基于对称相移键控混沌同步的高速密钥安全分发 
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The core of information security is secure communication. Shannon’s secret communication scheme, "one-time-pad, " requires that the key be discarded after use, the key length is larger than the information length, and the key generation cannot be predicted. The key distribution is mainly based on a mathematical algorithm. The security of the algorithm key depends on a computer’s computing power. Using brute force still threatens the security of this scheme. Physical-layer key distribution has high security, such as quantum key distribution, key distribution based on fiber laser, and key distribution based on channel noise. However, limited by the key distribution mechanism or the bandwidth of the physical entropy source, the key distribution rate of the above physical layer is only bit/skbit/s, which is difficult to meet the requirements of a high-speed communication rate.
A wideband chaotic laser can be produced by a semiconductor laser under an external disturbance. The time-domain waveform of a chaotic laser has the characteristics of large amplitude random fluctuations, which can generate high-speed physical random numbers from Gbit/s to Tbit/s. In addition, the parameter-matched laser can realize chaotic synchronization, and the high-speed correlated physical key can be generated by quantizing the chaotic synchronization waveform. Therefore, using a chaotic laser as a physical entropy source and combining it with chaotic synchronization is expected to achieve high-speed physical key distribution. Uchida et al. proposed a key distribution scheme based on optical feedback laser phase shift keying chaos synchronization and achieved a key distribution rate of approximately 180 kbit/s. Notably, the optical feedback laser constitutes a closed-loop structure. The random phase-shift-keying of the feedback optical path results in a long oscillation time for the laser to achieve chaos synchronization again, which results in a synchronization recovery time of tens of nanoseconds. Longer synchronization recovery time will limit the further improvement of the key distribution rate. To improve the key distribution rate, other methods have been proposed. For example, Jiang et al. proposed a key distribution scheme based on dynamic unpredictable post-processing, alternating step algorithm post-processing, and dynamic random polarization synchronization of vertical-cavity surface-emitting lasers. Xiang et al. proposed a key distribution scheme based on bandwidth enhanced chaotic synchronization. Notably, the above schemes use a chaotic laser to drive response laser to construct chaotic synchronization, but there is a residual correlation between the drive and response signals. Based on the correlation, an eavesdropper can directly obtain part of the relevant key through the drive signal, which causes the potential security threat of key distribution.
In this study, a scheme of high-speed secure key distribution based on symmetric-phase-shift-keying chaos synchronization is proposed and numerically demonstrated. The chaotic drive signal was injected symmetrically into the response lasers without external feedback through two unbalanced Mach-Zehnder (M-Z) interferometers. Phase-shift-keying chaos synchronization was realized by randomly modulating phase modulators in the M-Z interferometers. The chaotic temporal waveforms with phase-shift-keying synchronization were quantized into random bits, which were stored in the recorders together with the corresponding keying parameters. After exchanging and comparing the keying parameters, legitimate users retained random bits generated from synchronized chaos as shared keys to realize the key distribution.
In the proposed scheme, the unbalanced M-Z interferometer introduced a nonlinear transformation of delayed self-interference to the chaotic drive signal, which decreased the cross-correlation of drive and response signals to 0.25 (Fig. 3) and thus improved the security of key distribution. Moreover, the commonly driven response lasers without external feedback constituted an open-loop synchronization structure, which reduced the synchronization recovery time of dynamic phase-shift-keying chaos synchronization to 1.8 ns (Fig. 5) and thus improved the key distribution rate. We calculated the synchronization recovery time 5000 times and plotted the probability distribution, which showed that the open-loop synchronization structure had high stability in shortening the synchronization recovery time (Fig. 6). Then, we studied the mismatch effects of phase and intrinsic laser parameters on the chaos synchronization and bit error rate (BER) (Fig. 7) and evaluated the key distribution rate and security. In addition, we analyzed the BER of the key obtained by an eavesdropper intercepting the chaotic driving signal directly with or without an M-Z interferometer (Fig. 8).
In this study, we propose a high-speed secure key distribution scheme based on symmetric-phase-shift-keying chaotic synchronization. Using the delayed self-interference of an unbalanced M-Z interferometer, the correlation between the drive and response signals is reduced to 0.25. The eavesdropper cannot directly obtain part of the relevant key from the chaotic drive signal to improve the key distribution security. An open-loop chaotic synchronization structure is constructed by a commonly driven semiconductor laser without external cavity feedback, which avoid multiple oscillations of chaotic signals in the feedback external cavity in the closed-loop structure, shorten the recovery time of chaotic synchronization to 1.8 ns, and improve the key distribution rate. Finally, when the BER is 3.8×10-3, a high-speed secure key distribution at a rate of 1.28 Gbit/s is realized.
1 引言
信息安全的核心是保密通信。“一次一密”是绝对安全的保密通信[1]，其实现的关键在于高速密钥安全分发。随着计算机算力的不断提升，基于数学算法的密钥分发在原理上始终存在着被破解的安全隐患。物理层的密钥分发具有更高的安全性，如量子密钥分发[2-3]、基于光纤激光器的密钥分发[4]、基于信道噪声的密钥分发[5]等。然而，受密钥分发机制或物理熵源带宽的限制，上述物理层密钥分发速率仅为bit/s至kbit/s量级，难以满足高速通信的速率需求。
研究发现，半导体激光器在外部扰动下可产生宽带混沌激光，其输出光强具有大幅度随机起伏特征，可产生速率在Gbit/s～Tbit/s量级的高速物理随机数[6-7]。此外，参数匹配的激光器可实现混沌同步[8-10]，通过对混沌同步波形量化可产生相关的高速物理密钥[11]。因此，利用混沌激光作为物理熵源，并结合混沌同步，有望实现高速物理密钥分发。Uchida教授团队提出并实验验证了基于光反馈激光器相移键控混沌同步的密钥分发方案，实现了速率约180 kbit/s的密钥分发[12-13]。需要指出的是，光反馈激光器构成闭环结构，其反馈光路的随机相移键控导致激光器需要长时间振荡才可以再次实现混沌同步——同步恢复时间达数十纳秒，限制了密钥分发速率。国内学者在基于混沌同步的密钥分发速率提升方面开展了一些探索性的研究，例如：江宁教授课题组提出了基于动态不可预测后处理[14]、交替步进算法后处理[15]、垂直腔面发射激光器动态随机偏振同步的密钥分发方案[16]；项水英教授课题组提出一种基于带宽增强混沌同步的密钥分发方案[17]。值得注意的是，上述方案均利用混沌激光器驱动响应激光器构建混沌同步——驱动信号与响应信号之间存在残余相关性。基于该相关性，窃听者可直接通过驱动信号获取部分相关密钥，导致密钥分发安全性有所降低。此外，程孟凡教授课题组和张耀辉研究员课题组分别探索了基于模/数混合混沌同步与半导体超晶格混沌同步的密钥分发方案[18-19]。
本文提出一种基于对称相移键控混沌同步的高速密钥安全分发方案。在该方案中，混沌驱动信号经过非平衡马赫-曾德尔(M-Z)干涉仪产生基于延时自干涉的非线性变换，降低了驱动信号与响应信号之间的相关性，从而提高密钥分发的安全性。此外，响应激光器无外腔反馈，构成开环结构，可缩短混沌同步恢复时间，从而提高密钥分发速率。基于上述优点，通过数值模拟实现了速率为1.28 Gbit/s的高速密钥安全分发。
2 方案与理论模型
利用Lang-Kobayashi速率方程[20]建立密钥分发系统模型，并由四阶Runge-Kutta算法进行求解，即
激光器的参数设置如下：透明载流子密度N0＝1×10-6μm-3，载流子寿命τn＝2.3 ns，光子寿命τp＝1.6 ps，微分增益系数g＝6×10-3μm3·ns-1，线宽增强因子α＝6，增益饱和因子ε＝1×10-5μm3，自发辐射因子β＝1.0×10-3，激光器腔内往返时间τin＝7.3 ps，有源区体积V＝100 μm3。角频率ωD＝1.22×1015 rad/s，角频率失谐Δω＝0 GHz，阈值电流Ith＝9.8 mA，偏置电流 ID＝2.5Ith,IA，B＝1.2Ith，反馈强度Kf＝0.03，注入强度Kj＝0.2，反馈时延τf＝3 ns。M-Z干涉仪两臂的传输延时分别为：τj，1＝1 ns，τj，2＝2 ns。
3 结果与讨论
3.1 M-Z干涉仪非线性变换、驱动与响应相关性
首先，分析非平衡M-Z干涉仪对混沌驱动信号的非线性变换的影响。
图 2. 混沌驱动信号经过M-Z干涉仪(MZI)前、后的时序、频谱、互相关曲线。(a)(c)时序；(b)(d)频谱；(e)互相关曲线
Fig. 2. Time series, RF spectra, and cross correlation curve of chaotic driving signal before and after passing through the M-Z interferometer. (a)(c) Time series; (b)(d) RF spectra; (e) cross correlation curve
接下来，分析M-Z干涉仪的非线性变换对驱动与响应相关性的影响。
图 3. Kj＝0.2时，有、无M-Z干涉仪时的混沌时序及互相关曲线。(a)DFB-D的混沌时序和有/无M-Z干涉仪时DFB-A的混沌时序；(b)无M-Z干涉仪时DFB-D与DFB-A混沌时序的互相关曲线；(c)有M-Z干涉仪时DFB-D与DFB-A混沌时序的互相关曲线
Fig. 3. Time series and cross correlation curves with and without M-Z interferometer under Kj＝0.2. (a) Time series of DFB-D and DFB-A with or without M-Z interferometer; (b) cross correlation curve of chaotic signals from DFB-D and DFB-A without M-Z interferometer; (c) cross correlation curve of chaotic signals from DFB-D and DFB-A with M-Z interferometer
3.2 相移键控混沌同步
通信双方通过键控相位调制器实现相移键控混沌同步：双方相位差Δϕ＝0时，时序如
图 4. DFB-A和DFB-B的时序与关联点图。(a)(b)相位差Δϕ＝0；(c)(d)相位差Δϕ＝π
Fig. 4. Time series and scatter plots of DFB-A and DFB-B. (a)(b) Phase difference Δϕ＝0; (c)(d) phase difference Δϕ＝π
图 5. 相移键控码和DFB-A/DFB-B输出信号之间的短时互相关曲线。(a)相移键控码；(b)短时互相关曲线
Fig. 5. Phase-shift-keying code and short-time cross correlation curve between DFB-A and DFB-B. (a) Phase-shift-keying code; (b) short-time cross correlation curve
此外，为了评估缩短混沌同步恢复时间的稳定性，统计了5000个同步恢复时间，并绘制出
图 6. 同步恢复时间的概率统计分布
Fig. 6. Probability distribution of chaos synchronization recovery time
3.3 密钥分发
合法通信方使用双阈值量化[13]方法对混沌信号进行量化，从而产生随机密钥，然后交换键控参数并对比筛选出参数相同(即键控相位相同)时对应的随机密钥作为一致密钥，实现密钥分发。在双阈值量化方案中，设置上、下阈值(V＋与V-)，当采样点的幅值高于V＋和低于V-时，分别量化为1和0；当采样点的幅值处于上、下阈值之间或恰好落于阈值V＋与V-上时，则舍弃该采样点。与传统的单阈值量化相比，双阈值量化可以降低分发密钥的误码率(BER)，这是因为量化过程中舍弃了受噪声影响的部分混沌信号。
研究了相位参数失配和响应激光器内部参数失配对混沌同步以及不同保留率下密钥分发误码率的影响，在相同条件下，每个误码率测量5次。需要指出的是，保留率R＝1.0和R<1.0分别表示单阈值量化与双阈值量化。
图 7. 参数失配对混沌同步与误码率的影响。相位参数失配对(a)混沌同步性和(b)误码率的影响；(c)内部参数失配对混沌同步性的影响；保留率分别为(d)R＝1.0、(e)R＝0.5、(f)R＝0.2时，内部参数失配对误码率影响
Fig. 7. Influence of parameter mismatch on chaos synchronization and bit error rate (BER). Influence of phase parameter mismatch on (a) chaotic synchronization and (b) BER; (c) influence of intrinsic parameter mismatch on chaotic synchronization; influence of intrinsic parameter mismatch on BER under (d) R＝1.0, (e) R＝0.5, and (f) R＝0.2
进一步分析密钥分发的速率，计算公式为
最后，分析了在有/无M-Z干涉仪时，窃听者直接截取混沌驱动信号并在不同保留率下量化产生密钥的误码率，结果如
图 8. 有、无M-Z干涉仪时，窃听者直接量化混沌驱动信号产生密钥的误码率
Fig. 8. BER as a function of retained ratio when eavesdropper intercepts and quantizes the chaotic drive signal from key distribution system with and without M-Z interferometer
4 结论
提出一种基于对称相移键控混沌同步的高速密钥安全分发方案。利用非平衡M-Z干涉仪的延时自干涉对混沌驱动信号进行非线性变换，将驱动信号与响应信号之间的相关性降低至0.25，避免窃听者直接从混沌驱动信号中获取部分相关密钥，从而提高密钥分发的安全性；利用无外腔反馈的共驱半导体激光器构成开环混沌同步结构，避免了闭环结构中混沌信号在反馈外腔的多次振荡，缩短混沌同步恢复时间至1.8 ns，从而提高密钥分发速率。数值模拟研究了相移键控混沌同步，混沌同步恢复时间稳定性、相位参数失配和激光器内部参数失配对混沌同步质量与密钥分发误码率的影响，评估了密钥分发的误码率与安全性。最终，在误码率为3.8×10-3时，实现了速率为 1.28 Gbit/s的高速密钥安全分发。
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武超人, 高华, 王龙生, 郭园园, 王安帮, 王云才. 基于对称相移键控混沌同步的高速密钥安全分发[J]. 中国激光, 2022, 49(4): 0406001. Chaoren Wu, Hua Gao, Longsheng Wang, Yuanyuan Guo, Anbang Wang, Yuncai Wang. High-Speed Secure Key Distribution Based on Symmetric Phase-Shift-Keying Chaos Synchronization[J]. Chinese Journal of Lasers, 2022, 49(4): 0406001.
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