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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. [1] Shannon C E 1949 Bell Syst. Tech. J. 28 656
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[6] David P. Rosin, Damien Rontani, Daniel J. Gauthier 2013 Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87 040902
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Guo H, Liu Y, Dang A H, Wei W 2009 Chin. Sci. Bull. 54 3651
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[9] Ren M, Wu E, Liang Y, Jian Y, Wu G, Zeng H 2011 Phys. Rev. A 83 023820
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Zhou Q, Hu Y, Liao X F 2008 Acta Phys. Sin. 57 5413
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[11] 李念强 2016 博士学位论文 (成都: 西南交通大学)
Li N Q 2016 Ph. D. Dissertation (Chengdu: Southwest Jiaotong University) (in Chinese)
[12] Uchida A, Amano K, Inoue M, Hirano K, Naito S, Someya H, Oowada I, Kurashige T, Shiki M, Yoshimori S, Yoshimura K, Davis P 2008 Nat. Photon. 2 728
Google Scholar
[13] Reidier I, Aviad Y, Rosenblush M, Kanter I 2009 Phys. Rev. Lett. 103 024102
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[14] Kanter I, Aviad Y, Reidler I, Cohen E, Rosenbluh M 2010 Nat. Photon. 4 58
Google Scholar
[15] Harayama T, Sunada S, Yoshimura K, Davis P, Tsuzuki K, Uchida A 2011 Phys. Rev. A. 83 031803
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[16] Argyris A, Deligiannidis S, Pikasis E, Bogris A, Syvridis D 2010 Opt. Express 18 18763
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[17] Zhang J Z, Wang Y C, Liu M, Xue L G, Li P, Wang A B, Zhang M J 2012 Opt. Express 20 7496
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[22] Li P, Zhang J G, Sang L X, Liu X L, Guo Y Q, Guo X M, Wang A B, Shore K A, Wang Y C 2017 Opt. Lett. 42 2699
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Han T, Liu X L, Li P, Guo X M, Guo Y Q, Wang Y C 2017 Acta Phys. Sin. 66 124203
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Zhao D L, Li P, Liu X L, Guo X M, Guo Y Q, Zhang J G, Wang Y C 2017 Acta Phys. Sin. 66 050501
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[26] Tang X, Wu Z M, Wu J G, Deng T, Zhong Z Q, Chen J J, Xia G Q 2014 Laser Phys. Lett. 12 015003
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[30] Mu P H, Pan W, Xiang S Y, Li N Q, Liu X K, Zou X H 2015 Mod. Phys. Lett. B 29 1550142
[31] Wang Y, Xiang S Y, Wang B, Cao X Y, Wen A J, Hao Y 2019 Opt. Express 27 8446
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[32] Xiang S Y, Wang B, Wang Y, Han Y N, Wen A J, Hao Y 2019 J. Light. Technol. 37 3987
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[35] Li N Q, Pan W, Locquet A, Citrin D S 2015 Opt. Lett. 40 4416
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[36] Li S S, Li X Z, Chan S C 2018 Opt. Lett. 43 4751
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[37] Xiang S Y, Wen A J, Pan W, Lin L, Zhang H X, Zhang H, Guo X X, Li J F 2016 J. Light. Technol. 34 4221
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[38] Jiang N, Wang C, Xue C P, Li G L, Lin S Q, Qiu K 2017 Opt. Express 25 14359
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[39] Jiang X X, Liu D M, Cheng M F, Deng L, Fu S N, Zhang M M, Tang M, Shum P 2016 Opt. Lett. 41 1157
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[40] Ma Y T, Xiang S Y, Guo X X, Song Z W, Wen A J, Hao Y 2020 Opt. Express 28 1665
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[41] Zhou P, Fang Q, Li N Q 2020 Opt. Lett. 45 399
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[42] Rukhin A, Soto J, Nechvatal J, Smid M, Barker E, Leigh S, Levenson M, Vangel M, Banks D, Heckert A, Dary J, Vo S 2001 http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html [2020-11-21]
[43] Adams M J, Li N Q, Cemlyn B R, Susanto H, Henning I D 2017 Phys. Rev. A 95 053869
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[44] Fang Q, Zhou P, Mu P H, Li N Q 2021 IEEE J. Quantum Electron. 57 1200109.
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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(
$ \lambda /4 $ 为$ 1/4 $ 波片, PD1、PD2为光电转换器, ADC为模数转换器, LSB为最低有效位, XOR为异或处理)Figure 1. Schematic diagram of high quality random number generation based on the chaotic entropy source generated by ECSL and post-processed by phased-array semiconductor lasers (λ/4, 1/4 wave plate; PD1 and PD2, photo detector; ADC, analog-to-digital converter; LSB, least significant bit; XOR, exclusive OR).
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
Figure 2. 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.
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
Figure 3. 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.
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
Figure 4. 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.
图 5 散点图
Figure 5. Scatter diagram.
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
Figure 6. 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.
表 1 NIST统计测试结果
Table 1. Result of NIST statistical tests.
测试名称 P-value 概率 结果 频数 0.538182 0.992 通过 块内频数 0.239266 0.982 通过 累加 0.755819 0.994 通过 游程 0.140453 0.988 通过 块内最长游程 0.965860 0.988 通过 矩阵秩 0.281232 0.990 通过 离散傅里叶变换 0.206629 0.982 通过 非重叠模块匹配 0.020831 0.982 通过 重叠模块匹配 0.699313 0.984 通过 通用统计 0.510153 0.994 通过 近似熵 0.699313 0.994 通过 随机游动 0.443665 0.986 通过 随机游动变量 0.290158 0.983 通过 连续性 0.096578 0.984 通过 线性复杂度 0.340858 0.986 通过 
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[1] Shannon C E 1949 Bell Syst. Tech. J. 28 656
Google Scholar
[2] Durt T, Beimonte C, Lamoureux L P, Panajotov K, van den Berghe F, Thienpont H 2013 Phys. Rev. A. 87 022339
Google Scholar
[3] Williams C R S, Salevan J C, Li X W, Roy R, Murphy T E 2010 Opt. Express 18 23584
Google Scholar
[4] Guo H, Tang W Z, Liu Y, Wei W 2010 Phys. Rev. E. 81 051137
Google Scholar
[5] Ma X F, Xu F H, Xu H, Tan X Q, Qi B, Lo H K 2013 Phys. Rev. A. 87 062327
Google Scholar
[6] David P. Rosin, Damien Rontani, Daniel J. Gauthier 2013 Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87 040902
Google Scholar
[7] Li N Q, Kim B, Chizhevsky V N, Locquet A, Bloch M, Citrin D S, Pan W 2014 Opt. Express 22 6634
Google Scholar
[8] 郭弘, 刘钰, 党安红, 韦韦 2009 科学通报 54 3651
Google Scholar
Guo H, Liu Y, Dang A H, Wei W 2009 Chin. Sci. Bull. 54 3651
Google Scholar
[9] Ren M, Wu E, Liang Y, Jian Y, Wu G, Zeng H 2011 Phys. Rev. A 83 023820
Google Scholar
[10] 周庆, 胡月, 廖晓峰 2008 57 5413
Google Scholar
Zhou Q, Hu Y, Liao X F 2008 Acta Phys. Sin. 57 5413
Google Scholar
[11] 李念强 2016 博士学位论文 (成都: 西南交通大学)
Li N Q 2016 Ph. D. Dissertation (Chengdu: Southwest Jiaotong University) (in Chinese)
[12] Uchida A, Amano K, Inoue M, Hirano K, Naito S, Someya H, Oowada I, Kurashige T, Shiki M, Yoshimori S, Yoshimura K, Davis P 2008 Nat. Photon. 2 728
Google Scholar
[13] Reidier I, Aviad Y, Rosenblush M, Kanter I 2009 Phys. Rev. Lett. 103 024102
Google Scholar
[14] Kanter I, Aviad Y, Reidler I, Cohen E, Rosenbluh M 2010 Nat. Photon. 4 58
Google Scholar
[15] Harayama T, Sunada S, Yoshimura K, Davis P, Tsuzuki K, Uchida A 2011 Phys. Rev. A. 83 031803
Google Scholar
[16] Argyris A, Deligiannidis S, Pikasis E, Bogris A, Syvridis D 2010 Opt. Express 18 18763
Google Scholar
[17] Zhang J Z, Wang Y C, Liu M, Xue L G, Li P, Wang A B, Zhang M J 2012 Opt. Express 20 7496
Google Scholar
[18] Li P, Wang Y C, Zhang J Z 2010 Opt. Express 18 20360
Google Scholar
[19] Wu J G, Tang X, Wu Z M, Xia G Q, Feng G Y 2012 Laser Phys. 22 1476
Google Scholar
[20] Li X Z, Chan S C 2013 IEEE J. Quantum Electron. 49 829
Google Scholar
[21] Wang A B, Li P, Zhang J G, Zhang J Z, Li L, Wang Y C 2013 Opt. Express 21 20452
Google Scholar
[22] Li P, Zhang J G, Sang L X, Liu X L, Guo Y Q, Guo X M, Wang A B, Shore K A, Wang Y C 2017 Opt. Lett. 42 2699
Google Scholar
[23] Li P, Sun Y Y, Liu X L, Yi X G, Zhang J G, Guo X M, Guo Y Q, Wang Y C 2016 Opt. Lett. 41 3347
Google Scholar
[24] 韩韬, 刘香莲, 李璞, 郭晓敏, 郭龑强, 王云才 2017 66 124203
Google Scholar
Han T, Liu X L, Li P, Guo X M, Guo Y Q, Wang Y C 2017 Acta Phys. Sin. 66 124203
Google Scholar
[25] 赵东亮, 李璞, 刘香莲, 郭晓敏, 郭龑强, 张建国, 王云才 2017 66 050501
Google Scholar
Zhao D L, Li P, Liu X L, Guo X M, Guo Y Q, Zhang J G, Wang Y C 2017 Acta Phys. Sin. 66 050501
Google Scholar
[26] Tang X, Wu Z M, Wu J G, Deng T, Zhong Z Q, Chen J J, Xia G Q 2014 Laser Phys. Lett. 12 015003
Google Scholar
[27] Ran C, Tang X, Wu Z M, Xia G Q 2018 Laser Phys. 28 126202
Google Scholar
[28] 姚晓洁, 唐曦, 吴正茂, 夏光琼 2018 67 024204
Google Scholar
Yao X J, Tang X, Wu Z M, Xia G Q 2018 Acta Phys. Sin. 67 024204
Google Scholar
[29] Li N Q, Pan W, Xiang S Y, Zhao Q C, Zhang L Y 2014 IEEE Photon. Technol. Lett. 26 1886
Google Scholar
[30] Mu P H, Pan W, Xiang S Y, Li N Q, Liu X K, Zou X H 2015 Mod. Phys. Lett. B 29 1550142
[31] Wang Y, Xiang S Y, Wang B, Cao X Y, Wen A J, Hao Y 2019 Opt. Express 27 8446
Google Scholar
[32] Xiang S Y, Wang B, Wang Y, Han Y N, Wen A J, Hao Y 2019 J. Light. Technol. 37 3987
Google Scholar
[33] Xue C P, Jiang N, Qiu K, Lv Y X 2015 Opt. Express 23 14510
Google Scholar
[34] Zhao A K, Jiang N, Wang Y J, Liu S Q, Li B C, Qiu K 2019 Opt. Lett. 44 5957
Google Scholar
[35] Li N Q, Pan W, Locquet A, Citrin D S 2015 Opt. Lett. 40 4416
Google Scholar
[36] Li S S, Li X Z, Chan S C 2018 Opt. Lett. 43 4751
Google Scholar
[37] Xiang S Y, Wen A J, Pan W, Lin L, Zhang H X, Zhang H, Guo X X, Li J F 2016 J. Light. Technol. 34 4221
Google Scholar
[38] Jiang N, Wang C, Xue C P, Li G L, Lin S Q, Qiu K 2017 Opt. Express 25 14359
Google Scholar
[39] Jiang X X, Liu D M, Cheng M F, Deng L, Fu S N, Zhang M M, Tang M, Shum P 2016 Opt. Lett. 41 1157
Google Scholar
[40] Ma Y T, Xiang S Y, Guo X X, Song Z W, Wen A J, Hao Y 2020 Opt. Express 28 1665
Google Scholar
[41] Zhou P, Fang Q, Li N Q 2020 Opt. Lett. 45 399
Google Scholar
[42] Rukhin A, Soto J, Nechvatal J, Smid M, Barker E, Leigh S, Levenson M, Vangel M, Banks D, Heckert A, Dary J, Vo S 2001 http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html [2020-11-21]
[43] Adams M J, Li N Q, Cemlyn B R, Susanto H, Henning I D 2017 Phys. Rev. A 95 053869
Google Scholar
[44] Fang Q, Zhou P, Mu P H, Li N Q 2021 IEEE J. Quantum Electron. 57 1200109.
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