that when decoding raw information, we no longer employ the method of point-by-point reconstruction. Instead, we determine the corresponding original bit values for the entire time window by accumulating errors within the time window, which effectively eliminates the influence of RC during the transient period. However, for cumulative errors, we still adhere to strict comparison based on their difference, as a result of which, regardless of how much the two errors differ, we always restore the original signal based on their magnitudes. Such a method appears overly stringent. As shown in Fig. 8(a), we compare the E₁ − E₂ curve with the 0-line. When the curve is below the 0-line, we consider $RCindex1$ to have better predictive performance, indicating that the original digital signal at this point is classified as 0; otherwise, the classification is the opposite. However, there are instances when the difference between the E₁ − E₂ curve and the 0-line is very small. For example, in the 11th segment of Fig. 8(a), where t ∈ [5, 5.5], the E₁ − E₂ curve almost coincides with the 0-line. In fact, when the errors are relatively close, it does not necessarily mean that RC_index with the smaller error is the correct one. Comparing Figs. 8(a) and 8(b), we observe that the classifications in segments 7 and 11 are opposite to the actual values, and this discrepancy arises due to noise interference and inherent prediction errors in RC, both of which are unavoidable. To address this, we introduce the undetermined signal.

First, we establish a threshold, and for the error E_1,2 defined in Eq. (8), if the absolute difference between them exceeds this threshold, the original value is restored based on their magnitudes; otherwise, the corresponding original value for that time window is marked as undetermined. In Fig. 8(a), the undetermined regions are highlighted with a gray background. If the E₁ − E₂ curve falls within this gray area, it is marked as undetermined, and no decoding is attempted. If the E₁ − E₂ curve is above the gray area, the original digital signal is classified as 1; if below, it is classified as 0. Thus, segments 7, 9, and 11 in Fig. 8(a) are marked as undetermined. Although segment 9 could be correctly identified under strict comparison, this approach effectively reduces the BER.

In terms of security, chaotic communication is more difficult to crack compared to other traditional communication methods due to the unpredictability of chaotic systems. However, with the development of machine learning, an increasing number of data-driven dynamical system methods have been researched, making it theoretically possible to crack the system by eavesdropping on the received signal over an infinite period. Consequently, the security of a chaotic masking scheme based on the superposition principle may be compromised; however, extracting valid information from noisy chaotic signals remains a highly challenging task. In this paper, we employ an RC-based CSK framework: the received chaotic signal includes multiple similar chaotic systems, with the switching points being inconspicuous and the switching speed of the time window being very fast. Regardless of whether traditional synchronization methods or emerging data-driven methods are used, even if an eavesdropper intercepts the signal for an unlimited time, they still cannot reconstruct the original dynamic behavior. Therefore, our proposed design further enhances the system’s security, making it extremely difficult to crack and significantly improving the confidentiality and reliability of the communication.

This paper presents a reservoir computing-based chaotic shift keying framework, which enhances the resistance of the chaotic communication process to noise disturbances through the noise training of RC. In addition, compared to traditional synchronization methods, RC enables faster and simpler synchronization. Through various numerical experiments, we demonstrate that our proposed framework outperforms traditional chaotic synchronization methods significantly under strong noise and short time windows, and our work provides a possibility for the practical application of the CSK framework. However, we only apply RC to an early proposed communication method in chaotic communication. In the future, we hope to apply it to more communication structures, fully leveraging the potential of RC and machine-learning techniques.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12171350 and T2350003).

The authors have no conflicts to disclose.

Ji Xia: Methodology (equal); Software (equal); Writing – original draft (equal). Luonan Chen: Supervision (equal); Writing – review & editing (equal). Huan-Fei Ma: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Under two parameter sets, the MG system (A1) all exhibits chaotic behavior, with their attractor depicted in Fig. 9(a), and the attractors under both parameter sets overlap significantly in space, ensuring the security of the chaotic communication process.

During the data acquisition process, the fourth-order Runge–Kutta method is also employed for numerical discretization, with a time interval of 0.5. Unlike the Lorenz system, as shown in Fig. 9(e), the MG system requires a longer time to acquire ESP, as a result of which, here we set T = 100, meaning that each digit signal point corresponds to 200 discrete steps of the MG signal. In the simulations, we stipulate that the original 0–1 digital signal comprises 100 point values, thus necessitating 20 000 discrete steps of the MG signal, totaling a time requirement of 10 000. Figure 9(c) shows the received noisy MG signal, and Fig. 9(d) depicts the corresponding predictive MG signals with different indices.

The parameters of the pre-trained RC are shown in Table II. Due to the MG system’s smaller range, in numerical simulations, we add channel Gaussian white noise with a standard deviation of only 0.05, and the other processes are consistent with simulations of the Lorenz system, yielding results as shown in Fig. 9(f). From Fig. 9(e), it is still evident that the trends of the two error curves exhibit complementary features, effectively distinguishing between the values of the two data points. However, compared to the Lorenz system, the error curve fluctuations of the MG system are larger, which is because the MG system requires more time to acquire ESP. Consequently, after switching between the different systems, the errors in the first half of each time window increase significantly. Therefore, utilizing the MG system as a chaotic signal may result in a certain decrease in transmission efficiency.

The Lorenz attractors used in Sec. V are generated by different parameters, and all are illustrated in Fig. 10.

The choice of parameters in chaotic systems is crucial for secure communication, and we illustrate this with two simple examples using the Lorenz system. We fix one set of parameters as σ = 10, ρ = 28, $β=43$⁠, and keep σ = 10 and ρ = 28 constant. If another parameter set results in a high overlap between the two attractors, as shown in Fig. 11(a), our framework cannot effectively recover the original signal, even in a noise-free channel. Conversely, if the parameter set causes a large difference between the two attractors, the final decoding performance is also poor, which is because when generating chaotic signals, starting a new chaotic sequence with the end of the previous period as the initial condition prevents it from entering the chaotic attractor, making RC’s predictions ineffective. However, the choice of parameters currently relies on empirical methods, with no standardized approach; the parameters used in the main text are primarily based on Ref. 49.

In the pre-training of RC, we discussed several key hyper-parameters, including the input scale, the number of neurons within the reservoir, the spectral radius, and the regularization coefficient. We conducted exhaustive simulations and the results are shown in Fig. 12.

For both input scale and spectral radius, the final results initially decrease and then increase as these parameters grow. When the input scale is too small, RC is unable to receive sufficient effective information from the data. Conversely, if the input scaling is too large, the neurons reach a saturation state, limiting the RC’s effectiveness. Thus, only an optimal input scaling can maximize the RC’s potential. Similarly, when the spectral radius of the reservoir is zero, the RC degenerates into a single-layer feedforward neural network, namely, Extreme Learning Machine (ELM), which is highly susceptible to noise and yields poor training performance. As the spectral radius gradually increases, the RC transforms into an RNN, gaining memory capabilities and greater resilience to noise. However, when the spectral radius becomes too large, RC requires more time to achieve the ESP, resulting in a gradual decline in decoding performance within the same time window. As for the number of neurons within the reservoir and the regularization coefficient, their impact on the final result is not as significant. As long as these parameters remain within a suitable range, the decoding performance remains largely consistent. Therefore, to conserve computational resources, we set the number of neurons to 200 in the numerical simulations presented in the main text.

Through simple traversal experiments, we obtain the optimal hyper-parameters for RC, which are listed in Table I. Other parameters are based on Ref. 30. Similarly, RC’s hyper-parameters for training the MG system are obtained using the same method.