High-speed secure key distribution based on chaos synchronization in optically pumped QD spin-polarized VCSELs

Yu Huang; Yu Huang; Yu Huang; Pei Zhou; Pei Zhou; Pei Zhou; NianQiang Li; NianQiang Li

doi:10.1364/OE.426126

1. Introduction

Secure key distribution (SKD) has received enormous attention due to its important role in guaranteeing the security of encryption communication [1–3]. For decades, many key distribution schemes have been proposed and they can be classified into two categories. One is the mathematical algorithm-based key distribution, which may suffer from exhaustive attack. The other is the physical entropy source-based key distribution, which has information-theoretic security. The latter has attracted rising interest because it mainly relies on the probability of guessing the unpredictable physical phenomena rather than the computation ability. In recent years, some typical physical key distribution schemes have been proposed, such as quantum key distribution [4], fiber laser-based key distribution [5], and channel noise-based key distribution [6]. However, limited by the distribution mechanism or entropy bandwidth, these schemes cannot afford a high-speed SKD rate (∼Gb/s) for the current encryption communication.

Chaotic key distribution has been proposed as an alternative way to achieve high-speed SKD [7–11]. This type of key distribution is mainly based on random keys generation and chaos synchronization between mutually-coupled or between commonly-driven semiconductor lasers [12–16]. For the mutually-coupled lasers systems, the local signals bidirectionally coupled over the public channel may increase the possibility of information leakage [17–19]. However, for the commonly-driven lasers systems, the two users, i.e., the so-called Alice and Bob, exchange and compare the keying parameter over the public channel without any information of the local signal. Such a method greatly enhances the security of key distribution. Earlier, Uchida et al. proposed and experimentally demonstrated an SKD based on the dynamic characteristic of chaos synchronization in two optical-feedback lasers subject to symmetric random phase light injection [8,20,21]. The security of such key distribution depends on the practical difficulty of completely and precisely guessing the feedback phases of response lasers. However, an unsatisfactory key distribution rate, approximately 180 kb/s at a sampling rate of 2 Mb/s, was obtained due to the long synchronization recovery time. Later on, some efforts have been devoted to improving the key distribution rate by using a vertical-cavity surface-emitting laser (VCSEL) as a drive source due to its desired properties [22–25]. For example, Jiang et al. numerically demonstrated a high-speed SKD at Gb/s based on chaos synchronization of two VCSELs commonly driven by a random-polarization light [24]. However, limited by the optical feedback loop, the synchronization recovery time of the order of tens of nanoseconds was observed when the injection polarization angles mismatch is over 90°, which still limits the key distribution rate.

Spin-polarized VCSELs, as a novel type of spintronic devices, are known to exhibit superior properties compared with conventional VCSELs, such as a lower threshold, easier spin control of the lasing output, and much faster dynamics [26]. More importantly, it can yield satisfactory optical chaos without the need of any external perturbation [27,28], which indicates that there exists no time-delay signature (TDS) embedded in dynamics. In our previous reports, the stability and instability dynamics were uncovered in quantum well or quantum dot (QW/QD) spin-polarized VCSELs by using the direct simulations and the AUTO package [29–31]. Furthermore, a master-slave configuration for chaos communication was numerically implemented based on two optically pumped QW spin-polarized VCSELs [32]. It was demonstrated numerically that high-quality chaos synchronization and a message recovery at frequencies up to 4 GHz were achieved in a master-slave scheme consisting of two QW spin- polarized VCSELs. The above-mentioned reports indicate that significant progress has been made in studies of spin-polarized VCSELs operating in chaotic states. However, it is still open to explore the physical SKD based on chaos synchronization of QD spin-polarized VCSELs.

Herein, we propose a novel SKD scheme based on polarization-shift-keying chaos synchronization in commonly driven QD spin-polarized VCSELs without any external feedback. The polarization-shift-key is obtained by randomly and independently controlling the polarization ellipticity of the response lasers. Moreover, the commonly-driven QD spin-polarized VCSELs without any external feedback constitute an open-loop configuration. It has the advantage of a simple structure/low cost, where no feedback is needed, and more importantly, it yields a rather short synchronization recovery time- hundreds of picoseconds- and results in the enhancement of the key distribution rate. With these merits, we numerically demonstrate a 1.34 Gb/s key distribution with a bit error ratio (BER) lower than 3.8 × 10⁻³.

2. Scheme and theoretical model

Figure 1 shows the schematic diagram of the proposed SKD system. In the present scheme, three QD spin-polarized VCSELs are used, where one laser is used as the drive semiconductor laser (DSL) and the other two lasers act as the response semiconductor lasers (RSLA and RSLB). A simplified model of the spin-polarized VCSELs is shown in the red box of Fig. 1. The QD spin-polarized VCSELs are optically pumped and their output polarization can be readily controlled through an off-the-shelf polarization controller (PC). More details of the QD spin-polarized VCSELs were depicted in Ref. [10] and references therein. With the selected parameters, the DSL works in a chaotic regime with two coexisting contrary polarizations – right circular polarization (RCP) and left circular polarization (LCP) – and comparable average intensities. Its output is split into two branches by the optical coupler, and each of them is unidirectionally injected into the local RSLs. The RSLA and the RSLB also operate in chaotic RCP and LCP coexisting regimes and independently change the polarization ellipticity of each RSL with a PC controlled by a random control parameter generator (RCPG). By choosing the same random control parameter (polarization ellipticity) setting of Alice and Bob, high-quality chaos synchronization can be achieved between the RSLA and the RSLB, which is termed as polarization-shift-key chaos synchronization. We notice that all of the QD spin-polarized VCSELs are not deployed with any feedback loop, which is beneficial to enhancing the performance of the key distribution. On the one hand, there exists no TDS embedded in dynamics, indicating that the risk of information leakage from the DSL can be decreased. On the other hand, the open-loop configuration is deployed, which is beneficial to shortening the synchronization recovery time [33]. The outputs of RSLA and RSLB are detected by the inverse photodetector (IPD) and photodetector (PD), respectively. The difference signals between RCP and LCP are sampled and quantized as random bits with the single/dual-threshold quantization method [9,21]. After exchanging and comparing the keying parameter over a public channel, Alice and Bob only retain random bits generated from the synchronization chaos as the shared keys. In this scheme, the privacy of the key can be ensured due to the fact that the drive signal and the output of the response lasers are not transmitted over the public channel. Moreover, the keying parameters are independently and randomly selected, which extremely increases the difficulty of completely and precisely guessing the polarization ellipticity to reconstruct the chaos synchronization, and thus the security of key distribution is guaranteed.

Fig. 1. Schematic diagram of the proposed key distribution scheme. OI: Optical isolator; PBS: polarization beam splitter; (I)PD: (inverse) photodetector; RCPG: random control parameter generator; PC: polarization control. In red boxes, the simplified model of the QD spin-polarized VCSEL is showed.

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In the simulation, the rate equations based on the spin-flip model (SFM) are adopted and written as follows [28,30].

(1)$$\frac{{dE_\textrm{D}^ \pm }}{{dt}} = \kappa (n_{QD,\textrm{D}}^ \pm{-} 1)(1 + i\alpha )E_\textrm{D}^ \pm{-} ({\gamma _a} + i{\gamma _p})E_\textrm{D}^ \mp \textrm{ + }F_\textrm{D}^ \pm ,$$

(2)$$\frac{{dE_{\textrm{A,B}}^ \pm }}{{dt}} = \kappa (n_{QD,\textrm{A,B}}^ \pm{-} 1)(1 + i\alpha )E_{\textrm{A,B}}^ \pm{-} ({\gamma _a} + i{\gamma _p})E_{\textrm{A,B}}^ \mp \textrm{ - }i\Delta \omega E_{\textrm{A,B}}^ \pm{+} {k_{\textrm{inj}}}E_\textrm{D}^ \pm \textrm{ + }F_{\textrm{A,B}}^ \pm ,$$

(3)$$\frac{{dn_{WL,\textrm{D,A,B}}^ \pm }}{{dt}} = {\eta _ \pm }{\gamma _n} + h\frac{{{\gamma _n}}}{2} - {\gamma _0}n_{WL,\textrm{D,A,B}}^ \pm [\frac{{h - n_{QD,\textrm{D,A,B}}^ \pm }}{{2h}}] \mp {\gamma _j}(n_{WL,\textrm{D,A,B}}^ +{-} n_{WL.\textrm{D,A,B}}^ - ),$$

(4)$$\begin{aligned} \frac{{dn_{QD,\textrm{D,A,B}}^ \pm }}{{dt}} &= {\gamma _o}\frac{{n_{WL,\textrm{D,A,B}}^ \pm }}{h}(h - n_{QD,\textrm{D,A,B}}^ \pm ) - {\gamma _n}(h + n_{QD,\textrm{D,A,B}}^ \pm ) \mp {\gamma _j}(n_{QD,\textrm{D,A,B}}^ +{-} n_{QD,\textrm{D,A,B}}^ - )\\ &\textrm{ } - 2{\gamma _n}n_{QD,\textrm{D,A,B}}^ \pm {|{E_{\textrm{D,A,B}}^ \pm } |^2} \end{aligned}$$

where the subscripts D, A, B stand for the DSL, RSLA, and RSLB, and ${E^ + }$ (${E^ - }$) represents the complex electric field of the RCP (LCP). The normalized carrier variable is defined for the conduction band carrier concentration n with the superscript + (–) representing spin-up (spin-down), and subscript WL (QD) representing the wetting layer (quantum dot ground state). The parameters of this model are defined as follows: κ is the optical decay field, ${\gamma _n}$ is the carrier recombination rate, α is the linewidth enhancement factor, and h is the normalized gain coefficient, ${\gamma _j}$ is the spin relaxation rate, ${\gamma _o}$ is the carrier capture rate, ${\gamma _p}$ is the linear birefringence rate, and ${\gamma _a}$ is the dichroism. The total normalized pump power (${\eta}$) and pump polarization ellipticity (P) are separately defined by $\eta = {\eta _ + } - {\eta _ - }$ and $P = ({{\eta_ + } - {\eta_ - }} )/({{\eta_ + } + {\eta_ - }} )$, where ${\eta _ + }\; $ (${\eta _ - }$) corresponds to the right (left) circularly polarized component of the pump. The injection parameters include the injection rate ${k_{\textrm{inj}}}$, and angular detuning frequency Δω, i.e., Δω = 2πΔf, where Δf is the linear frequency detuning between the DSL and the RSL. The spontaneous emission noise is modeled by the following Langevin sources: $F_{\textrm{D},\textrm{A},\textrm{B}}^{} = \sqrt {{\beta _{sp}}}\xi_{\mathrm{D}, \mathrm{A}, \mathrm{B}}^{\pm}$, where $\xi_{\mathrm{D}, \mathrm{A}, \mathrm{B}}^{\pm}$ are independent Gaussian white noise sources with zero mean and unit variance, and ${\beta _{sp}}$ is the spontaneous emission rate [34].

To quantify the synchronization quality, the cross-correlation coefficient (CC) is calculated as follows [35,36]:

(5)$$\textrm{C}{\textrm{C}_{\textrm{DA|DB|AB}}} (\Delta t) = \frac{{\left\langle {[{I_{\textrm{D|D|A}}}(t\textrm{ + }\Delta t) - \left\langle {{I_{\textrm{D|D|A}}}(t + \Delta t)} \right\rangle ] \cdot [{I_{\textrm{A|B|B}}}(t) - \left\langle {{I_{\textrm{A|B|B}}}(t)} \right\rangle ]} \right\rangle }}{{\sqrt {\left\langle {{{\left|{{I_{\textrm{D|D|A}}}(t + \Delta t) - \left\langle {{I_{\textrm{D|D|A}}}(t + \Delta t)} \right\rangle } \right|}^2}} \right\rangle \cdot \left\langle {{{\left|{{I_{\textrm{A|B|B}}}(t) - \left\langle {{I_{\textrm{A|B|B}}}(t)} \right\rangle } \right|}^2}} \right\rangle } }},$$

where $\Delta t$ is the time shift, $\left\langle \cdot \right\rangle $ is the time average, and $I(t )= {|{E(t )} |^2}$ denotes the chaotic intensity time series. The value CC = 1 stands for perfect synchronization, while CC = 0 indicates no synchronization.

3. Results and discussion

The dynamics of the free-running QD spin-polarized VCSELs are studied by numerically solving Eqs. (1)–(4) with the fourth-order Runge-Kutta algorithm. Figure 2(a) depicts a typical two-parameter bifurcation diagram in the plane of the pump intensity η and the pump ellipticity P. In our previous work [30], we found that the chaos regimes always appear in a smaller value of γ_p and γ_j. Thus, here the values of γ_p and γ_j are set as 20 ns⁻¹ and 10 ns⁻¹, respectively. The other parameters are listed in the caption of Fig. 2. The two-parameter bifurcation diagram contains four distinct regions: (I) the white area corresponds to the stable region, where the QD spin-polarized VCSELs operate in continuous-wave emission, (II) the blue area refers to an uninterrupted period-one state, where the oscillation frequencies are determined by birefringence rather than the relaxation oscillation (RO) frequency, (III) the green area stands for the period-two oscillation, and (IV) this region accounts for the complicated dynamics shown with colors other than the three mentioned. Furthermore, a symmetrical phenomenon that the shape of each state is almost mirror image about P = 0 can be found in the bifurcation diagram, which is consistent with the previous report [30]. In the present work, we mainly focus on the study of chaos synchronization in chaotic regions. To confirm chaotic regions, the 0–1 test is employed to separate chaotic regions from non-chaotic regions [37], and the results are shown in Fig. 2(b). A value close to 1 refers to a chaotic state shown in red. A value close to 0, indicating non-chaotic states, is displayed in green. Comparison with mapping results in Fig. 2(a) shows that the regions of complicated dynamics identiﬁed via our bifurcation analysis overlap with those obtained from the 0–1 test for chaos. In our previous work, the effects of key parameters on the chaos dynamics of QD spin-polarized VCSELs were systematically investigated by using the direct simulations and the AUTO package [30]. These results show that the chaotic regimes can be greatly enhanced by increasing the values of some key parameters, such as ${\gamma _o}$, h, and α. In addition, Al-Seyab et al. numerically demonstrated the stability of spin-polarized VCSELs can be affected in the presence of an axial magnetic field, which may cause more rich chaos dynamics and/or the local changes of chaos regions at the same parameter [38].

Fig. 2. (a) Bifurcation diagram and (b) 0–1 test. The other parameters are $\kappa= 250\; \textrm{n}{\textrm{s}^{ - 1}}$, $\; {\gamma _n} = 1\; \textrm{n}{\textrm{s}^{ - 1}}$, ${\gamma _j} = 10\; \textrm{n}{\textrm{s}^{ - 1}}$, ${\gamma _o} = 600\; \textrm{n}{\textrm{s}^{ - 1}}$, ${\gamma _a} = 0$, ${\gamma _p} = 20\; \textrm{n}{\textrm{s}^{ - 1}}$, and $h = 1.665$.

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For the SKD scheme, two synchronization properties must be required. One is that the correlation between the drive signals and each of the response outputs is low for all parameter values. The other is that high-quality chaos synchronization can be achieved between the RSLs if and only if the parameter values are identical and vice versa [21]. To this end, the synchronization quality is evaluated using Eq. (5). Here, the value of polarization ellipticity of DSL (P_D) is set as 0, where $\eta = 2.5$. We notice that when the P_D and the P_A (P_B) are set as positive (negative values), the correlation between the DSL and the RSLs in LCP is higher (lower) than that in PCP. The different correlation in LCP and RCP is attributed to the fact that the intensity of LCP is lower than that of RCP in the QD spin-polarized VCSELs under the scenario of positive P_A and P_B and vice versa. Moreover, based on the same reason, the values of polarization ellipticity of RSLs (P_A and P_B) are set as either positive or negative. For the sake of simplicity, the values of polarization ellipticity of RSLs are set as either 0.4 or −0.4, where the QD spin-polarized VCSELs works in chaotic regions, as shown in Fig. 2(a). For this set of parameter values, the bandwidth of chaos output of the DSL, RSLA, and RSLB is estimated as 9.24 GHz, 12.38 GHz, and 13.54 GHz, respectively. Here the bandwidth is defined as the range between DC and the frequency that contains 80% of the spectral power [22]. Such larger bandwidth provides the possibility to increase the SKD rate to the Gb/s level. Figure 3 displays a global view of the maximum CC (CC^max) in the parameter space of the injection parameters, where ${P_\textrm{A}} = {P_\textrm{B}} = 0.4$. Figures 3(a1)-(a3) correspond to the correlation between the DSL and the RSLA in RCP, LCP, and total intensity, respectively, while Figs. 3(b1)-(b3) represent the synchronization quality between the RSLA and the RSLB in RCP, LCP, and total intensity, respectively. In Figs. 3(a1) and 3(b1), we can see that the high-quality chaos synchronization (CC^max>0.99) can be obtained between the RSLA and RSLB with increasing k_inj, while the cross-correlation between the DSL and RSLA is maintained at a relatively low level (CC^max<0.6). Without loss of generality, we consider the scenario of $\mathrm{\Delta }f = 0$. In this case, when the k_inj varies in a range of 30 ns⁻¹< k_inj < 50 ns⁻¹, the values of CC^max between the RSLA and RSLB are greater than 0.99, and in the meantime, the values of CC^max between the DSL and RSLA are lower than 0.6. This situation of the synchronization quality is suitable for the SKD based on chaos synchronization. Similar synchronization results are seen in LCP and total intensity as shown in Figs. 3(a2), (b2) and (a3), ( b3), respectively. Furthermore, it is can be seen that the CC^max between the DSL and the RSLA in RCP and LCP is slightly higher than that of the total intensity, which is displayed in Fig. 3(a3). This is mainly because the RSLA has uncoordinated average intensities in RCP and LCP. A similar phenomenon is also found under the scenario of ${P_\textrm{A}} = {P_\textrm{B}} ={-} 0.4$ (not shown here). In addition, asymmetry synchronization with respect to zero detuning is discovered in all of the results, which is determined by the nonzero linewidth enhancement factor [23,32,39].

Fig. 3. Maps of the maximum CC between (a1-a3) DSL and RSLA, (b1-b3) RSLA and RSLB in the (Δf, k_inj)-plane, where $\eta = 2.5$, ${P_\textrm{D}} = 0$, and ${P_\textrm{A}} = {P_\textrm{B}} = 0.4$. (a1, b1) RCP, (a2, b2) LCP, and (a3, b3) total intensity. The other parameters are the same as those in Fig. 2.

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Figure 4 shows an example of the cross-correlation between the DSL and the RSLs, as well as that between the RSLA and the RSLB, where ${k_{\textrm{inj}}} = 35\; \textrm{n}{\textrm{s}^{ - 1}}$, and $\Delta f = 0$. In Figs. 4(a1)-(a3) and 4(b1)-(b3), one can expect a lower correlation (CC^max<0.6) can be achieved between the DSL and the RSLs in RCP, LCP, and total intensity. In addition, we also find that the value of CC^max in total intensity is lower than that in RCP and LCP, which is in line with the results shown in Fig. 3. Under such a scenario, the privacy of key distribution can be guaranteed since the correct key is difficult to be extracted from injection signals for the attacker. Limited by the injecting-locking mechanism, the correlation of the drive signals and response outputs cannot be completely eliminated. This phenomenon was also observed in the edge-emitting semiconductor lasers and the conventional VCSEL-based systems for the same mechanism [21,24]. Furthermore, the cross-correlation between the RSLA and the RSLB with polarization mismatch is shown in Figs. 4(c1)-(c3). It is can be seen that the values of CC^max in RCP, LCP, and total intensity are in the vicinity of 0.4. Later on, the CC between the RSLA and the RSLB is also calculated at the same polarization ellipticity, and the corresponding results are shown in Figs. 4(d1)-(d4). As can be seen that the values of CC^max between the RSLA and the RSLB are close to 1 in the RCP, LCP, and total intensity, indicating that high-quality chaos synchronization can be obtained when the polarization ellipticity is matched. A similar phenomenon (not shown here) is also found with the polarization ellipticity of −0.4.

Fig. 4. CC between (a) DSL and RSLA, (b) DSL and RSLB, (c) RSLA and RSLB with mismatch polarization ellipticity, where ${P_\textrm{A}} = 0.4$, and ${P_\textrm{B}} ={-} 0.4$, (d) RSLA and RSLB with match polarization ellipticity, where ${P_\textrm{A}} = 0.4$, and ${P_\textrm{B}} = 0.4$. The other parameters are ${k_{\textrm{inj}}} = 35\; \textrm{n}{\textrm{s}^{ - 1}}$, $\Delta f = 0$, $\eta = 2.5$, ${P_\textrm{D}} = 0$, $\kappa= 250\; \textrm{n}{\textrm{s}^{ - 1}}$, $\; {\gamma _n} = 1\; \textrm{n}{\textrm{s}^{ - 1}}$, ${\gamma _j} = 10\; \textrm{n}{\textrm{s}^{ - 1}}$, ${\gamma _o} = 600\; \textrm{n}{\textrm{s}^{ - 1}}$, ${\gamma _a} = 0$, ${\gamma _p} = 20\; \textrm{n}{\textrm{s}^{ - 1}}$, and $h = 1.665$.

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In spin-VCSELs, the polarization ellipticity P is an experimentally accessible parameter, which can be readily controlled through an off-the-shelf polarization controller. Moreover, the chaotic dynamic regime can be changed by adjusting the value of P [30]. Thus, we consider the effect of polarization ellipticity mismatch (denoted as $\Delta P$) on the cross-correlation between the RSLA and the RSLB with the same keying polarization parameter set as either 0.4 or −0.4. Here, we fix the P_A, while the P_B varies, and the detuning value is defined as the parameter value of the RSLB minus that of the RSLA. In Fig. 5(a), the CC^max is calculated to illustrate the variation of chaos synchronization quality between the RSLA and the RSLB, where ${P_\textrm{A}} = 0.4$. It is can be seen that the synchronization quality decreases significantly with increasing $|{\Delta P} |$ in RCP, LCP, and total intensity, and the values of CC^max are over 0.8 in the polarization ellipticity detuning range of −0.02<$\Delta P$<0.03. Furthermore, we also show the case of a negative polarization ellipticity ${P_\textrm{A}} ={-} 0.4$ in Fig. 5(b), and a similar phenomenon is seen again. The values of CC^max are over 0.8 in the polarization ellipticity detuning range of −0.03<$\Delta P$<0.04. Moreover, we can also find that the decreasing trend of the synchronization quality is asymmetrical with respect to $\Delta P$ = 0 and that with respect to $P = 0$. This is mainly attributed to the P can impact the chaotic regime of QD spin-polarized VCSELs [30]. These results mean that the chaos synchronization quality in the proposed SKD system is extremely sensitive to the polarization ellipticity mismatch. Under such a scenario, the security of our proposed SKD can be enhanced.

Fig. 5. CC^max between the RSLA and the RSLB as a function of $\Delta P$, where (a) ${P_\textrm{A}} = 0.4$ and (b) ${P_\textrm{A}} ={-} 0.4$. The other parameters are the same as those in Fig. 4.

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To gain more details on the polarization-keying-shift chaos synchronization, we study the dynamic correlations between the RSLA and the RSLB, as shown in Fig. 6. The windows for each keying parameter and the short-time correlation are set as 100 ns and 10 ns, respectively. In Fig. 6, the dynamics CC^max is calculated in RCP, LCP, and total intensity to observe the dynamic variation of synchronization quality between the RSLA and the RSLB. One can expect that the outputs of the RSLs in RCP, LCP, and total intensity can well synchronize only when an almost identical polarization-shift is used for the RSLA and the RSLB. When the polarization ellipticities are not precisely matched, the CC^max degrades obviously. In addition, the dynamic CC^max between the DSL and the RSLA is also shown (red dashed curves). The average values of the CC^max between the DSL and the RSLA in RCP, LCP, and total intensity are in the vicinity of 0.5, which is also in accordance with the results shown in Fig. 4. It is worth mentioning that the polarization ellipticity of QD spin-polarized VCSELs is introduced as private keying parameters and shortening the duration time of each keying parameter can increase the number of keying parameters. Under such a scenario, it is very difficult for the attacker to simultaneously operate numerous private keying parameters, and thus the security of key distribution can be enhanced compared with the mutually-coupled systems or the commonly-driven synchronization chaos without private parameter keying systems [40,41]. As a compromise, the generation rate of the shared keys will be inevitably decreased due to the long duration time of the keying parameter reducing the ratio of available synchronization time, which is discussed below. Therefore, there is a trade-off between the security of key distribution and the key rate.

Fig. 6. (a) Temporal-varying random polarization ellipticity (P_A and P_B); (b-d) short-term cross-correlation between the DSL and the RSLA (red dashed curves) and that between the RSLA and the RSLB (black solid curves).

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Furthermore, we investigate the recovery time from the non-synchronization state to the synchronization state. As shown in Figs. 7(a1)-(c1), the views of the temporal waveforms of RSLs are enlarged, where 500 ns is the start time with the same polarization ellipticity (green dash line). To precisely evaluate the synchronization recovery time, the temporal waves are normalized and the recovery time is defined as the difference of normalization temporal wave within 10% [42]. It is can be seen that the synchronization recovery time is separately 400.2 ps, 200 ps, and 390 ps in RCP, LCP, and total intensity. Moreover, we find that the synchronization recovery time in RCP and total intensity is nearly twice as large as that in LCP. Furthermore, the synchronization time is measured over twenty times, and the average value and standard deviation are displayed in Table 1. Repeated simulations indicate that a larger standard deviation for the synchronization recovery time is found in RCP and LCP than that in total intensity. This is mainly because the inverse polarization ellipticity of the RSLs is applied as keying parameters, where the intensity of RCP (LCP) contained in the RSL with a positive (negative) P is a little stronger than that of LCP (RCP). Such a phenomenon is also observed in the conventional VCSELs, where the different polarization of the injection light (i.e., with injection polarization angle mismatch being over 45°) induces an obviously larger standard deviation for the synchronization recovery time than that of the same polarization of injection light [24]. Despite the synchronization recovery time has an obvious fluctuation, it is still kept in the magnitude of subnanosecond and about 2 orders of magnitude smaller than that of achieved in the conventional VCSELs [24]. This is attributed to the fact that the QD spin-polarized VCSELs without any external feedback constitute an open-loop synchronization configuration. In previous reports, the synchronization recovery time was observed linearly depending on the external feedback time in the closed-loop configuration, which induces a larger synchronization recovery time [8,20,21,33]. In our present scheme, the synchronization recovery time is independent of the external feedback time and thus yields a short synchronization recovery time of hundreds of picoseconds.

Fig. 7. Temporal waveforms of RSLA and RSLB switching from the non-synchronized state to the synchronized state in the (a1) RCP, (b1) LCP, and (c1) total intensity. Difference of the normalized temporal waveforms in the (a2) RCP, (b2) LCP, and (c2) total intensity.

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Table 1. Synchronization recovery time is calculated twenty times.

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Based on the private dynamic synchronization between the RSLA and the RSLB, we now turn attention to the feasibility of the proposed SKD scheme. The difference between the RCP signal and the LCP signal is served as the correlated physical random source instead of the single intensity or total intensity, which decreases the possibility of keying leakage [24]. The difference signals with polarization-shift-keying synchronization are sampled and quantized into random bits sequence. By changing and comparing the random control parameter, Alice and Bob retain the bits generated in the synchronization chaos as the shared secret key and discard the useless bits generated in the non-synchronization chaos and synchronization recovery process. Here, the dual-threshold quantization method is adopted to generate the shared key with a sampling frequency of 5 GHz [42]. With this method, the difference signals larger than the upper-threshold are quantized as bits “1” and those smaller than the lower-threshold are quantized as bits “0”, while those between the upper-threshold and the lower-threshold are discarded. Despite such a method will inevitably reduce the retained ratio of the shared key, it can minish the BER of the key distribution compared to the conventional single-threshold technology since the noise effect can be eliminated [9,21,42]. Moreover, to improve the randomness of the shared key, the bit-wise exclusive-OR (XOR) operation, widely employed in random bit generation, is used to enhance the inherent randomness and thus further eliminate residual correlations [12,43]. Subsequently, the randomness of the shared keys is tested by using the typical NIST 800–22 [44], and the results are shown in Table 2. Here, each test is evaluated using 1000 samples of 1 Mbit sequences with a significance level of α_H = 0.01. As we can see that all of the 15 NIST tests are passed with a P-value larger than 0.0001 and a success ratio greater than 0.979, which indicates good statistical randomness of the generated bits by using the proposed method.

Table 2. Results of NIST 800–22 tests of secret keys from RSLs.

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To explore the effect of the inevitable mismatch between the RSLA and the RSLB on the BER of key distribution, we first show the evolution of chaos synchronization as a function of the mismatch of intrinsic parameters in QD spin-polarized VCSELs. Here, the intrinsic parameters (κ, ${\gamma _o}$, α, h, ${\gamma _n}$, ${\gamma _p}$, ${\gamma _j}$) for the DSL and the RSLA are fixed, while those of the RSLB are varied. The mismatched parameters of RSLB are written as ${\kappa _\textrm{B}} = ({1\pm\mu } ){\kappa _\textrm{A}}$, ${\gamma _{o,\textrm{B}}} = (1\pm \mu ){\gamma _{o,\textrm{A}}}$, ${\alpha _\textrm{B}} = ({1\pm \mu } ){\alpha _\textrm{A}}$, ${h_\textrm{B}} = ({1\pm \mu } ){h_\textrm{A}}$, ${\gamma _{n,\textrm{B}}} = (1\pm \mu ){\gamma _{n,\textrm{A}}}$, ${\gamma _{p,\textrm{B}}} = (1 \pm \mu){\gamma _{p,\textrm{A}}}$, ${\gamma _{j,\textrm{B}}} = (1\pm \mu ){\gamma _{j,\textrm{A}}}$ where the μ stands for the mismatch ratio. We notice that the mismatch of the dichroism ${\gamma _a}$ does not be considered because the ${\gamma _a}$ was set as 0 and the fairly impact was provided in the previous report [23]. Figure 8(a) presents the variation of chaos synchronization between the RSLA and the RSLB with mismatch in single parameter and of the whole parameters mentioned above. As can be seen from Fig. 8(a), the high-quality synchronization with the CC^max larger than 0.94 between the RSLA and the RSLB can be maintained in the single parameter mismatch range up to tens of percent. This is mainly because the injection-locking effect induced by the symmetric injection from DSL, which drives the evolutions of RSLA and RSLB towards that of DSL [9]. In addition, there are some interesting asymmetries between the positive and negative mismatch, i.e. α (blue rhombus). It appears that a smaller value of α will prevent synchronization for a negative mismatch. This is mainly attributed to the fact that reducing the α will decrease the coupling effect between the amplitude and phase of the electric field [23,45]. The phenomenon of asymmetries in synchronization is enlarged in the scenario associated with the whole parameter mismatch, which is consistent with that in the conventional VCSELs for a similar mechanism [24]. Based on the evolution of chaos synchronization quality, the BER of key distribution is calculated with different retained ratios as shown in Figs. 8(b)-(f). The retention ratio r equals to real random bit number divided by the maximum random bit generation number [46]. The retained ratios r = 0.2, 0.4, 0.6, and 0.8 denote the results obtained by using the dual-threshold quantization. As a comparison, the single-threshold quantization method (denoted as r = 1) is also used to evaluate the BER of key distribution. From Figs. 8(b)-(f), it is seen that the BER increases monotonically with increasing the intrinsic parameter mismatch ratio in both the single parameter mismatch and the whole parameter mismatch. However, the BER can be reduced with a smaller retained ratio, e.g., when r = 0.2, the BER of key distribution is lower than the forward error correction (HD-FEC) threshold (3.8 × 10⁻³) in both parameter mismatch scenarios with the ratio being up to tens of percent. By contrast, as shown in Fig. 8(f), a BER above the HD-FEC threshold can be observed in the whole parameter mismatch range up to tens of percent under the scenario of r = 1. These results indicate that a low r corresponds to a better BER performance, which is at the cost of lower key distribution. Thus, there is a trade-off between the immunity to synchronization error and key rate. As a compromise, a BER below the HD-FEC threshold can be obtained when the whole parameter mismatch is controlled within −3%∼5% under the case of r = 0.6, as shown in Fig. 8(d).

Fig. 8. (a) Chaos synchronization and (b-f) BER as a function of the mismatch of QD spin-polarized VCSELs intrinsic parameters. r = 0.2, 0.4, 0.6, 0.8, and 1 represent quantization with different retained ratios. The dotted line stands for the HD-FEC threshold.

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Finally, we evaluate the generation rate of the shared key. In the current scheme, the probability of a polarization ellipticity match is 0.5 and one-bit quantizing technology is adopted from each sample. Therefore, the theoretical key generation rate can be written as $0.5 \cdot r \cdot R \cdot {f_s}$ [42]. In this formula, r is the retained ratio of the share keys, ${f_s}$ is the down-sampling rate of extracting keys from the difference signals as aforementioned, and R is the ratio of available synchronization time, which is mainly determined by $({\tau _c} - {\tau _s} - {\tau _r})/{\tau _c}$ [42]. Here, ${\tau _c}$ is the duration time of each keying parameter, ${\tau _s}$ is the time for changing the optoelectronic switch once, and ${\tau _r}$ is the recovery time from the non-synchronization state to the synchronization state. From this formula, one can see that increasing the value of R is beneficial to the improvement of the key generation rate. Specifically, two categories can be employed to improve the key generation rate. One approach enhances the duration time of each keying parameter to increase the value of R, but the private control parameters will be inevitably reduced, which may decrease the security of key distribution. Therefore, shortening the synchronization recovery time may be the most promising way to enlarge the value of R, and finally, the key generation rate can be significantly improved. In the previous reports, limited by the external feedback time, the synchronization recovery time up to a few tens of nanoseconds can be observed in the closed-loop configuration [20,24]. Herein, the short synchronization recovery time of hundreds of picoseconds can be obtained that benefits from the open-loop synchronization configuration. To quantitatively evaluate the key generation rate, the worst synchronization recovery time is adopted with ${\tau _r}$ = 814.5 ps as shown in Table 1. With the above settings and investigations, the final key generation rate is calculated to be 1.34 Gb/s, where r = 0.6, ${\tau _s}$ = 10 ns [42], ${\tau _c}$ = 100 ns, and ${f_s}$ = 5 GHz. We notice that the key generation rate is not limited at Gb/s level and it can be further improved by properly increasing the sampling rate or adopting the multi-bit quantizing technology [13,47]. Compared with the key distribution systems based on conventional VCSELs, our scheme has a simple structure due to the complex dynamics found in a QD spin-polarized VCSELs without any external feedback, which enhances the performance of the key distribution. Specifically, the TDS is not introduced, which decreases the risk of information leakage from the drive signal. On the other hand, the open-loop configuration can be constituted to obtain short synchronization recovery time-hundreds of picoseconds, indicating that a generation rate of the key at Gb/s level is easily achieved at the same during time of the keying parameter. Additionally, without reducing the generation rate of the key, the short synchronization recovery time can support a relatively short during time of the keying parameter, which increases the number of key parameters and thus ensures the security of key distribution.

4. Conclusion

In conclusion, we have numerically demonstrated a novel SKD scheme based on polarization-shift-keying chaos synchronization of QD spin-polarized VCSELs without any external feedback. The results show that chaos synchronization between Alice and Bob is very sensitive to the mismatch of polarization ellipticity. Moreover, the open-loop configuration induces a short synchronization recovery time of hundreds of picoseconds supporting the achievement of high-speed key distribution. Based on this, we have obtained a 1.34 Gb/s SKD with a BER below 3.8 × 10⁻³. The proposed SKD scheme provides a new potential way to implement high-performance key distribution.

Funding

National Natural Science Foundation of China (62004135, 62001317); Natural Science Research Project of Jiangsu Higher Education Institutions of China (20KJA416001, 20KJB510011); Natural Science Foundation of Jiangsu Province (BK20200855); Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), P. R. China (IPOC2020A012); State Key Laboratory of Advanced Optical Communication Systems Networks, China (2021GZKF003); Project of Key Laboratory of Radar Imaging and Microwave Photonics (Nanjing University of Aeronautics and Astronautics), Ministry of Education (RIMP2020001); Startup Funding of Soochow University (Q415900119).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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	RCP	LCP	Total intensity
Average	470.4 ps	484.9 ps	408.1 ps
Standard deviation	322.6 ps	329.6 ps	289.7 ps

	RSLA		RSLB
Statistical test	P-value	Proportion	P-value	Proportion	Results
Frequency	0.554420	0.9890	0.854708	0.9840	Success
Block Frequency	0.308561	0.9920	0.474986	0.9900	Success
Cumulative Sums	0.375313	0.9880	0.486588	0.9820	Success
Runs	0.452173	0.9890	0.052610	0.9820	Success
Longest Run	0.649612	0.9910	0.645448	0.9920	Success
Rank	0.880145	0.9910	0.208837	0.9940	Success
FFT	0.474986	0.9810	0.081510	0.9820	Success
Non-Overlapping Template	0.055010	0.9850	0.666245	0.9800	Success
Overlapping Template	0.745908	0.9920	0.455937	0.9900	Success
Universal	0.941144	0.9900	0.440975	0.9860	Success
Approximate Entropy	0.624627	0.9830	0.236810	0.9940	Success
Random Excursions	0.930992	0.9850	0.417734	0.9894	Success
Random Excursions Variant	0.930992	0.9851	0.327854	0.9798	Success
Serial	0.846338	0.9860	0.649612	0.9940	Success
Linear Complexity	0.512137	0.9970	0.286836	0.9960	Success

	RCP	LCP	Total intensity
Average	470.4 ps	484.9 ps	408.1 ps
Standard deviation	322.6 ps	329.6 ps	289.7 ps

	RSLA		RSLB
Statistical test	P-value	Proportion	P-value	Proportion	Results
Frequency	0.554420	0.9890	0.854708	0.9840	Success
Block Frequency	0.308561	0.9920	0.474986	0.9900	Success
Cumulative Sums	0.375313	0.9880	0.486588	0.9820	Success
Runs	0.452173	0.9890	0.052610	0.9820	Success
Longest Run	0.649612	0.9910	0.645448	0.9920	Success
Rank	0.880145	0.9910	0.208837	0.9940	Success
FFT	0.474986	0.9810	0.081510	0.9820	Success
Non-Overlapping Template	0.055010	0.9850	0.666245	0.9800	Success
Overlapping Template	0.745908	0.9920	0.455937	0.9900	Success
Universal	0.941144	0.9900	0.440975	0.9860	Success
Approximate Entropy	0.624627	0.9830	0.236810	0.9940	Success
Random Excursions	0.930992	0.9850	0.417734	0.9894	Success
Random Excursions Variant	0.930992	0.9851	0.327854	0.9798	Success
Serial	0.846338	0.9860	0.649612	0.9940	Success
Linear Complexity	0.512137	0.9970	0.286836	0.9960	Success

High-speed secure key distribution based on chaos synchronization in optically pumped QD spin-polarized VCSELs

Abstract

1. Introduction

2. Scheme and theoretical model

3. Results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (2)

Equations (5)

Optics Express