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Abstract
We propose and numerically demonstrate a novel secure key distribution (SKD) scheme by using dynamically synchronized semiconductor lasers (SLs) subject to common dual injections from two mutually coupled SLs. The performance of hybrid chaos synchronization, complexity of chaotic signals, chaos-based key distribution, and the privacy of SKD scheme are systematically discussed. It is shown that high-quality hybrid chaos synchronization of zero lag and lead lag can be both achieved between two local lasers under different injection delay conditions, whereas low cross correlations are observed among the driving lasers and the local lasers. By randomly perturbing the injection delays with four independent random sequences, the outputs of local SLs can be dynamically synchronized. Extracting the outputs in the synchronization time slots of zero lag and lead lag, synchronous entropy sources are obtained and used to generate keys with high consistency at local ends of Alice and Bob, which are robust to the parameter mismatches of local lasers to some extent. Moreover, large BER is calculated in two types of typical illegal attacks, which demonstrates the security of the proposed scheme. This work proposed a high-level secure key distribution solution to one-time pad communication.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, with the development of quantum computer technology, more and more algorithmic encryption has been cracked, which means that mathematical algorithm encryption is not computationally secure, and the premise is that the attacker’s computing ability is limited. Physical encryption having information-theoretic security that does not assume constraints on computing ability has drawn great attentions [1]. In order to realize one-time pad encryption in which the unconditional security has been demonstrated, a physical layer key distribution with high-level security is required so that the users can share the keys [2–5]. Quantum key distribution (QKD), which provides absolute theoretical security, is an ideal SKD scheme, but the application of QKD to commercial optical communication systems is still challenging [6,7]. Moreover, to the best of our knowledge, the maximum key generation rate of QKD is limited to around 10 Mb/s, which cannot support high-speed one-time pad encryption.
From a different perspective, synchronous chaotic sources can be applied to private key distribution due to the characteristics of broadband spectra and noise-like waveforms [8–16]. The local SLs’ chaos synchronization induced by the external injection from a third laser was most frequently adopted in the SKD schemes [10–16]. However, in most of the previously reported studies, high values of cross correlation were observed between the driving SL and the local ones because of injecting-locking effects, for this reason, a complex post-processing method was demanded to enhance the security level of SKD scheme but lead to increments of difficulty and complexity in practical application. Studies on improving the cross correlations between injecting chaotic signal and local generated chaos are still rarely reported. Therefore, it is meaningful to decrease the correlation of driving source and local entropy sources in Alice and Bob and improve the security of the chaos-based SKD system.
In this work, a novel SKD scheme is proposed and demonstrated in virtue of randomly perturbed hybrid chaos synchronization. The results show that dynamic hybrid chaotic synchronization of zero lag and lead lag is observed between the local lasers when the injection delays are controlled by four independent random bit sequences. By splicing the waveforms in synchronous time slots and utilizing the spliced waveforms as the entropy sources, key distribution with inconsistent rate around 2×10−5 is achieved at the ends of Alice and Bob, which is robust to the parameter mismatch of local lasers to some extent. On the other hand, the low cross correlations are calculated between the outputs of driving lasers and the slave local lasers, which indicate that inconsistent keys would be obtained for Eve even if an identical post processing is applied to the injecting chaos for intercepting keys. Moreover, the synchronization of two local lasers is very sensitive to the mismatch of injection delays, as such, it is quite difficult for Eve to rebuild an interception system that is synchronized to Alice and Bob.
2. Scheme for key distribution and theoretical model
Figure 1 illustrates the proposed SKD scheme. Here, the structure of the SKD scheme can be regarded as a combination of two parts. One is the double-channel driving sources generated by two mutually coupled SLs. The output of the SL1 (SL2) is firstly power-adjusted by the variable optical attenuator (VOA), then split into two beams wherein one beam is utilized as chaotic coupling between SL1 and SL2 and another beam is directly outputted as the injecting signal to Alice and Bob. With the increase of coupling strength, the output states of SL1 and SL2 would evolve from stable to quasiperiodic (multiple periodic), and then evolve to chaotic states. When the driving lasers SL1 and SL2 work in chaotic states, the outputs of SL1 and SL2 are used as the driving signals to induce chaos synchronization in Alice and Bob. The other part of the proposed scheme is the local key generators which consist of symmetrically injected SL3 and SL4, and the post processing technology. The built-in SLs are subject to dual driving signals from SL1 and SL2. The injecting delays of SL2 to local SL3 and SL4 are respectively controlled by two random sequences R1 and R2 that are generated by two independent random bit generators (RBG), and those of SL1 to the local SL3 and SL4 are controlled by two independent random sequences R3 and R4. The injection delays in the time slots of bits of Rm (m = 1, 2, 3, 4) equaling 1 are 0.1ns larger than those in the time slots of bits equaling 0. Obviously, in the time slots of bits in R1 identical to those in R2 and bits in R3 identical to those in R4, zero-lag chaos synchronization can be observed due to identical dual-path external injections. Additionally, in the time slots of bits in R1-R2≠ 0 equaling those in R3-R4≠ 0, the dual-path injections from SL1 and SL2 to SL3 lead (R1-R2 = 1) or lag (R1-R2=-1) with respect to the injections to SL4, for this reason, lead-lagged chaos synchronization between SL3 and SL4 can be achievable. Then, the output of SL3 (SL4) is divided by a coupler into two parts, one firstly go through a fiber delay line (DL) with a time delay of 0.3ns and then detected by the photodetector (PD), and the other part directly detected by an inverse photodetector (IPD). Based on this, delayed self-difference signals with symmetric distribution are obtained as demonstrated in [17]. Since the delay time of 0.3ns is much shorter than the length of each time slot controlled by random bits, the self-difference operation does not cause effect to the synchronization states, as such, chaos synchronization of zero lag and lead lag are also observed in the time slots of R1 = R2&R3 = R4 and R1-R2 = R3-R4=±1. Exchanging Rm (m = 1, 2, 3, 4) in Alice and Bob, only the self-difference series in the time slots of R1 = R2 and R3 = R4 (zero-lag chaos synchronization), R1-R2 = R3-R4=±1 (lead-lagged chaos synchronization) are preserved, while the those in other time slots are discarded. For the time slots of lead-lagged synchronization, the time series are shifted to reach zero-lag chaos synchronization. After that, the preserved signals in Alice and Bob are always isochronously synchronized. The spliced identical outputs can be regarded as the entropy sources, and then sent to the post processing technology. Firstly, the self-difference signals are sampled by a dual-threshold analog-to-digital converter (ADC), which is implemented by denoting the samples above the high threshold as “1”, those below the low threshold given as “0”, and symbol “T” stands for the samples between the two thresholds. Next, a frequently-used logical exclusive-OR (XOR) operation is performed for randomness improvement. Note that, the XOR results of “T” and “0”/“1” are assumed as “T”. Currently, the obtained SequA and SequB are composed of bits “0”, “1”, and “T”. Then, after exchanging the position information PTA and PTB of invalid symbols, Alice and Bob discard the bits in the positions of $P_{\rm TA}\bigcup P_{\rm TB}$. Finally, shared consistent keys are obtained at the ends of Alice and Bob.
Fig. 1. Schematic of the SKD in virtue of hybrid chaos synchronization. SL: semiconductor laser, FC: fiber coupler, VOA: variable optical attenuator, OI: optical isolator, OS: optical switch, (I)PD: (inverse) photodetector, DL: fiber delay line, ADC: analog-to-digital converter.
For numerical purposes, the typical Lang-Kobayashi rate equations that are frequently employed to model the semiconductor laser suffering external perturbation are taken and modified as follows [18–20]:
driving lasers SL1 and SL2:
The chaos synchronization of two local SLs mainly depends on the symmetrical external disturbance. In our proposed scheme, since SL3 and SL4 always work under the identical injections from the driving lasers, dual-injection induced chaos synchronization can be achievable. When the SL3 and SL4 are synchronous, the relationship of variables can be given by:
Here, the mutually-coupled SL1 and SL2 without self-feedback are desynchronized due to the asymmetrical operation mechanism. More specifically, the SL1 and SL2 work under the injections of each other, as such, no synchronization could be observed between them. Based on this, the synchronization condition in Eq. (8) can be simplified as:
To satisfy the synchronization conditions given in Eq. (9), the SKD system should meet the requirements:
To quantitatively describe the correlation property for the chaotic signals, the frequently-adopted cross-correlation function (CCF) is introduced and written as [25–27]:
The frequently used permutation entropy (PE) is employed to evaluate the complexity of the chaotic signals [21,28–31]. We take the time series {xt, t = 1, 2, …, T} and reconstruct a d-dimensional space Xt = [x(t), x(t+τ), …, x(t+(d−1)τ)], where d = 6 and τ denote the embedding dimension and the embedding time delay, respectively. The vector Xt is constructed by arranging elements of ${{\{ }{x_t}{\} }_{t = }}{_{\textrm{1,}}{_{\ldots ,T}}}$ increasing order ${x_{t + ({r_1} - 1)\tau }} \le {x_{t + ({r_2} - 1)\tau }} \le \ldots \le {x_{t + ({r_d} - 1)\tau }}$, and any Xt is uniquely mapped onto an ordinal pattern Ω=(r1, r2, …, rd) out of d! possible permutations. For the permutation Ω of order d, the probability distribution P = p(Ω) of the ordinal patterns is:
3. Results and discussions
3.1 Chaos synchronization of local lasers
The temporal waveforms and cross correlation functions of two local SLs in different scenarios are shown in Fig. 2. In the first row, it is shown that similar fluctuations are noticed with zero lag in the waveforms of the built-in SLs, and the corresponding C34 as large as 1 is achieved when lag time equals 0, which demonstrate that zero-lag chaos synchronization is obtained for SL3 and SL4 as the external injections from SL1 and SL2 are simultaneously injected to them. Moreover, in the scenario of R1-R2 = R3-R4 = 1, τ13-τ14= τ23-τ24= 0.1ns (τ13 = 4.5ns, τ14 = 4.4ns, τ23 = 4.3ns, τ24 = 4.2ns), SL3 and SL4 are lead-lagged synchronized with C34(0.1) =0.9404. The results agree well with the above theoretical derivations. Namely, two chaos synchronization patterns are realized in the local lasers of our scheme under different injecting conditions. Taking the synchronous outputs of SL3 and SL4 as the entropy sources, highly consistent keys can be obtained under a suitable post processing technology. However, when τ13-τ14 ≠ τ23-τ24, the SL3 and SL4 are asymmetrically injected by the driving signals, as such, the value of cross correlation for SL3 and SL4 is as low as 0.11 as shown in Fig. 2(c3). Under such a scenario, the desynchronized outputs of SL3 and SL4 cannot be used as the entropy sources to generate keys. For this reason, only when the injection delays meet the condition of τ13-τ14 = τ23-τ24 can the outputs of local lasers be synchronous and utilized as the entropy sources.
Fig. 2. Temporal waveforms and CCF between the chaotic signals of SL3 and SL4 in the cases with Δt = 0 (first row), Δt ≠ 0 (second row), and τ13-τ14 ≠ τ23-τ24 (third row).
Figure 3 presents the cross correlations between external driving signals and local lasers. As illustrated in Fig. 3, the chaotic outputs of two driving SLs are asynchronous with low correlation of 0.4774, and due to the asymmetric properties, C13 = 0.265, C23 = 0.252 are observed among two driving sources and the entropy sources, which indicate that the driving signals cannot be used as the substitutes to generate keys that are synchronized with Alice and Bob. For this reason, the eavesdroppers cannot obtain correct keys even if an identical post processing is employed to the driving signals. Thus, the security of our SKD scheme is effectively ensured.
In order to discuss the effect of injection strength on cross correlation between two local lasers and those between driving chaotic signals and entropy sources, Fig. 4 illustrates the CC versus injection strength in the cases with different pump currents. As shown in Fig. 4(a), compared with the lag synchronization scenario, zero-lag chaos synchronization of higher quality can be achieved within a broader parameter interval of injection strength. Nevertheless, with a properly large injection strength, we can simultaneously observe two types of chaos synchronization between local SLs under different injecting delay conditions. It is noticed that large value of C34 is also achieved in the region of low injection strength. However, the corresponding outputs of SL3 and SL4 do not exhibit chaotic dynamic behaviors, and for this reason, the local lasers are not chaotic synchronized during the low-injecting operation range. Meanwhile, for the scenarios of 1.5Ith and 1.7Ith, the development trends of cross correlations are obviously modified as presented in Figs. 4(b)–4(c). On the one hand, a larger injection strength is required to realize chaotic synchronization between entropy sources compared with the case with lower pump currents. Regarding to the cross correlations of driving signals and outputs of local lasers, the values of C13, C14, C23, and C24 always maintain below 0.4 in the cases with different pump currents. In particular, for a fixed injection strength, the correlation values of injecting chaos and the local SLs’ outputs decrease as increasing the pump current, and even decrease to below 0.3 in some regions. The results indicate that by appropriately setting the injection strength and working current, not only can local SLs be well synchronized, but also low cross correlations are obtained between external injecting signals and the local outputs.
Fig. 4. Cross-correlation coefficients C13 (square), C14 (circle), C23 (upper-triangle), C24 (down-triangle), zero lag C34 (diamond) with Δt = 0, and lead lag C34 (left-triangle) with Δt ≠ 0 versus injections strength with (a) I = 1.3Ith, (b) I = 1.5Ith, (c) I = 1.7Ith.
To obtain the optimal parameter range for simultaneous achievements of high-quality chaos synchronization of entropy sources as well as the low CC of driving SLs and local SLs in Alice and Bob, Fig. 5 illustrates the influences of injection strength and coupling strength on two dimensional maps of C13, C14, C23, C24, and C34 in the cases of lead-lagged and zero lag chaos synchronization. As shown in the first two columns of Fig. 5, cross correlations below 0.3 are realized between the driving SLs and the local ones over a wide dynamic operation range. Meanwhile, satisfactory lead-lagged and zero-lag chaos synchronization of local SLs are respectively observed in wide parameter spaces of coupling strength and injection strength as seen in Figs. 5(c) and 5(f). By taking the intersection of the low cross correlation regions of C13, C14, C23, C24 and the synchronous region of C34, the optimal dynamic operation range for achieving simultaneous chaos synchronization of local lasers and low cross correlations of driving lasers and local lasers is obtained.
Fig. 5. Two dimensional maps of (a) C13, (b) C14, (c) lead lag C34 with Δt ≠ 0, (d) C23, (e) C24, and (f) zero lag C34 with Δt = 0 as the functions of injection strength and coupling strength with pump current I = 1.5Ith.
The complexity of the generated chaotic signals can be evaluated by the largest Lyapunov exponent (LLE) and permutation entropy (PE) [31]. The LLEs and PE curves of chaotic signals outputted by driving SL1, SL2, and local SL3, SL4 are presented in Fig. 6. As shown in Fig. 6(a), when the coupling strength between SL1 and SL2 is very weak, the LLE1 and LLE2 are very close to 0, and the corresponds PE values are very low since the SL1 and SL2 work in quasiperiodic (multiple periodic) states. As the coupling strength increases, the LLEs and PE values gradually increase which indicate that the driving SLs work in chaotic states and the complexity of the driving chaotic signals is improved. The changes of LLEs are small when the coupling strength σ12 is larger than 9ns−1. In regard to the LLE3, LLE4 and PE values of chaotic signals outputted by SL3 and SL4, the LLEs are always maintained at a relatively high level. This is because SL3 and SL4 work in chaotic states even with a small injection strength under chaotic injections from SL1 and SL2. Besides, the PE values increase from 0.98 to over 0.99 with the increase of injection strength, which means high-level complexity of chaotic signals outputted from local lasers. The results agree well with those in Refs. [31,32].
Fig. 6. (a) The largest Lyapunov exponents LLE1 (blue), LLE2 (red) and PE1 (blue), PE2 (blue) versus coupling strength, (b) LLE3 (red), LLE4 (red) and PE3 (blue), PE4 (blue) versus injection strength. The squares, circles, up triangles, and down triangles correspond to the results of chaotic signals generated by SL1, SL2, SL3, and SL4, respectively.
As aforementioned, the synchronization regime of entropy source is determined by the injection delays from the driving lasers to the local lasers. To systematically study the influences of injection delays on chaotic synchronization quality, Fig. 7 illustrates the results of cross correlation coefficient C34 in different synchronization scenarios. For the zero lag synchronization case of τ13=τ14 = 4.5ns, τ23=τ24 = 4.3ns and the lead-lagged synchronization case of τ13 = 4.5ns, τ14 = 4.4ns, τ23 = 4.3ns, τ24 = 4.2ns, Figs. 7(a), 7(c) and 7(b), 7(d) present C34 versus τ13’= τ13 + u and τ23’= τ23 + u with u${\in}$ [−1ns,1ns] by fixing the values of τ14 and τ24. The insets located in the upper right of Figs. 7(a) and 7(b) are within the range of [−0.01ns, 0.01ns] with a step of 1ps. For further investigation on the influences of injection delays on the synchronization quality, Fig. 7(e) illustrates the two-dimensional maps of C34 as the functions of double injection delay mismatches τ13=τ23 + u1, τ23=τ24 + u2. It is shown in Figs. 7(a)–7(d) that high-quality chaos synchronization of the entropy sources outputted by SL3 and SL4 can be only observed when u = 0, that is, strictly meeting the conditions of τ13-τ14 = τ23-τ24 = 0 and τ13-τ14 = τ23-τ24 = 0.1ns in zero lag synchronization and lead-lagged synchronization scenarios. When the mismatch of injection delays deviates from 0, the cross correlation decreases sharply, indicating obvious synchronization degradation in the injection delay mismatched cases. Regarding to the mismatches of double injection delays presented by Fig. 7(e), obviously, zero lag chaos synchronization can be observed when τ14=τ13 and τ24=τ23, and when the non-zero mismatch of τ24 and τ23 equals the mismatch of τ14 and τ13, lead-lagged chaos synchronization can be available. The results agree very well with the theoretical derivation given in section 2. This is because when the mismatches of double injection delays are equal, the double external injections can be regarded as single one that arrives earlier or later in SL4 compared with that injected to SL3. Under this condition, the leader/laggard chaos synchronization can be achieved between SL3 and SL4, as reported in [33]. Nevertheless, in the region of u1≠u2, low correlations are observed between SL3 and SL4. Therefore, it can be concluded that, by adjusting the injection delays, two types of chaos synchronization patterns would appear in our SKD scheme, and the consistency of the entropy sources has stringent requirement on the injection delays from driving signals to the local lasers. As for illegal attack, even Eve gets the dual-path injections of SL1 and SL2 from the public channel, she cannot intercept the injection delays τ13 (τ14) and τ23 (τ24), and for this reason, it is quite difficult for Eve to rebuild a copy slave SL and break into our key distribution by generating synchronous entropy source. Hence, the security of the key distribution system can be undoubtedly guaranteed.
Fig. 7. (a)-(d) Cross correlation coefficient C34 versus mismatches of injection delays τ13 and τ23, (e) two dimensional maps of C34 versus mismatches between τ13 and τ14 as well as τ23 and τ24.
Fig. 8. (a)-(d) Random sequences R1, R2, R3, R4 controlling injection delays τ13, τ14, τ23, τ24, (e) CC curve of entropy sources with random injection delays, and (f) average CC within each time slot.
3.2 Secure key distribution based on hybrid synchronization
As mentioned above, two types of synchronization patterns are observed in our proposed key distribution scheme. When the injection delays dynamically vary, zero-lag and lead-lagged chaos synchronization can be found in different time slots of same outputs, which is referred as hybrid chaos synchronization. Based on the hybrid synchronization of entropy sources, this section studies the chaos-based key distribution.
Figure 8 illustrates the dynamic chaos synchronization under dynamic injection delays that are controlled by four independent random bit sequences. Here, the rates of random sequences R1, R2, R3 and R4 are 50 Mb/s. It is shown that in the time slots denoted by light blue, R1 = R2, R3 = R4, τ13= τ14, τ23= τ24, the zero-lag chaos synchronization with stable CC is obtained between the self-difference outputs of SL3 and SL4. On the other hand, in the time slots denoted by light orange, R1-R2= R3-R4≠ 0, τ13-τ14= τ23-τ24≠ 0, lead-lagged chaos synchronization is achieved. Although the average CC in the time slots of lead-lagged chaos synchronization is lower than those of zero-lag chaos synchronization, satisfied average CC over 0.9 can be achievable as presented in Fig. 8(f). Nevertheless, in other time slots with R1-R2≠ R3-R4, τ13-τ14≠ τ23-τ24, low correlations are found between the self-difference signals. We retain the time series in the time slots of R1-R2 = R3-R4 = 0, and R1-R2 = R3-R4≠0, and discard those in other time slots. More importantly, the retained zero-lag synchronous time series in the time slots of R1-R2 = R3-R4 = 0 are directly kept, while the lead-lagged synchronized time series in the time slots of R1-R2 = R3-R4≠0 are shifted by τ13-τ14 to make them synchronized with zero lag. Consequently, zero-lag synchronization can be realized between the cropped entropy sources.
Based on the cropped entropy sources with high-level cross correlation, highly consistent keys are obtained by utilizing a post-processing technology to the entropy sources. The generated key rate can be roughly estimated by p.r.fs, where p = p1 + p2, p1 and p2 stand for the probabilities that the outputs are synchronized with zero lag and a lag in each time slot, p is the total synchronization probability, r is the retaining rate of the dual-threshold ADC, and fs denotes the sampling rate. Here, the usually used NIST 800-22 test method reported in [34–36] is employed to evaluate the randomness of the generated keys. In our key distribution scheme, the probability of zero-lag chaos synchronization (R1 = R2, R3 = R4) is p1 = 0.25, the lead-lagged synchronization probability (R1-R2 = R3-R4≠0) is p2 = 0.125, the sampling rate is set as 1Gb/s, the retaining rate equals 0.6, and 1000 sequences with a length of 1 Mbit are used for test. Table 1 shows the randomness test results of distributed keys of 0.225 Gb/s. Table 1 shows the randomness test results of distributed keys of 0.225 Gb/s. With the significance level of 0.01, the test results consisting of 15 items are all “success” with P-value beyond 0.0001 and proportion over 0.983, demonstrating the randomness of the shared keys.
Table 1. NIST 800-22 test results of the 0.225 Gb/s distributed keys
In the above discussions, the entropy sources in Alice and Bob are from two identical local lasers. However, it is hard to find two completely identical SLs in practice. Thus, we explore the dependences of hybrid chaos synchronization quality between SL3 and SL4 as well as the BER of the generated keys on the parameter mismatch between them, and the results are shown in Fig. 9. Here, parameter mismatches are introduced by keeping the intrinsic parameters of SL3 fixed, while varying those for SL4 with respect to SL3, mathematically, α’=α(1 + u), g’=g(1 + u), τp’=τp(1 + u), τe’=τe(1-u), N0’=N0(1-u), s’=s(1-u), where u is the mismatch ratio from −10% to 10% [37]. When u = 0, high-quality hybrid chaos synchronization of cross correlation C34 = 0.99 can be achieved between matched SL3 and SL4. As the u deviates from 0, the corresponding cross correlation gradually decreases. Nevertheless, hybrid chaos synchronization with C34 beyond 0.95 is observed over a wide mismatch range of [−8%, 10%]. As far as BER performance, the maximum tolerable BER in our proposed SKD scheme is the hard decision forward error correction (HD-FEC) threshold of 3.8×10−3. It is demonstrated that for a fixed relatively-large remaining rate r = 0.8, the smaller mismatches, the lower BER, and SKD BER below 3.8×10−3 can be achieved in the mismatch range of [−1%, 1.5%]. By decreasing the remaining rate, it is observed that, the tolerant ability is significantly improved in contrast with the higher remaining rate case for the same entropy source. Specifically, under the scenario of r = 0.2, the optimization operation range is further extended beyond [−10%, 10%]. The wide tolerant mismatch range indicates good robustness of our proposed key distribution scheme. It is worth noting that in the scenarios with fixed cross correlation, better BER performance means lower key distribution rate. Overall, in the proposed SKD scheme, good robustness is realized by properly choosing the retention ratio of double-threshold sampling.
Fig. 9. CC of entropy sources (black) and the bit error ratio (BER) of the generated keys with dual-thresholds ADC retaining rates of 0.2 (magenta), 0.4 (green), 0.6 (red), and 0.8 (blue) versus intrinsic parameter mismatch between SL3 and SL4.
Next, we turn to discuss the security of our proposed key distribution system. Although the driving signals available to Eve is lowly correlated to the entropy sources, they are directly transmitted over the public links. Thus, it is deserved to discuss the possible illegal attack that is implemented by directly applying an identical self-difference operation and post-processing technology to the injecting chaotic signals outputted by SL1 and SL2. Meanwhile, the entropy sources in our scheme are not transmitted over the public links, the security of our key distribution scheme mainly relies on the difficulty of rebuilding the key distribution system for Eve, namely, identifying the whole parameter set in structure and operation. However, the injection delays of one-way injection from driving SLs to local SLs are difficult to be precisely estimated according to the lengths of partially exposed fiber links. Without loss of generality, we suppose Eve knows the structure of local lasers and some operation parameters, for instance, the injection strength, pump current, etc. Then, another illegal attack is performed by injecting the obtained driving signals to a rebuilt attack SLE that is assumed to be identical to the local SL4 with different injection delays different from our scheme, here, τ1E = 4.5ns, τ2E = 4.1ns, then applying identical post-processing technology to the SLE. Figure 10 illustrates the cross correlation coefficients between legal entropy sources and illegal sources and the BER in two types of attacks as the function of parameter mismatch of SL3. It is demonstrated that low cross correlations are observed in two types of interceptions over the whole range of parameter mismatch of SL3. As demonstrated in last section, this is because the synchronization is very sensitive to the values of injection delays, and only when satisfy the specific conditions can entropy sources are synchronized. Correspondingly, the BER14, BER24, and BERE4 of two illegal attacks is always much higher than the HD-FEC threshold. As such, the proposed key distribution scheme is against two possible illegal interceptions, indicating the high-level security of our proposed scheme. Moreover, the injection delay controlling with four independent random bit sequences would introduce a tailor to the original chaotic source generated by Alice and Bob, which further increases the level of security of our SKD system.
Fig. 10. CC and BER of keys generated by SL1 and SL4 (green), SL2 and SL4 (red), and rebuilt SLE and SL4 (blue) versus parameter mismatch of SL4.
Previously reported approaches mainly focused on the key distribution based on synchronous chaotic lasers driving by a third laser [10–12,38,39]. In most of the reported literature, a dynamic parameter-controlling method was employed in Alice and Bob to obtain the dynamically synchronous laser chaos, then sampling and post procession were implemented. Only the bits in the parameter matched time slots were retained, while the bits in the parameter mismatched time slots were discarded. The adoption of parameter controlling method can randomly select the time slots of chaos sources to improve the privacy of key distribution. Although the third-laser driving scheme can generate high-quality synchronous laser chaos to share synchronous keys, the cross correlation coefficients between driving signals and the local signals can be as large as 0.67, as demonstrated in Ref. [12]. The large cross correlation would increase the possibility of key interception by Eve. Thus, we explore the key distribution scheme based on common dual chaotic injections. Compared with the traditional third-laser driving-based SKD schemes, the cross correlations of two driving signals and the local entropy sources can be lower than 0.3. As such, the Eve cannot obtain any useful information even if the dual driving signals are obtained, which guarantees the security of our SKD scheme. Moreover, to the best of our knowledge, only one type of chaos synchronization was observed in the entropy sources of reported SKD schemes, while the entropy sources generated in our scheme are composed of zero-lag and lead-lagged synchronized chaos signals, which would further improve the privacy of the proposed SKD scheme.
4. Conclusion
In summary, a novel key distribution scheme is numerically demonstrated based on dynamic hybrid chaos synchronization of local SLs which is perturbed by randomly controlling the injection delays from driving signals to local SLs. The results show that zero lag and lead lag chaos synchronization as well as desynchronization can be achieved in different time slots of complex chaotic signals generated in Alice and Bob when the injection delays of driving signals are randomly perturbed, while cross correlations of the driving lasers and the local lasers are below 0.3. By exchanging the random controlling sequences to identify zero-lag and lead-lag chaotic synchronization time slots and splicing local SLs’ outputs as the entropy sources according to the identified time slots, Alice and Bob can generate consistent random keys which are robust to the parameter mismatches of local lasers to some extent. Moreover, since the synchronization of local lasers is sensitive to the mismatch of injection delays, the privacy of entropy sources is guaranteed, indicating the high-level security of our proposed SKD scheme.
Funding
National Natural Science Foundation of China (61671119, 61805031, 62171087); Sichuan Science and Technology Program (2021JDJQ0023); Fundamental Research Funds for the Central Universities (ZYGX2019J003); Exchange Project for Key Lab of Optical Fiber Sensing and Communications (ZYGX2021K010); Science and Technology Commission of Shanghai Municipality (SKLSFO2020-05).
Acknowledgments
The authors appreciate Prof. Nianqiang Li and Dr. Shuangquan Gu from Soochow University, China, for their useful help and suggestion in our simulations.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data used to support the findings of this study are available from the corresponding author upon request.
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