Abstract

This paper reports the experimental investigation of two different approaches to random bit generation based on the chaotic dynamics of a semiconductor laser with optical feedback. By computing high-order finite differences of the chaotic laser intensity time series, we obtain time series with symmetric statistical distributions that are more conducive to ultrafast random bit generation. The first approach is guided by information-theoretic considerations and could potentially reach random bit generation rates as high as 160 Gb/s by extracting 4 bits per sample. The second approach is based on pragmatic considerations and could lead to rates of 2.2 Tb/s by extracting 55 bits per sample. The randomness of the bit sequences obtained from the two approaches is tested against three standard randomness tests (ENT, Diehard, and NIST tests), as well as by calculating the statistical bias and the serial correlation coefficients on longer sequences of random bits than those used in the standard tests.

© 2014 Optical Society of America

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2013

M. C. Soriano, J. Garcia-Ojalvo, C. R. Mirasso, I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductor lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[CrossRef]

T. Durt, C. Belmont, L. P. Lamoureux, K. Panajotov, F. Van den Berghe, H. Thienpont, “Fast quantum-optical random-number generators,” Phys. Rev. A 87(2), 022339 (2013).
[CrossRef]

D. P. Rosin, D. Rontani, D. J. Gauthier, “Ultrafast physical generation of random numbers using hybrid Boolean networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(4), 040902 (2013).
[CrossRef] [PubMed]

X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, H. Lo, “Postprocessing for quantum random-numbers generators: Entropy evaluation and randomness extraction,” Phys. Rev. A 87(6), 062327 (2013).
[CrossRef]

N. Oliver, M. C. Soriano, D. W. Sukow, I. Fischer, “Fast random bit generation using a chaotic laser: approaching the information theoretic limit,” IEEE J. Quantum Electron. 49(11), 910–918 (2013).
[CrossRef]

X. Fang, B. Wetzel, J. Merolla, J. M. Dudley, L. Larger, C. Guyeux, J. M. Bahi, “Noise and chaos contributions in fast random bit sequence generated from broadband optoelectronic entropy sources,” IEEE Trans. Circuits Syst. I 99, 1–14 (2013).

X. Z. Li, S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. 49(10), 829–838 (2013).
[CrossRef]

T. Yamazaki, A. Uchida, “Performance of random number generators using noise-based superluminescent diode and chaos-based semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 19(4), 0600309 (2013).
[CrossRef]

A. B. Wang, P. Li, J. G. Zhang, J. Z. Zhang, L. Li, Y. C. Wang, “4.5 Gbps high-speed real-time physical random bit generator,” Opt. Express 21(17), 20452–20462 (2013).
[CrossRef] [PubMed]

S. Priyadarshi, Y. Hong, I. Pierce, K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external-cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013).
[CrossRef]

2012

J. Z. Zhang, Y. C. Wang, L. G. Xue, J. Y. Hou, B. B. Zhang, A. B. Wang, M. J. Zhang, “Delay line length selection in generating fast random numbers with a chaotic laser,” Appl. Opt. 51(11), 1709–1714 (2012).
[CrossRef] [PubMed]

Y. Akizawa, T. Yamazaki, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, P. Davis, “Fast random number generation with bandwidth-enhanced chaotic semiconductor lasers at8×50Gb/s, ” IEEE Photon. Technol. Lett. 24(12), 1042–1044 (2012).
[CrossRef]

J. G. Wu, X. Tang, Z. M. Wu, G. Q. Xia, G. Y. Feng, “Parallel generation of 10 Gbits/s physical random number streams using chaotic semiconductor lasers,” Laser Phys. 22(10), 1476–1480 (2012).
[CrossRef]

X. Z. Li, S. C. Chan, “Random bit generation using an optically injected semiconductor laser in chaos with oversampling,” Opt. Lett. 37(11), 2163–2165 (2012).
[CrossRef] [PubMed]

T. Harayama, S. Sunada, K. Yoshimura, J. Muramatsu, K. Arai, A. Uchida, P. Davis, “Theory of fast nondeterministic physical random-bit generation with chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(4), 046215 (2012).
[CrossRef] [PubMed]

T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(1 Pt 2), 016211 (2012).
[CrossRef] [PubMed]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, G. Van der Sande, “Fast random bits generation based on a single chaotic semiconductor ring laser,” Opt. Express 20(27), 28603–28613 (2012).
[CrossRef] [PubMed]

J. Z. Zhang, Y. C. Wang, M. Liu, L. G. Xue, P. Li, A. B. Wang, M. J. Zhang, “A robust random number generator based on differential comparison of chaotic laser signals,” Opt. Express 20(7), 7496–7506 (2012).
[CrossRef] [PubMed]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20(3), 3241–3249 (2012).
[CrossRef] [PubMed]

2011

R. M. Nguimdo, P. Colet, L. Larger, L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. 107(3), 034103 (2011).
[CrossRef] [PubMed]

X. Li, A. B. Cohen, T. E. Murphy, R. Roy, “Scalable parallel physical random number generator based on a superluminescent LED,” Opt. Lett. 36(6), 1020–1022 (2011).
[CrossRef] [PubMed]

V. N. Chizhevsky, “Fast generation of random bits based on polarization noises in a semiconductor vertical-cavity laser,” Opt. Spectrosc. 111(5), 689–694 (2011).
[CrossRef]

T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A 83(3), 031803 (2011).
[CrossRef]

N. Oliver, M. C. Soriano, D. W. Sukow, I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36(23), 4632–4634 (2011).
[CrossRef] [PubMed]

2010

K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18(6), 5512–5524 (2010).
[CrossRef] [PubMed]

A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express 18(18), 18763–18768 (2010).
[CrossRef] [PubMed]

P. Li, Y. C. Wang, J. Z. Zhang, “All-optical fast random number generator,” Opt. Express 18(19), 20360–20369 (2010).
[CrossRef] [PubMed]

C. R. S. Williams, J. C. Salevan, X. W. Li, R. Roy, T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express 18(23), 23584–23597 (2010).
[CrossRef] [PubMed]

H. Guo, W. Tang, Y. Liu, W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81(5), 051137 (2010).
[CrossRef] [PubMed]

B. Qi, Y.-M. Chi, H.-K. Lo, L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. Lett. 35(3), 312–314 (2010).
[CrossRef] [PubMed]

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, M. Rosenbluh, “An optical ultrafast random number generator,” Nat. Photonics 4(1), 58–61 (2010).
[CrossRef]

M. A. Wayne, P. G. Kwiat, “Low-bias high-speed quantum random number generator via shaped optical pulses,” Opt. Express 18(9), 9351–9357 (2010).
[CrossRef] [PubMed]

V. N. Chizhevsky, “Symmetrization of single-sided or nonsymmetrical distributions: The way to enhance a generation rate of random bits from a physical source of randomness,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(5), 050101 (2010).
[CrossRef] [PubMed]

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, M. Rosenbluh, “Towards the generation of random bits at terahertz rates based on chaotic semiconductor laser,” J. Phys. Conf. Ser. 233, 012002 (2010).
[CrossRef]

2009

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009).
[CrossRef]

I. Reidler, Y. Aviad, M. Rosenbluh, I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[CrossRef] [PubMed]

T. Honjo, A. Uchida, K. Amano, K. Hirano, H. Someya, H. Okumura, K. Yoshimura, P. Davis, Y. Tokura, “Differential-phase-shift quantum key distribution experiment using fast physical random bit generator with chaotic semiconductor lasers,” Opt. Express 17(11), 9053–9061 (2009).
[CrossRef] [PubMed]

2008

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[CrossRef]

T. E. Murphy, R. Roy, “Chaotic lasers: The world’s fastest dice,” Nat. Photonics 2(12), 714–715 (2008).
[CrossRef]

2007

2005

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[CrossRef] [PubMed]

2004

F. Y. Lin, J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004).
[CrossRef]

2002

N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002).
[CrossRef]

J. T. Gleeson, “Truly random number generator based on turbulent electroconvection,” Appl. Phys. Lett. 81(11), 1949 (2002).
[CrossRef]

2001

T. Stojanovski, J. Pihl, L. Kocarev, “Chaos-based random number generators–Part II: practical realization,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48(3), 382–385 (2001).
[CrossRef]

1998

C. Masoller, N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external cavity semiconductor laser,” Phys. Rev. A 57(2), 1313–1322 (1998).
[CrossRef]

1997

W. T. Holman, J. A. Connelly, A. B. Dowlatabadi, “An integrated analog/digital random noise source,” IEEE Trans. Circuits Syst. I 44(6), 521–528 (1997).
[CrossRef]

1990

J. Mørk, J. Mark, B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65(16), 1999–2002 (1990).
[CrossRef] [PubMed]

1948

C. E. Shannon, “A mathematical theory of communication,” Bell System Technical Journal 27, 379–423 and 623–656 (1948).

Abraham, N. B.

C. Masoller, N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external cavity semiconductor laser,” Phys. Rev. A 57(2), 1313–1322 (1998).
[CrossRef]

Aida, H.

Akizawa, Y.

Y. Akizawa, T. Yamazaki, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, P. Davis, “Fast random number generation with bandwidth-enhanced chaotic semiconductor lasers at8×50Gb/s, ” IEEE Photon. Technol. Lett. 24(12), 1042–1044 (2012).
[CrossRef]

Amano, K.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009).
[CrossRef]

T. Honjo, A. Uchida, K. Amano, K. Hirano, H. Someya, H. Okumura, K. Yoshimura, P. Davis, Y. Tokura, “Differential-phase-shift quantum key distribution experiment using fast physical random bit generator with chaotic semiconductor lasers,” Opt. Express 17(11), 9053–9061 (2009).
[CrossRef] [PubMed]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
[CrossRef]

Annovazzi-Lodi, V.

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Arai, K.

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T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(1 Pt 2), 016211 (2012).
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T. Durt, C. Belmont, L. P. Lamoureux, K. Panajotov, F. Van den Berghe, H. Thienpont, “Fast quantum-optical random-number generators,” Phys. Rev. A 87(2), 022339 (2013).
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Cohen, A. B.

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I. Kanter, Y. Aviad, I. Reidler, E. Cohen, M. Rosenbluh, “An optical ultrafast random number generator,” Nat. Photonics 4(1), 58–61 (2010).
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I. Kanter, Y. Aviad, I. Reidler, E. Cohen, M. Rosenbluh, “Towards the generation of random bits at terahertz rates based on chaotic semiconductor laser,” J. Phys. Conf. Ser. 233, 012002 (2010).
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R. M. Nguimdo, P. Colet, L. Larger, L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. 107(3), 034103 (2011).
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Davis, P.

Y. Akizawa, T. Yamazaki, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, P. Davis, “Fast random number generation with bandwidth-enhanced chaotic semiconductor lasers at8×50Gb/s, ” IEEE Photon. Technol. Lett. 24(12), 1042–1044 (2012).
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T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(1 Pt 2), 016211 (2012).
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T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A 83(3), 031803 (2011).
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K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18(6), 5512–5524 (2010).
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Dowlatabadi, A. B.

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T. Durt, C. Belmont, L. P. Lamoureux, K. Panajotov, F. Van den Berghe, H. Thienpont, “Fast quantum-optical random-number generators,” Phys. Rev. A 87(2), 022339 (2013).
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X. Fang, B. Wetzel, J. Merolla, J. M. Dudley, L. Larger, C. Guyeux, J. M. Bahi, “Noise and chaos contributions in fast random bit sequence generated from broadband optoelectronic entropy sources,” IEEE Trans. Circuits Syst. I 99, 1–14 (2013).

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N. Oliver, M. C. Soriano, D. W. Sukow, I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36(23), 4632–4634 (2011).
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M. C. Soriano, J. Garcia-Ojalvo, C. R. Mirasso, I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductor lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
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A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
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Guyeux, C.

X. Fang, B. Wetzel, J. Merolla, J. M. Dudley, L. Larger, C. Guyeux, J. M. Bahi, “Noise and chaos contributions in fast random bit sequence generated from broadband optoelectronic entropy sources,” IEEE Trans. Circuits Syst. I 99, 1–14 (2013).

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T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(1 Pt 2), 016211 (2012).
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Y. Akizawa, T. Yamazaki, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, P. Davis, “Fast random number generation with bandwidth-enhanced chaotic semiconductor lasers at8×50Gb/s, ” IEEE Photon. Technol. Lett. 24(12), 1042–1044 (2012).
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T. Harayama, S. Sunada, K. Yoshimura, J. Muramatsu, K. Arai, A. Uchida, P. Davis, “Theory of fast nondeterministic physical random-bit generation with chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(4), 046215 (2012).
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K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18(6), 5512–5524 (2010).
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K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18(6), 5512–5524 (2010).
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K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009).
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T. Honjo, A. Uchida, K. Amano, K. Hirano, H. Someya, H. Okumura, K. Yoshimura, P. Davis, Y. Tokura, “Differential-phase-shift quantum key distribution experiment using fast physical random bit generator with chaotic semiconductor lasers,” Opt. Express 17(11), 9053–9061 (2009).
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A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
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W. T. Holman, J. A. Connelly, A. B. Dowlatabadi, “An integrated analog/digital random noise source,” IEEE Trans. Circuits Syst. I 44(6), 521–528 (1997).
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K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009).
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A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
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T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(1 Pt 2), 016211 (2012).
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Lamoureux, L. P.

T. Durt, C. Belmont, L. P. Lamoureux, K. Panajotov, F. Van den Berghe, H. Thienpont, “Fast quantum-optical random-number generators,” Phys. Rev. A 87(2), 022339 (2013).
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X. Fang, B. Wetzel, J. Merolla, J. M. Dudley, L. Larger, C. Guyeux, J. M. Bahi, “Noise and chaos contributions in fast random bit sequence generated from broadband optoelectronic entropy sources,” IEEE Trans. Circuits Syst. I 99, 1–14 (2013).

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20(3), 3241–3249 (2012).
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Li, L.

Li, P.

Li, X.

Li, X. W.

Li, X. Z.

X. Z. Li, S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. 49(10), 829–838 (2013).
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X. Z. Li, S. C. Chan, “Random bit generation using an optically injected semiconductor laser in chaos with oversampling,” Opt. Lett. 37(11), 2163–2165 (2012).
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X. Fang, B. Wetzel, J. Merolla, J. M. Dudley, L. Larger, C. Guyeux, J. M. Bahi, “Noise and chaos contributions in fast random bit sequence generated from broadband optoelectronic entropy sources,” IEEE Trans. Circuits Syst. I 99, 1–14 (2013).

Mikami, T.

T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(1 Pt 2), 016211 (2012).
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M. C. Soriano, J. Garcia-Ojalvo, C. R. Mirasso, I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductor lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013).
[CrossRef]

L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20(3), 3241–3249 (2012).
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Naito, S.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009).
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Okumura, H.

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N. Oliver, M. C. Soriano, D. W. Sukow, I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36(23), 4632–4634 (2011).
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Oowada, I.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
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T. Durt, C. Belmont, L. P. Lamoureux, K. Panajotov, F. Van den Berghe, H. Thienpont, “Fast quantum-optical random-number generators,” Phys. Rev. A 87(2), 022339 (2013).
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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of ultrafast RBG based on optical chaos. Laser: a distributed feedback (DFB) laser diode; OC: optical circulator; VA, variable optical attenuator; FC, 85: 15 fiber coupler; PD, photodiode; OSC, 40 GHz real-time oscilloscope; HFD, high-order finite differences; LSBs, least significant bits.

Fig. 2
Fig. 2

Statistical properties of laser chaos. (a) PDF and (b) autocorrelation function of the chaotic waveforms. The blue curve in (a) denotes the fitted Gaussian.

Fig. 3
Fig. 3

PDFs over the range of digitization levels for (a) 7 LSBs, (b) 6 LSBs, (c) 5 LSBs, and (d) 4 LSBs retained from each 8-bit sample.

Fig. 4
Fig. 4

(a) Characteristics of the calculated floating-point numbers based on 50th-order finite differences. (a) Time series and (b) its associated statistical distribution. The dashed lines stand for the mean value of the time series. The blue line in (b) represents a fitted perfect Gaussian distribution.

Fig. 5
Fig. 5

The results of the statistical bias B and serial autocorrelation coefficient C k for the physical RBG at a speed of 160 Gb/s. (a) B as a function of the number of generated bits N; (b) first 200 serial autocorrelation coefficients for the binary sequence of 10 Gbit length. The values of C k were calculated by ensemble averaging over 1× 10 4 sequences of 1 Mbit length each.

Fig. 6
Fig. 6

The results of the statistical bias B and serial autocorrelation coefficient C k for the physical-based pseudo RBG at a speed of 2.2 Tb/s. (a) B as a function of the number of generated bits N; (b) first 200 serial autocorrelation coefficients for the binary sequence of 50 Gbit length. The values of C k were calculated by ensemble averaging over 5× 10 4 sequences of 1 Mbit length each.

Tables (2)

Tables Icon

Table 1 Results of NIST statistical tests for physical random bits. The results have been performed using 1000 samples of 1 Mbit data and a significance level α=0.01 , for “Success”, the P value (uniformity of p values) should be larger than 0.0001 and the proportion should be in the 0.99±0.0094392 range [56]. For the tests that produce multiple P values and proportions, the worst case is shown.

Tables Icon

Table 2 Results of NIST statistical tests for ultrafast physical-based pseudo random bits.

Equations (2)

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B= b i 0.5,
C k = ( b i b i )( b i+k b i ) ( b i b i ) 2 ,

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