Abstract

The nondeterministic property of the optoelectronic random bit generator (RBG) based on laser chaos are experimentally analyzed from two aspects of the central limit theorem and law of iterated logarithm. The random bits are extracted from an optical feedback chaotic laser diode using a multi-bit extraction technique in the electrical domain. Our experimental results demonstrate that the generated random bits have no statistical distance from the Brownian motion, besides that they can pass the state-of-the-art industry-benchmark statistical test suite (NIST SP800-22). All of them give a mathematically provable evidence that the ultrafast random bit generator based on laser chaos can be used as a nondeterministic random bit source.

© 2016 Optical Society of America

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References

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  1. A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, and L. E. Bassham III, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” National Institute of Standards and Technology, Special Publication 800–22, Revision 1a, April 2010, NIST Statistical Tests Suite, [Online]. Available: http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html
  2. J. Walker, “Hotbits: Genuine random numbers generated by radioactive decay,” [Online]. Available: http://www.fourmilab. ch/hotbits .
  3. RANDOM.ORG, [Online]. Available: http://www.random.org/ .
  4. B. Jun and P. Kocher, “The Intel random number generator,” White Paper Prepared for Intel Corporation, Cryptography Research Inc., 1999, http://www.cryptography.com/resources/whitepapers/IntelRNG.pdf .
  5. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008).
    [Crossref]
  6. I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
    [Crossref] [PubMed]
  7. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010).
    [Crossref]
  8. A. Wang, P. Li, J. Zhang, J. Zhang, L. Li, and Y. Wang, “4.5 Gbps high-speed real-time physical random bit generator,” Opt. Express 21(17), 20452–20462 (2013).
    [Crossref] [PubMed]
  9. P. Li, Y. C. Wang, and J. Z. Zhang, “All-optical fast random number generator,” Opt. Express 18(19), 20360–20369 (2010).
    [Crossref] [PubMed]
  10. N. Oliver, M. C. Soriano, D. W. Sukow, and 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]
  11. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and 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]
  12. A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and 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]
  13. X. Z. Li and 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]
  14. N. Li, B. Kim, V. N. Chizhevsky, A. Locquet, M. Bloch, D. S. Citrin, and W. Pan, “Two approaches for ultrafast random bit generation based on the chaotic dynamics of a semiconductor laser,” Opt. Express 22(6), 6634–6646 (2014).
    [Crossref] [PubMed]
  15. X. Tang, Z. M. Wu, J. G. Wu, T. Deng, J. J. Chen, L. Fan, Z. Q. Zhong, and G. Q. Xia, “Tbits/s physical random bit generation based on mutually coupled semiconductor laser chaotic entropy source,” Opt. Express 23(26), 33130–33141 (2015).
    [Crossref] [PubMed]
  16. M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
    [Crossref]
  17. M. Virte, E. Mercier, H. Thienpont, K. Panajotov, and M. Sciamanna, “Physical random bit generation from chaotic solitary laser diode,” Opt. Express 22(14), 17271–17280 (2014).
    [Crossref] [PubMed]
  18. Diehard Test Suite, [Online]. Available: http://www.stat.fsu.edu/pub/diehard/ .
  19. J. Walker, Ent: A Pseudorandom Number Sequence Test Program [Online]. Available: http://www.fourmilab.ch/random/
  20. Y. Wang and T. Nicol, “Statistical properties of pseudo random sequences and experiments with PHP and Debian OpenSSL,” in Proc. ESORICS 2014, Wroclaw, Poland, Sept. 2014, pp. 454–471.
    [Crossref]
  21. Y. Wang, “Resource bounded randomness and computational complexity,” Theor. Comput. Sci. 237(1-2), 33–55 (2000).
    [Crossref]
  22. W. Feller, “The fundamental limit theorems in probability,” Bulletin of AMS 51(11), 800–833 (1945).
    [Crossref]
  23. P. Erdös and M. M. Kac, “On certain limit theorems of the theory of probability,” Bulletin of AMS 52(4), 292–303 (1946).
    [Crossref]
  24. A. Y. Khinchin, “On a theorem of probability calculation,” Fundam. Math. 6, 9–20 (1924).
  25. W. Feller, Introduction to Probability Theory and its Applications ( John Wiley & Sons Inc., 1968).
  26. J. A. Clarkson and C. R. Adams, “On definitions of bounded variation for functions of two variables,” Tran. AMS35, 824–854(1933).
    [Crossref]

2015 (2)

2014 (2)

2013 (3)

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

N. Oliver, M. C. Soriano, D. W. Sukow, and 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. Z. Li and 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]

2012 (1)

2010 (3)

2009 (1)

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

2008 (1)

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

2000 (1)

Y. Wang, “Resource bounded randomness and computational complexity,” Theor. Comput. Sci. 237(1-2), 33–55 (2000).
[Crossref]

1946 (1)

P. Erdös and M. M. Kac, “On certain limit theorems of the theory of probability,” Bulletin of AMS 52(4), 292–303 (1946).
[Crossref]

1945 (1)

W. Feller, “The fundamental limit theorems in probability,” Bulletin of AMS 51(11), 800–833 (1945).
[Crossref]

1924 (1)

A. Y. Khinchin, “On a theorem of probability calculation,” Fundam. Math. 6, 9–20 (1924).

Adams, C. R.

J. A. Clarkson and C. R. Adams, “On definitions of bounded variation for functions of two variables,” Tran. AMS35, 824–854(1933).
[Crossref]

Amano, K.

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

Argyris, A.

Aviad, Y.

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

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

Bloch, M.

Bogris, A.

Bolk, J.

Chan, S. C.

X. Z. Li and 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]

Chen, J. J.

Chizhevsky, V. N.

Citrin, D. S.

Clarkson, J. A.

J. A. Clarkson and C. R. Adams, “On definitions of bounded variation for functions of two variables,” Tran. AMS35, 824–854(1933).
[Crossref]

Cohen, E.

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

Danckaert, J.

Davis, P.

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

Deligiannidis, S.

Deng, T.

Erdös, P.

P. Erdös and M. M. Kac, “On certain limit theorems of the theory of probability,” Bulletin of AMS 52(4), 292–303 (1946).
[Crossref]

Fan, L.

Feller, W.

W. Feller, “The fundamental limit theorems in probability,” Bulletin of AMS 51(11), 800–833 (1945).
[Crossref]

Fischer, I.

N. Oliver, M. C. Soriano, D. W. Sukow, and 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]

Hirano, K.

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

Inoue, M.

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

Kac, M. M.

P. Erdös and M. M. Kac, “On certain limit theorems of the theory of probability,” Bulletin of AMS 52(4), 292–303 (1946).
[Crossref]

Kanter, I.

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

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

Khinchin, A. Y.

A. Y. Khinchin, “On a theorem of probability calculation,” Fundam. Math. 6, 9–20 (1924).

Kim, B.

Kurashige, T.

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

Leijtens, X.

Li, L.

Li, N.

Li, P.

Li, X. Z.

X. Z. Li and 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]

Locquet, A.

Mercier, E.

Naito, S.

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

Nguimdo, R. M.

Nicol, T.

Y. Wang and T. Nicol, “Statistical properties of pseudo random sequences and experiments with PHP and Debian OpenSSL,” in Proc. ESORICS 2014, Wroclaw, Poland, Sept. 2014, pp. 454–471.
[Crossref]

Oliver, N.

N. Oliver, M. C. Soriano, D. W. Sukow, and 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]

Oowada, I.

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

Pan, W.

Panajotov, K.

Pikasis, E.

Reidler, I.

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

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

Rosenbluh, M.

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

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

Sciamanna, M.

Shiki, M.

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

Shore, K. A.

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

Someya, H.

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

Soriano, M. C.

N. Oliver, M. C. Soriano, D. W. Sukow, and 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]

Sukow, D. W.

N. Oliver, M. C. Soriano, D. W. Sukow, and 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]

Syvridis, D.

Tang, X.

Thienpont, H.

Uchida, A.

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

Van der Sande, G.

Verschaffelt, G.

Virte, M.

Wang, A.

Wang, Y.

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

Y. Wang, “Resource bounded randomness and computational complexity,” Theor. Comput. Sci. 237(1-2), 33–55 (2000).
[Crossref]

Y. Wang and T. Nicol, “Statistical properties of pseudo random sequences and experiments with PHP and Debian OpenSSL,” in Proc. ESORICS 2014, Wroclaw, Poland, Sept. 2014, pp. 454–471.
[Crossref]

Wang, Y. C.

Wu, J. G.

Wu, Z. M.

Xia, G. Q.

Yoshimori, S.

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

Yoshimura, K.

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

Zhang, J.

Zhang, J. Z.

Zhong, Z. Q.

Bulletin of AMS (2)

W. Feller, “The fundamental limit theorems in probability,” Bulletin of AMS 51(11), 800–833 (1945).
[Crossref]

P. Erdös and M. M. Kac, “On certain limit theorems of the theory of probability,” Bulletin of AMS 52(4), 292–303 (1946).
[Crossref]

Fundam. Math. (1)

A. Y. Khinchin, “On a theorem of probability calculation,” Fundam. Math. 6, 9–20 (1924).

IEEE J. Quantum Electron. (2)

N. Oliver, M. C. Soriano, D. W. Sukow, and 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. Z. Li and 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]

Nat. Photonics (3)

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

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

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

Opt. Express (7)

A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and 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, and J. Z. Zhang, “All-optical fast random number generator,” Opt. Express 18(19), 20360–20369 (2010).
[Crossref] [PubMed]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and 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]

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

N. Li, B. Kim, V. N. Chizhevsky, A. Locquet, M. Bloch, D. S. Citrin, and W. Pan, “Two approaches for ultrafast random bit generation based on the chaotic dynamics of a semiconductor laser,” Opt. Express 22(6), 6634–6646 (2014).
[Crossref] [PubMed]

M. Virte, E. Mercier, H. Thienpont, K. Panajotov, and M. Sciamanna, “Physical random bit generation from chaotic solitary laser diode,” Opt. Express 22(14), 17271–17280 (2014).
[Crossref] [PubMed]

X. Tang, Z. M. Wu, J. G. Wu, T. Deng, J. J. Chen, L. Fan, Z. Q. Zhong, and G. Q. Xia, “Tbits/s physical random bit generation based on mutually coupled semiconductor laser chaotic entropy source,” Opt. Express 23(26), 33130–33141 (2015).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

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

Theor. Comput. Sci. (1)

Y. Wang, “Resource bounded randomness and computational complexity,” Theor. Comput. Sci. 237(1-2), 33–55 (2000).
[Crossref]

Other (9)

W. Feller, Introduction to Probability Theory and its Applications ( John Wiley & Sons Inc., 1968).

J. A. Clarkson and C. R. Adams, “On definitions of bounded variation for functions of two variables,” Tran. AMS35, 824–854(1933).
[Crossref]

Diehard Test Suite, [Online]. Available: http://www.stat.fsu.edu/pub/diehard/ .

J. Walker, Ent: A Pseudorandom Number Sequence Test Program [Online]. Available: http://www.fourmilab.ch/random/

Y. Wang and T. Nicol, “Statistical properties of pseudo random sequences and experiments with PHP and Debian OpenSSL,” in Proc. ESORICS 2014, Wroclaw, Poland, Sept. 2014, pp. 454–471.
[Crossref]

A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, and L. E. Bassham III, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” National Institute of Standards and Technology, Special Publication 800–22, Revision 1a, April 2010, NIST Statistical Tests Suite, [Online]. Available: http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html

J. Walker, “Hotbits: Genuine random numbers generated by radioactive decay,” [Online]. Available: http://www.fourmilab. ch/hotbits .

RANDOM.ORG, [Online]. Available: http://www.random.org/ .

B. Jun and P. Kocher, “The Intel random number generator,” White Paper Prepared for Intel Corporation, Cryptography Research Inc., 1999, http://www.cryptography.com/resources/whitepapers/IntelRNG.pdf .

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Figures (13)

Fig. 1
Fig. 1 Experimental setup. DFB-LD, distributed feedback laser diode; PC, polarization controller; OC, optical coupler; VOA, variable optical attenuator; M, fiber mirror; EDFA, erbium-doped fiber amplifier; PD, photodetector; 8-bit ADC, 8-bit analog-to-digital converter; LSBs, least significant bits.
Fig. 2
Fig. 2 (a) Measured temporal waveform, (b) RF spectrum and (c) AC function of the chaotic signal.
Fig. 3
Fig. 3 Random bit extraction procedure from the chaotic signal. Blue dots denote the sample points.
Fig. 4
Fig. 4 (a) Bias of the generated 80 Gb/s random bit sequence with different lengths n = 1, …, 16 Mbits, where the solid line is its three-standard-deviation line, 3σbias = (3n-1/2)/2. (b) AC coefficients as a function of the delay bit for the random bit sequence with a length n = 16 Mbits where the solid line is its three-standard-deviation line, 3σAC = 3n-1/2.
Fig. 5
Fig. 5 Typical results of NIST statistical tests. Using 1000 samples of 1-Mb data and significance level D = 0.01, for “Success,” the P-value (uniformity of p-values) should be larger than 0.0001 and the proportion should be greater than 0.9805608. The numbers on the horizontal axis represent 15 different statistical tests in the NIST test suite, which are named as ‘Frequency’, ‘Block frequency’, ‘Cumulative sums’, ‘Runs’, ‘Longest-run’, ‘Rank’, ‘FFT’, ‘Non-periodic templates’, ‘Overlapping templates’, ‘Universal’, ‘Approximate entropy’, ‘Random excursions’, ‘Random excursions variant’, ‘Serial’ and ‘Linear Complexity’, respectively.
Fig. 6
Fig. 6 Density functions for distributions μnU with n = 219, …, 225.
Fig. 7
Fig. 7 Density functions for distributions μnRlaser for n = 219, …, 225 with 6000 bit strings from our generator
Fig. 8
Fig. 8 LIL plot for the laser chaos based RBG with 6000×17 MB strings
Fig. 9
Fig. 9 Normalized distributions for the decimal quantization values (in integer representation) generated by retaining m-LSBs for cases: (a) m = 1, (b) m = 2, (c) m = 3, (d) m = 4, (e) m = 5, (f) m = 6, (g) m = 7 and (h) m = 8.
Fig. 10
Fig. 10 AC functions of the digitized binary sequences from (a) interleaved m-LSBs and (b) independent m-th LSB: (a-i) 1-LSBs, (a-ii) 2-LSBs, (a-iii) 3-LSBs, (b-i) 1-st LSB, (b-ii) 2-nd LSB and (b-iii) 3-rd LSB. The insets in (a-iii) and (b-iii) are the magnified ranges corresponding to the TD signature of the chaotic dynamics.
Fig. 11
Fig. 11 Simulated RF spectrum and AC function for two different time-delays of chaotic signals. (a-i) and (a-ii) correspond to the RF spectrum and AC function for the chaotic signal with a time-delay of τ = 100.000 ns, whereas (b-i) and (b-ii) are the counterparts for the other chaotic signal with the same incommensurable time-delay of τ = 105.453 ns as the chaotic laser in experiment.
Fig. 12
Fig. 12 AC functions of the independent m-th LSB binary sequences from the two simulated chaotic signal with different TD signature levels, where m = 1, 2, 3, 4, 5, 6, 7 and 8. Note, the first and second columns correspond to the chaotic signal with a high TD signature level of 0.140 in Fig. 8(a), while the third and fourth columns correspond to the other chaotic signal with a low TD signature level of 0.098 in Fig. 8(b). Inset numbers from i to viii in the plot corresponds to m value from 1 to 8 in order.
Fig. 13
Fig. 13 Extracted TD signature levels from Fig. 12 as a function of m-th LSB, where m = 1, 2, 3, 4, 5, 6, 7 and 8. Note, the dots and squares corresponds to the TD signature levels in Fig. 12(a) and Fig. 12(b), respectively.

Equations (4)

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S(x)= i=0 | x |1 x[ i ] and S (x)= 2S(x)| x | | x |
S lil ( ξn )= 2 i=0 n1 ξ[ i ]n 2nlnlnn
dE(t) dt = 1+iα 2 [ G n [N(t) N 0 ] 1+ε | E(t) | 2 1 τ p ]E(t)+ κ f E(tτ)exp(iwτ)
dN(t) dt = J e N(t) τ e G n [N(t) N 0 ] 1+ε | E(t) | 2 | E(t) | 2

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