We propose a type of semiconductor laser (SL) network that supports flexible chaos synchronization and multipoint-to-multipoint communications by using one-way isolation (OWI). The properties of chaos synchronization, influences of coupling strength and time delay mismatches on the quality of chaos synchronization, and the performance as well as the security of the SL network-based chaotic communications are systematically discussed. The numerical results demonstrate that, with the introduction of OWI, flexible chaos synchronization can be easily achieved in arbitrary-size SL clusters over wide parameter spaces of coupling strength and current factor. Based on the high-quality flexible chaos synchronization, satisfactory performance for Gb/s chaotic communications can be achieved in arbitrary-size clusters in the SL networks. Moreover, it is also indicated that in the SL networks, the security of intra-cluster communications can be guaranteed in three aspects. Firstly, the eavesdroppers cannot intercept any useful information by using a typical illegal attack. Secondly, due to the OWI, the chaotic carriers are only transmitted in the corresponding clusters, not transmitted among clusters, as such the security can be further improved. Thirdly, the high sensitivity of cross-correlation coefficient to the injection delay mismatches indicates that the injection delays of idle SLs to communicating SLs can be regarded as the keys of the communication clusters. The proposed scheme offers an alternative solution to flexible secure network-type communications.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Laser chaos has attracted more and more attention due to its unpredictability and broadband bandwidth [1–7]. In 1990, chaos synchronization was firstly demonstrated between two circuits . Since then, many efforts have been devoted to carrying out chaos-based communication in unidirectional and bidirectional transmission systems [9–16]. However, most of the previously reported chaotic communication systems only focused on the point-to-point transmission, while the multipoint-to-multipoint chaotic communication is rarely discussed.
Until recent years, a few chaos synchronization and communication schemes in the semiconductor laser (SL) networks with complex topologies are proposed [17–22]. Zhang et al. theoretically and numerically investigated the cluster chaos synchronization in symmetric SL network, by dividing the SLs into a set of clusters . Xiang et al. numerically studied the synchronization and complexity properties of a hierarchical tree-type network composed of mutually coupled SLs . Li et al. numerically demonstrated the point-to-multipoint communications in a star-type SL network, the adjacent communication in a ring SL network, and the wavelength-division-multiplexing (WDM) based chaos communication in the star-type SL network by modulating the bias currents with pseudorandom digital messages [21,22]. In these schemes, chaos synchronization can be only achieved among symmetric SLs of the networks, and consequently, chaos communication can be only realized between a few specific SLs that are structurally symmetric. However, it is difficult to flexibly construct communication systems among any SLs, for this reason, the flexibility of multi-user chaos communication systems is limited. Therefore, it is of great significance to enhance the flexibility of multipoint-to-multipoint chaos communication in SL networks.
In this work, a type of SL network is proposed by applying the one-way isolation (OWI) to a global SL network in which each SL is mutually coupled with the other SLs, and the properties of the flexible isochronal chaos synchronization and bidirectional chaotic communication in the proposed SL network are systematically investigated. To the best of our knowledge, this is the first time to enhance the flexibility of network-type chaos synchronization and communication. In contrast with previously reported works, the flexibility of our proposed SL network is significantly improved with the contribution of OWI because both of high-quality chaos synchronization and secure bidirectional communications can be realized among any SLs of the global networks, instead of several specific SLs, and moreover, chaotic communications are simultaneously achieved in two independent communication clusters. Additionally, due to the effective isolation to chaotic carriers from communication clusters to idle cluster, the security of communication clusters is ensured. The proposed SL networks are also against illegal interceptions over the public links even in the low-rate transmission cases, as such the system security can be further enhanced.
2. Theoretical model
The topology of mutually coupled global SL network is depicted in Fig. 1(a). In the global semiconductor lasers network, each SL is identically and mutually coupled with the other SLs. That is, all of the SLs play exactly the same structural role. In this work, for the sake of simplicity, we take a 6-SLs global network for instance. Under a traditional scenario, all 6 SLs are regarded as a 6-SLs cluster and only the communication among 6 SLs is allowed when all the SLs are synchronized, while the partial communications within 6 synchronous SLs could not be supported with security ensured, because the mutual couplings between communication SLs and idle SLs would threaten the safety of communication carriers. Nevertheless, in our proposed scheme, we introduce a novel method referred to as OWI to achieve flexible chaos synchronization and security-enhanced chaotic communication. We divide the SLs in the global network into two parts that respectively contain communication SLs and idle SLs. The OWI is performed by preserving the injections from the idle SLs to the communication SLs, but cutting off those in the opposite directions. Under such a case, due to the symmetric breaking the global network can be divided into two SLs clusters, namely a communication cluster and an idle cluster where the intra-cluster SLs are structurally symmetric while the inter-cluster SLs are asymmetrical to each other. Similarly, two or more independent communication SL clusters can be obtained by performing OWI twice or multiple times in the global SL network. In general, by introducing the OWI to a global SL network, the global SL network can be divided into a set of arbitrary-size clusters that support intra-cluster chaotic communication. For the exemplary global 6-SLs network shown in Fig. 1(a), it can be divided into 2-SLs, 3-SLs, 4-SLs, and 5-SLs communication clusters as shown in Figs. 1(b)–1(e). Furthermore, Fig. 1(f) displays another available case in which two independent 2-SLs clusters simultaneously exist in a SL network. Here, the network topologies for same-size clusters composed of different communication SLs are similar, and for this reason the exemplary communication SLs in Figs. 1(b)–1(f) are taken for instance. Therefore, with the OWI, flexible message exchange among arbitrary 2 SLs, 3 SLs, 4 SLs, 5 SLs, and 6 SLs (traditional case) in the SL networks can be achieved based on the intra-cluster chaos synchronization, and simultaneous communications in two independent communication clusters can also be supported. Meanwhile by using OWI, the security of chaos communications in clusters can be guaranteed since the chaotic carriers are only limited in the corresponding SL clusters.
The Lang-Kobayashi rate equations are modified to describe the dynamics of the SLs in the network. Considering the couplings inside the SL networks, the mathematical model can be expressed as follows [23–26]:
3. Properties of chaos synchronization
High-quality chaos synchronization is the basis of successful chaotic communication. For the exemplary global 6-SLs network (6-SLs cluster) shown in Fig. 1(a), Fig. 2 presents the temporal waveforms and corresponding spectra for different coupling strength cases. When σ(1)=6ns-1, six inconsistent waveforms are observed, which indicate no synchronization is achieved. Correspondingly, for the spectrum of the exemplary SL (SL1), the energy is concentrated nearby the relaxation oscillation frequency, resulting in an effective bandwidth of 17.5 GHz. As the increase of coupling strength, global isochronous synchronization is realized among all the 6 SLs with a relatively large σ(1). Moreover, the energy distribution in the spectrum is more uniform, and the power spectrum is flatter. Specifically, some low-frequency components appear when σ(1)=18ns-1. Here, the 6 SLs of the global SL network can be regarded as a cluster of 6 SLs. Therefore, it is demonstrated that the coupling strength influences the chaos synchronization of 6-SLs cluster, and high-quality chaos synchronization is achievable among all of the SLs in the global SL network, by properly selecting coupling strength. As for the communication clusters of 2 SLs, 3SLs, 4 SLs, 5 SLs, and even two simultaneous communication clusters, the waveforms of intra-cluster SLs keep consistent by choosing proper coupling strengths. The results are similar to that of the 6-SLs communication cluster and not shown here for the sake of simplicity.
To further investigate the effects of key parameters on the chaos synchronization errors and obtain the synchronous parameter regions of different-size communication clusters, Fig. 3 presents the influences of the coupling strength and current factor on the intra-cluster RMS, for the scenarios of 2-SLs, 3-SLs, 4-SLs, 5-SLs, 6-SLs communication clusters and two co-existing 2-SLs communication clusters shown in Fig. 1. Here, the white dotted lines denote the boundaries of the synchronous regions (RMS<0.01) and the desynchronization ones (RMS>0.01), and the deep blue parameter regions (denoted by CS) indicate the high-quality chaos synchronization. It is shown that chaos synchronization can be achieved over wide parameter spaces of coupling strength and current factor. For the different OWI-SL networks shown in Figs. 1(a)–1(f), the topologies are different from each other, as such the parameter regions for intra-cluster chaos synchronization are distinct from each other. It is worth mentioning that, in the two co-existing 2-SLs clusters case, the parameter regions for chaos synchronization is obviously smaller than that of one 2-SLs cluster scenario. This is because the corresponding total external injections from the idle SLs are weaker compared with the 2-SLs cluster case under same injection strength. Since the parameter regions for chaos synchronization in different SL clusters are different, proper adjustment to the coupling strength is required if the network topology is changed. Over all, by properly selecting the parameters of current factor and coupling strength, intra-cluster chaos synchronization can be flexibly achieved in arbitrary communication clusters of different sizes.
In the above discussions, the intra-cluster communication SLs are assumed to be coupled with identical coupling strengths and homogeneous coupling time delays. It is valuable to discuss the influences of mismatched coupling strengths and time delays on the quality of chaos synchronization, and the results are presented in Fig. 4. For simplicity, the mismatches of coupling strength and time delay , which are mathematically expressed as: σ’=σ(1+uσ), τ’=τ(1+uτ), are introduced to the common injections from an idle SL to two exemplary intra-cluster SLs of different-size communication clusters, and the CC of the mismatched intra-cluster SLs in different clusters is calculated. In particular, the parameter mismatches of the 6-SLs communication cluster which is not injected by any idle SL, are introduced to the common injection from a third communication SL to a pair one. As seen in Fig. 4(a), high-quality chaos synchronization with CC larger than 0.95 can be achieved in all of the different-size clusters over a wide range of coupling-strength mismatch. The synchronization characteristics of different clusters are different due to the structural differences among them as seen in Fig. 1. Regarding to the mismatch of time delay, as shown in Fig. 4(b), it is obviously observed that the communication clusters are sensitive to the mismatch of injection time delays or the intra-cluster injection time delays. Especially in the two independent 2-SLs clusters, the mismatched common injections from an idle SL account for a half of the total idle-cluster injections, and the corresponding proportion of the time-delay mismatch part is much higher than those in other communication clusters. Thus, the CC of the two-communication-cluster scenario with mismatched time delays is apparently smaller than those of one-communication-cluster cases. Therefore, we can conclude that the chaos synchronization of intra-cluster SL pairs is robust to the coupling-strength mismatch of common injections to some extent, while well-matched injection time delays from the idle SLs and well-matched coupling delays within communication clusters are desired for the achievement of high-quality chaos synchronization, indicating that the distances between communication SLs should be adjusted to be homogeneous and as fixed as possible to achieve stable chaos synchronization. Moreover, for the eavesdroppers, it is difficult to adjust the injection delays to be close to the private injection delays in our SL network, even if they obtain the chaotic outputs of idle cluster. Accordingly, the injection time delays from the idle clusters to the communication clusters can be regarded as the keys, which can further enhance the security of the communication clusters.
4. Multipoint-to-multipoint communications
Next, we turn to implement intra-cluster bidirectional chaotic communications and discuss the communication performance for different-size SL clusters in the SL networks. For the sake of simplicity, we take the 2-SLs cluster scenario in Fig. 1(b) for instance to demonstrate the feasibility of the proposed SL networks. Figure 5 illustrates the communication process for the 2-SLs cluster scenario. Under such a scenario, with the OWI, the global 6-SLs network is divided into two SL clusters, namely a 2-SLs communication cluster and an idle cluster composed of 4 idle SLs that are not involved for communication. The privacy of chaotic carriers is effectively protected since the carriers are only transmitted within the corresponding communication clusters. Regarding to the transmission from SL1 to SL2, the message m1(t) is firstly modulated into the chaotic carrier outputted by SL1, which is mathematically expressed by E1m(t)=E1(t)[1 + 0.05m1(t)], where the modulation index is chosen as 0.05 which is small enough to guarantee that the message is successfully hidden in the chaotic carrier [31,32]. After that, the modulated chaotic carrier (chaos + message) is transmitted to the receiver end. On the receiver end, the message is decoded by filtering the subtraction of |E1m|2 and |E2|2. The decryption process is presented as: m1’(t)=LPF[|E1m|2-|E2|2], wherein LPF denotes a filtering process using a low-pass fourth-order Butterworth filter with a cut-off frequency equaling to the message bit rate R. For the transmission from SL2 to SL1, the processes are similar and not stated here for the sake of simplicity.
The recovered massages m1’(t) and m2’(t) in the 2-SLs intra-cluster transmission cases of 1 Gb/s, 5 Gb/s, and 10 Gb/s are presented in Fig. 6. As shown in Figs. 6(a1) and 6(a2), the exchanged massages are correctly decoded in the 1 Gb/s case. With the increase of message bit rate R, bigger fluctuations are observed in the waveforms of the messages recovered by receivers compared with the original messages. This is because the high-frequency components of chaotic carriers are more susceptible to noise than the low-frequency components and the corresponding synchronization performance of high-frequency components is relatively poor, as such the communication performance of low-rate transmission is better than that of high-rate cases. Therefore, it can be concluded that the quality of recovered messages in 2-SLs intra-cluster is influenced by the message bit rate.
The quality of the decrypted messages can also be evaluated by the eye diagrams. Figure 7 shows the eye diagrams of recovered messages m1’(t) and m2’(t) as well as the intercepted messages m1e(t) and m2e(t) in the 2-SLs intra-cluster communication. Here, we consider the typical attack scenario of directly filtering the signals transmitted over public links by using a low pass ﬁlter with a cut-off frequency equaling to R [33,34]. It is assumed that the eavesdroppers know the bit rate of message. As presented in Fig. 7(a), widely-open eyes are observed in the 1-Gb/s case for the legal receivers. As the bit rate of message increases, the eye diagrams have narrower openings, which means the performance of low-R transmission case is better than that of high-R case for two main reasons. On the one hand, due to the chaotic filtering effect, more low-frequency noise in low-speed transmissions are filtered compared with the high-speed transmission cases. On the other hand, the effective bandwidth of synchronized chaotic carrier is around 30 GHz which limits the transmission capacity. For the illegal attack scenarios, the eyes are completely closed and indistinguishable in the transmission cases of different rates as shown in Figs. 7(d)–7(f). Thus, it is concluded that the messages can be successfully recovered by the legal receivers in the 2-SLs intra-cluster communication, and the security of communication system is guaranteed since the eavesdroppers cannot obtain any useful information through the public links.
Then, the communication performance is systematically and intuitively investigated in the 2-SLs, 3-SLs, 4-SLs, 5-SLs, 6-SLs, and two co-existing 2-SLs communication clusters. Figure 8 depicts the dependence of Q-factors of legal received messages and illegal intercepted messages in the transmission clusters on the message bit rate R. The definition of Q-factor is identical to that in Ref. , and satisfactory communication performance is obtained when the Q-factor value is greater than 6. For the sake of simplicity, the messages sent by the same transmitter are assumed to be identical, namely, the message sent by SL1 is denoted by m1, and that sent by SL2 is denoted by m2, etc. For the 2-SLs intra-cluster communication, it is shown in Fig. 8(a) that the Q-factors of recovered messages decrease as the R increases. Even so, satisfactory performance (Q>6) of chaos communication is still achieved with bit rate of message over 8 Gb/s. For communications in larger clusters given in Figs. 8(b)–8(e), the larger the R, the smaller the Q-factors, and the development trends are similar to that of 2-SLs intra-cluster communication. Moreover, it is found that the communication performance in clusters of smaller sizes is better than that in larger clusters. This is because more messages are simultaneously exchanged between SLs in the larger clusters, and consequently, the quality of chaos synchronization and communication performance would be easier to be influenced. Considering a possible communication scenario: simultaneous intra-cluster communication in two communication clusters, as shown in Fig. 8(f), intra-cluster communications of good performance are simultaneously achieved in two 2-SLs clusters of the SL network. The result indicates that two independent communication clusters can be simultaneously supported in the proposed SL networks with OWI. Besides, it is also noticed that the communication performance of two independent 2-SLs clusters is very similar. This is because the two clusters are completely symmetric to each other as seen in Fig. 1(f). For the illegal attack in the intra-cluster communications, the values of Q-factors all maintain at a very low level (Q<2.2) over the whole range of transmission bit rate since the uniform energy distribution of chaotic carriers presented in Fig. 2(c2) ensures the message can be successfully hidden even in the low-R scenarios. Compared with the traditional point-to-point communication which is easy to be attacked (Q>6) in the low-rate range [11,35], the security of intra-cluster communications in our proposed SL networks is significantly enhanced. Thus, it is demonstrated that by using OWI, Gb/s communications are successfully realized in the arbitrary-size clusters of the SL networks, and the flexibility and security of communication clusters are greatly improved. Furthermore, in terms of the OWI-SL network with larger size, multipoint-to-multipoint secure communications among more users can also be achieved, and the results are not presented here for the sake of simplicity.
In the above discussions, we assume that messages transmissions are simultaneously performed in all SL pairs of a communication cluster, and the messages sent by the same transmitter (SL) are identical. To further validate the generality of our proposed scheme, we also discuss two types of intra-cluster communications, which are more common in practice. One type is that not all pairwise SLs in the clusters communicate simultaneously, and it is performed by assuming that messages are only exchanged between SL1 and SL2, SL2 and SL3, while SL1 and SL3 are not communicating. Here, false messages are embedded into the chaotic carriers transmitted between SL1 and SL3 to avoid exposing of synchronized carrier . The other type is that the transmitted messages of different transmitter and receiver pairs are different, and the message transmitted from SLi to SLj is denoted by mij. The network-type communication of 3-SL cluster is taken for instance to demonstrate two types of common scenarios, and the corresponding Q-factors as the function of message bit rate are illustrated in Fig. 9. The results show that satisfactory transmission performance is realized beyond 6 Gb/s in two common scenarios, and the results are similar to those under two assumptions shown in Fig. 8(b). Thus, we can conclude that the proposed SL networks under OWI support partial intra-cluster communication and communication with different messages between different intra-cluster SL pairs. Moreover, under such scenarios, the low-value Q-factors mean that the eavesdroppers cannot obtain any useful information, and with the contribution of OWI, the privacy of the carriers is ensured because the chaotic carriers are only transmitted in the corresponding communication cluster, which further improve the security of flexible multipoint-to-multipoint communications.
Up to present, relevant previously reported approaches mainly focused on the chaos synchronization in fixed SL networks, in which chaos synchronization can be achieved in fixed SL clusters [18–20]. In our previous work  and Ref. , cluster chaos synchronization was investigated in different types of SL networks, including the small-world network, random network, the Nepal power grid network, etc., and the results showed that isochronous chaos synchronization can be achieved among intra-cluster SLs. In Ref. , hierarchical chaos synchronization was demonstrated in tree-type SL networks where SLs belonging to the same layer can be synchronized. In these reported works, only the intra-cluster symmetric SLs can be synchronized while the SLs of different clusters are asynchronous. Besides, it is difficult to change the network topologies to satisfy different demands, and consequently, the flexibility of chaos synchronization in SL networks is limited. Compared with the existing methods of chaos synchronization, our SL networks can support high-quality chaos synchronization between any SLs with the contribution of OWI, which indicate that the flexibility of the chaos synchronization in SL networks is greatly improved. Additionally, the intra-cluster chaos synchronization of all the different-size clusters is robust to the mismatch of coupling strength to some extent, and the sensitivity of intra-cluster chaos synchronization to the mismatch of external injection time delays means that the time delays of the injections from idle SLs to the communication SLs can be employed as the keys. For this reason, the privacy of chaos synchronization among any SLs can be ensured. On the basis of flexible chaos synchronization, secure key distributions and chaotic communications in arbitrary-size SL clusters can be available.
Regarding to the network-type chaotic communication, broadcast message transmissions from the center SL to the side SLs in a star-type SL network as well as the bidirectional chaotic communication between neighboring SLs in a ring SL network was numerically achieved with message bit rate of several hundreds of Mbit/s in Ref. . Besides, in Ref. , WDM-based communication system was constructed in a star-type SL network, and Gb/s secure communication was realized between two equal parts composed of side SLs. A complex network composed of two-layer chaotic nodes is proposed and demonstrated in Ref. , where secure chaotic communication is achieved between pairs of matching nodes by using one-time-one-cipher encryption method. In our previous work, intra-cluster message exchange in the SL network composed of 12 SLs was investigated with bit rate over Gbit/s based on cluster chaos synchronization . In most of these reported schemes, message can be only exchanged among specific users which are symmetric to each other, and the various transmission demands in real world could not be satisfied. For the purpose of realizing flexible chaotic communication, this paper systematically discusses the multipoint-to-multipoint transmissions in the communication clusters composed of different number of users, by using OWI. With respect to the existing multi-user communication schemes, our work proposes a novel OWI method to realize the flexible multipoint-to-multipoint communications, in which the communicating SLs are not limited to several specific SLs. Moreover, the security of the proposed communication system is demonstrated from three aspects. Firstly, by using a typical attack method, the eye diagrams of the intercepted messages are completely close and indistinguishable. Secondly, the chaotic carriers are only transmitted within the communication clusters since the injections from the communication clusters to the idle cluster are cut off with the introduction of OWI, and for this reason eavesdroppers are unable to simultaneously obtain the unmodulated carriers and the modulated carriers to recover the correct messages. Thirdly, as demonstrated in Fig. 4(b), the injection time delays from idle SLs to the communication SLs can be regarded as the keys of the communication system, and consequently the security can be obviously enhanced.
In summary, flexible chaos synchronization and multipoint-to-multipoint communications are systematically investigated in SL networks by using one-way isolation. In the proposed SL networks, it is demonstrated that chaos synchronization can be realized in the arbitrary-size clusters over a wide parameter range. Besides, the quality of chaos synchronization in different clusters is all robust to the mismatch of coupling strengths to some extent but sensitive to the mismatched time delays. Based on high-quality intra-cluster chaos synchronization, messages bidirectionally transmitted in arbitrary-size SL clusters are successfully recovered with bit rate over several Gb/s, and simultaneous communications in two independent 2-SL communication clusters of a SL network are also supported. Moreover, the security of flexible intra-cluster message transmissions is guaranteed because the communications in arbitrary-size SL clusters are all against interceptions over the public links during the whole range of message bit rate. Meanwhile, with the contribution of one-way isolation, on the one hand, the privacy of the chaotic carriers is protected since the idle SLs cannot obtain any chaotic injections from the communication SLs, on the other hand, the security of communication clusters is further improved because the injection time delays from the idle SLs to the communication SLs can be used as the keys. Compared with the previously reported communication schemes, the flexibility of multipoint-to-multipoint communications is obviously enhanced since message exchange among any SLs is achievable. Our work provides a novel multipoint-to-multipoint communication scheme to realize flexible secure multiple-user communications.
National Natural Science Foundation of China (61671119, 61805031); Fundamental Research Funds for the Central Universities (ZYGX2019J003).
The authors declare no conflicts of interest.
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