1.School of Optoelectronic Science and Engineering, Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China 2.Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province, Key Lab of Modern Optical Technologies of Education Ministry of China, Soochow University, Suzhou 215006, China 3.Institute of Science and Technology for Opto-Electornic Information, Yantai University, Yantai 264005, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant Nos. 62004135, 62001317), the Natural Science Research Project of Jiangsu Higher Education Institutions, China (Grant No. 20KJA416001), and the Startup Funding of Soochow University, China (Grant No. Q415900119).
Received Date:02 December 2020
Accepted Date:19 December 2020
Available Online:13 May 2021
Published Online:20 May 2021
Abstract: With the rapid development of the computer technology and communication technology, as well as the popularization of the Internet, information security has received much attention of all fields. To ensure the information security, a large number of random numbers must be generated. It is well accepted that random numbers can be divided into physical random numbers and pseudo random numbers. The pseudo random numbers are mainly generated based on algorithms, which can be reproduced once the seed is decoded. The physical random numbers are extracted from physical entropies. While the bandwidth of the traditional physical entropy source is quite small, the bit rate of generated physical random numbers is limited. In the literature, a lot of methods have been proposed to produce high-quality and high-speed random number sequences with the chaotic entropy source, which exhibits wide bandwidth, large amplitude and random fluctuations. Usually, a semiconductor laser with optical feedback, i.e, an external-cavity semiconductor laser (ECSL), is chosen as a chaotic entropy source to generate a chaotic signal output. However, the chaotic signal output has a high time delay characteristic, which is not conducive to the production of high-quality random numbers. In this paper, to produce high-quality chaos with time-delay signature (TDS) being well suppressed, we propose to employ an integration-oriented phased-array semiconductor laser to post-process the original chaos generated by an ECSL. It is shown that the proposed laser array is effective in TDS suppression, which improves the quality of optical chaos. After certain necessary post-processing, high-speed and high-quality random number sequences can be achieved. In this paper, we employ the conventional post-processing techniques, which include an 8-bit analog-to-digital converter (ADC) for sampling and quantization, and m-bits least significant bit (m-LSB) and exclusive OR (XOR) for removing bias. The simulation results show that the random number sequences obtained from the chaotic entropy source comprised of an ECSL and phased-array semiconductor lasers have uniform distribution characteristic and their scatter diagram contains no obvious pattern. Meanwhile, the obtained random number sequences can pass all tests of the standard randomness benchmark, NIST SP 800-22. Additionally, based on the extensibility of phased-array semiconductor lasers, random number generators that can generate parallel random numbers are achievable. Keywords:semiconductor lasers/ phased-array semiconductor lasers/ laser chaos/ random number
3.结果与讨论利用四阶龙格-库塔算法对该系统速率方程进行数值求解, 得到激光器输出的混沌信号. 在本文数值模拟中, 相关参数取值如下[41]: αH = 5, a = 4 μm, adiff = 2.5 × 10–16 cm2, γN = 1.0 ns–1, τp = 1.53 ps, N0 = 1 × 1018 cm–3, n = 3.4, P = 1.5Pth. 除非特别说明, 我们选择激光器阵列中A、B节点间的分离比(d/a)为0.5, 其中d为A、B节点间距离的1/2, a为激光器节点宽度的1/2, 波导参数选择带增益引导的反折射率引导, 具体定义和参数可参照我们前期的工作[43]. 不失一般性, 我们选取反馈强度kf = 5 ns–1和反馈时延τf = 1 ns, 此时光反馈半导体激光器工作在混沌状态, 其强度时间序列如图2(a1)所示. 通过计算强度时间序列的自相关(autocorrelation function, ACF), 我们发现在反馈时延τf = 1 ns及其倍数处ACF出现峰值, 如图2(a2)所示. 通过观察图2(a3)给出的频谱, 同样可以发现等间隔的峰值, 频率间隔等于反馈时延τf的倒数. 它们表明此原始混沌信号存在周期性, 不利于获取高品质随机数. 研究发现通过注入到单个激光器或激光器阵列可有效改善此时延特性, 特别地, 在同等注入条件下, 激光器阵列可在更大的参数空间内实现时延隐藏. 图2(b)和2(c)分别为光反馈半导体激光器产生的混沌信号注入单个激光器和两节点激光器阵列后的混沌熵源及其ACF与频谱特征. 上述图中选择以注入参数kinj = 30 ns–1与Δf = –30 GHz为例. 通过比较可以发现, 同等条件下, 激光器阵列更适合用于后处理混沌熵源, 其ACF和频谱均无明显峰值, 此结果与我们之前报道的结果具有一致性[41]. 图 2 激光器输出混沌信号的时间序列(左列), 自相关函数谱(中列), 功率谱(右列) (a) 光反馈半导体激光器; (b) 注入激光器; (c) 注入激光器阵列 Figure2. Time series (left column), autocorrelation function (middle column), and power spectra (right column) of the chaotic signal output by laser: (a) ECSL; (b) injection to a single laser A; (c) injection to phased-array lasers.
在我们另外的工作中, 详细研究了不同波导参数、注入参数和耦合参数对于混沌熵源ACF特征的影响[44]. 图3以带增益引导的反折射率引导波导为例, 给出了时延处的ACF峰值随着注入参数和阵列中激光器分离比d/a的演化情况. 红色代表时延处ACF峰值明显的情况, 而蓝色则表示时延被抑制或消除. 从图3可见, 当分离比较小时, 注入强度不宜过大, 在负频率失谐区域更易实现时延隐藏; 随着分离比的增大, 激光器阵列更易实现时延隐藏. 因此, 通过合理设计, 激光器阵列能够有效提升光反馈半导体激光器产生的混沌熵源的性能. 本文重点证明采用激光器阵列后处理的混沌熵源获取高品质随机数的可行性, 对于时延特征的详细分析不再赘述, 可参照我们的其他同步工作. 图 3 经过激光器阵列后处理混沌熵源的ACF时延处峰值随着注入参数和激光器分离比d/a的演化情况 (a) d/a = 0.2; (b) d/a = 0.4; (c) d/a = 0.6; (d) d/a = 1.0 Figure3. The evaluation of the ACF peak value located around the feedback delay of the chaotic entropy source that is processed by the phased-array in the plane of injection parameters for several values of laser separation: (a) d/a = 0.2, (b) d/a = 0.4, (c) d/a = 0.6, (d) d/a = 1.0.
接着, 分析利用激光器阵列处理后的混沌熵源输出经过图1所示后处理产生的二进制序列的特性. 正如前面提到的, 在给定仿真参数下, 由于没考虑增益饱和效应和器件或仪器带宽受限, 混沌信号的统计直方图服从近似的指数分布, 远离理想的高斯分布, 并不利于随机数的直接提取, 这里采用混沌熵源与其延迟特定时间后的混沌信号作差. 所得的混沌熵源的统计直方图如图4(a)所示, 其分布的两边存在较长的尾巴. 将激光器输出的混沌信号经过8位ADC后转变为8位二进制序列, 其中ADC的采样速率为20 GHz. 这里以8位二进制量化序列中提取3位LSB拼接为例, 其随机数生产速率可达60 Gb/s (采用高阶差分和拓展激光器阵列节点数目, 可轻松实现Tb/s量级速率的随机数产生), 所得序列的分布如图4(b)所示, 此时分布的均匀性有所改善, 但仍能看到一些量化比特位出现的概率比其他的高一些. 最后采用常规的XOR处理, 得到如图4(c)所示的近似理想的均匀分布, 适合直接用于获取随机数. 图 4 激光器B输出的混沌信号量化后的统计直方图 (a) 8位ADC输出; (b) 3-LSB输出; (c) XOR输出 Figure4. Statistical histogram of the quantized chaotic signal of the laser B: (a) The output of 8 bit ADC; (b) the output of 3-LSB; (c) the output of XOR.
表1NIST统计测试结果 Table1.Result of NIST statistical tests.
最后, 我们强调激光器阵列后处理光反馈激光器产生混沌信号的另一个优势, 即是其可同时获取多路相关或不相关的随机数序列. 通过选择参数, 激光器阵列中A、B节点的混沌输出均不具有时延特征. 如以kinj = 7 ns–1, Δf = –30 GHz为例, 结果如图6所示, 激光器阵列中节点A、B均实现高维混沌输出, 而且在ACF图中无时延特征峰值. 通过必要后处理, 很容易得到两路高品质的随机数序列. 其实激光器阵列的可扩展性强, 可以实现多节点甚至大型节点的激光器阵列, 因此本文结果可为实时产生多路并行的高速高品质的随机数序列提供思路. 图 6 激光器输出的时间序列与自相关函数 (a) A激光器输出的时间序列; (b) A激光器输出的自相关函数; (c) B激光器输出的时间序列; (d) B激光器输出的自相关函数 Figure6. Time series and autocorrelation function of the lasers: (a) Time series of laser A; (b) autocorrelation function of laser A; (c) time series of laser B; (d) autocorrelation function of laser. B.
图 1 基于激光器阵列后处理的混沌熵源获取高品质随机数的示意图(
图 2 激光器输出混沌信号的时间序列(左列), 自相关函数谱(中列), 功率谱(右列) (a) 光反馈半导体激光器; (b) 注入激光器; (c) 注入激光器阵列
图 3 经过激光器阵列后处理混沌熵源的ACF时延处峰值随着注入参数和激光器分离比d/a的演化情况 (a) d/a = 0.2; (b) d/a = 0.4; (c) d/a = 0.6; (d) d/a = 1.0
图 4 激光器B输出的混沌信号量化后的统计直方图 (a) 8位ADC输出; (b) 3-LSB输出; (c) XOR输出
图 5 散点图
图 6 激光器输出的时间序列与自相关函数 (a) A激光器输出的时间序列; (b) A激光器输出的自相关函数; (c) B激光器输出的时间序列; (d) B激光器输出的自相关函数