Abstract

A quantum random number generator (QRNG) can generate true randomness by exploiting the fundamental indeterminism of quantum mechanics. Most approaches to QRNG employ single-photon detection technologies and are limited in speed. Here, we experimentally demonstrate an ultrafast QRNG at a rate over 6 Gbits/s based on the quantum phase fluctuations of a laser operating near threshold. Moreover, we consider a potential adversary who has partial knowledge on the raw data and discuss how one can rigorously remove such partial knowledge with postprocessing. We quantify the quantum randomness through min-entropy by modeling our system and employ two randomness extractors - Trevisan’s extractor and Toeplitz-hashing - to distill the randomness, which is information-theoretically provable. The simplicity and high-speed of our experimental setup show the feasibility of a robust, low-cost, high-speed QRNG.

© 2012 OSA

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  1. N. Meteopolis and S. Ulam, “The monte carlo method,” J. Am. Stat. Assoc.44, 335–341 (1949).
  2. C. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proc. of IEEE Inter. Conf. on Computer Systems and Signal Processing, 175–179 (IEEE Press, 1984).
  3. B. Schneier and P. Sutherland, Applied Cryptography: Protocols, Algorithms, and Source Code in C (John Wiley & Sons, 1995).
  4. 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. Photonics2, 728–732 (2008).
    [CrossRef]
  5. I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett.103, 24102 (2009)
    [CrossRef]
  6. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4(1), 58–61 (2010).
    [CrossRef]
  7. C. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express18, 23584–23597 (2010).
    [CrossRef] [PubMed]
  8. X. Li, A. Cohen, T. Murphy, and R. Roy, “Scalable parallel physical random number generator based on a superluminescent LED,” Opt. Lett.36, 1020–1022 (2011).
    [CrossRef] [PubMed]
  9. T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum.71, 1675–1679 (2000).
    [CrossRef]
  10. J. Dynes, Z. Yuan, A. Sharpe, and A. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett.93, 031109 (2008).
    [CrossRef]
  11. M. Wayne and P. Kwiat, “Low-bias high-speed quantum number generator via shaped optical pulses,” Opt. Express18, 9351–9357 (2010).
    [CrossRef] [PubMed]
  12. M. Fürst, H. Weier, S. Nauerth, D. Marangon, C. Kurtsiefer, and H. Weinfurter, “High speed optical quantum random number generation,” Opt. Express18, 13029–13037 (2010).
    [CrossRef] [PubMed]
  13. M. Wahl, M. Leifgen, M. Berlin, T. Rhlicke, H.-J. Rahn, and O. Benson, “An ultrafast quantum random number generator with provably bounded output bias based on photon arrival time measurements,” Appl. Phys. Lett.98, 171105 (2011).
    [CrossRef]
  14. S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature464, 1021–1024 (2010).
    [CrossRef] [PubMed]
  15. 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, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
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  19. B. Qi, Y. Chi, H.-K. Lo, and Q. Li, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. Lett.35, 312–314 (2010).
    [CrossRef] [PubMed]
  20. H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. Rev. E81, 051137 (2010).
    [CrossRef]
  21. H. Takesue, S. Nam, Q. Zhang, R. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over a 40-dB channel loss using superconducting single-photon detectors,” Nat. Photonics1, 343–348 (2007).
    [CrossRef]
  22. M. N. Wegman and J. L. Carter, “New hash functions and their use in authentication and set equality,” J. Comput. Syst. Sci.22, 265–279 (1981).
    [CrossRef]
  23. L. Trevisan, “Extractors and Pseudorandom Generators,” J. ACM48, 860–879 (2001).
    [CrossRef]
  24. R. Shaltiel, “Recent developments in explicit constructions of extractors,” Bull. Eur. Assoc. Theor. Comput. Sci.77, 67–95 (2002).
  25. C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron.18259–264, (1982).
    [CrossRef]
  26. K. Vahala and A. Yariv, “Occupation fluctuation noise: A fundamental source of linewidth broadening in semiconductor lasers,” Appl. Phys. Lett.43, 140 (1983)
    [CrossRef]
  27. The measured accuracy of the temperature controller is 0.01°C. The fluctuations of the setpoint temperature of the PLC-MZI are smaller than 0.01°C during a few hours.
  28. K. Petermann, Laser Diode Modulation and Noise (Springer, 1988).
    [CrossRef]
  29. A practical laser presents some classical noises, such as occupation fluctuations [26] and 1/f noise (see Electron. Lett., 19, 812, 1983). These classical noises are power independent [26].
  30. To experimentally determine γ, the key idea is that when the laser is operated at a significant high power level, the classical noise part (C in Eq. (3)) will dominate over the quantum fluctuations part (QP in Eq. (3)). It consists of three steps: a) at an optical power level Po, we measured the variance of Vpr(t) as σ12. b) the laser was operated to its maximal power (around 25 mW for our DFB laser diode) and an optical attenuator (JDS Uniphase HA1) was applied right after the laser to attenuate the output power down to Po, in which the variance of Vpr(t) was measured as σ22. From σ12 and σ22, we could derive the experimental value γ=σ12−σ22σ22 at power Po. c) the process was repeated at different power levels and the experimental results were shown in Fig. 3.
  31. X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, and H.-K. Lo, under preparation (2011).
  32. There are mainly five spikes around 0, 100, 200, 500, and 650 MHz. These frequencies are all within practical broadcast radio bands (see http://www.fcc.gov/oet/spectrum ).
  33. To reduce the correlations and ensure the independence between adjacent samples, the sampling time (1 ns) has been chosen to be larger than the sum of PLC-MZI time difference (500 ps) and detector response time (200 ps). For details, see Ref. [19].
  34. We remark that in a practical system, it will be interesting for future research to investigate how to determine an optimal ADC range, which can maximize the extractable randomness.
  35. In information theory, the channel capacity of a given channel is the limiting information rate that can be achieved with arbitrarily small error probability by the noisy-channel coding theorem. For a more detailed discussion, see Thomas M. Cover and Joy A. Thomas, Elements of Information Theory (John Wiley & Sons, 2006).
  36. The final security parameter of randomness extractor (i.e. statistical distance between output distribution and a perfect-random distribution) is a function of input data size n. In the infinite key limit, the output of randomness extractor is determined by the min-entropy. In general, randomness extractors are quite efficient (close to 100% for a reasonable input data size, such as 100Mbits). See [31] for a rigorious discussion.
  37. H. Krawczyk, in Advances in Cryptology - CRYPTO’94, Lecture Notes in Computer Science, 893, 129–139 (Springer-Verlag, 1994).
    [CrossRef]
  38. For demonstration purpose, we use pseudo-random number generator of Matlab to generate the seed constructing Toeplitz matrix. In the future, we plan to generate the seed from either some well-developed QRNGs (such as Ref. [16]) or pre-stored random bits generated by our own QRNG system. Note that Toeplitz-hashing allows the re-use of the seed in subsequent applications (see details in [31]).
  39. R. Raz, O. Reingold, and S. Vadhan, in Proc. of the 31st Annual ACM Symposium on Theory of Computing, 149–158 (1999).
  40. http://www.stat.fsu.edu/pub/diehard/
  41. http://csrc.nist.gov/groups/ST/toolkit/rng/
  42. P. L’Ecuyer and R. Simard “TestU01: AC library for empirical testing of random number generators,” ACM Trans. Math. Softw.33, 22 (2007).
  43. F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, arXiv:1109.0643 (2011).
  44. T. Symul, S. Assad, and P. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett.98, 231103 (2011).
    [CrossRef]
  45. M. Jofre, M. Curty, F. Steinlechner, G. Anzolin, J. P. Torres, M. W. Mitchell, and V. Pruneri, “True random numbers from amplified quantum vacuum,” Opt. Express19, 20665–20672 (2011).
    [CrossRef] [PubMed]

2011

T. Symul, S. Assad, and P. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett.98, 231103 (2011).
[CrossRef]

M. Wahl, M. Leifgen, M. Berlin, T. Rhlicke, H.-J. Rahn, and O. Benson, “An ultrafast quantum random number generator with provably bounded output bias based on photon arrival time measurements,” Appl. Phys. Lett.98, 171105 (2011).
[CrossRef]

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

M. Jofre, M. Curty, F. Steinlechner, G. Anzolin, J. P. Torres, M. W. Mitchell, and V. Pruneri, “True random numbers from amplified quantum vacuum,” Opt. Express19, 20665–20672 (2011).
[CrossRef] [PubMed]

2010

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

M. Wayne and P. Kwiat, “Low-bias high-speed quantum number generator via shaped optical pulses,” Opt. Express18, 9351–9357 (2010).
[CrossRef] [PubMed]

M. Fürst, H. Weier, S. Nauerth, D. Marangon, C. Kurtsiefer, and H. Weinfurter, “High speed optical quantum random number generation,” Opt. Express18, 13029–13037 (2010).
[CrossRef] [PubMed]

C. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express18, 23584–23597 (2010).
[CrossRef] [PubMed]

S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature464, 1021–1024 (2010).
[CrossRef] [PubMed]

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, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[CrossRef]

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4(1), 58–61 (2010).
[CrossRef]

H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. Rev. E81, 051137 (2010).
[CrossRef]

2009

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett.103, 24102 (2009)
[CrossRef]

R. H. Hadeld, “Single-photon detectors for optical quantum information applications,” Nat. Photonics3, 696–705 (2009).
[CrossRef]

2008

J. Dynes, Z. Yuan, A. Sharpe, and A. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett.93, 031109 (2008).
[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. Photonics2, 728–732 (2008).
[CrossRef]

2007

H. Takesue, S. Nam, Q. Zhang, R. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over a 40-dB channel loss using superconducting single-photon detectors,” Nat. Photonics1, 343–348 (2007).
[CrossRef]

P. L’Ecuyer and R. Simard “TestU01: AC library for empirical testing of random number generators,” ACM Trans. Math. Softw.33, 22 (2007).

2002

R. Shaltiel, “Recent developments in explicit constructions of extractors,” Bull. Eur. Assoc. Theor. Comput. Sci.77, 67–95 (2002).

2001

L. Trevisan, “Extractors and Pseudorandom Generators,” J. ACM48, 860–879 (2001).
[CrossRef]

2000

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum.71, 1675–1679 (2000).
[CrossRef]

1983

K. Vahala and A. Yariv, “Occupation fluctuation noise: A fundamental source of linewidth broadening in semiconductor lasers,” Appl. Phys. Lett.43, 140 (1983)
[CrossRef]

1982

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron.18259–264, (1982).
[CrossRef]

1981

M. N. Wegman and J. L. Carter, “New hash functions and their use in authentication and set equality,” J. Comput. Syst. Sci.22, 265–279 (1981).
[CrossRef]

1949

N. Meteopolis and S. Ulam, “The monte carlo method,” J. Am. Stat. Assoc.44, 335–341 (1949).

Achleitner, U.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum.71, 1675–1679 (2000).
[CrossRef]

Acin, A.

S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature464, 1021–1024 (2010).
[CrossRef] [PubMed]

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, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[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. Photonics2, 728–732 (2008).
[CrossRef]

Anzolin, G.

Assad, S.

T. Symul, S. Assad, and P. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett.98, 231103 (2011).
[CrossRef]

Aviad, Y.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4(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, 24102 (2009)
[CrossRef]

Bennett, C.

C. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proc. of IEEE Inter. Conf. on Computer Systems and Signal Processing, 175–179 (IEEE Press, 1984).

Benson, O.

M. Wahl, M. Leifgen, M. Berlin, T. Rhlicke, H.-J. Rahn, and O. Benson, “An ultrafast quantum random number generator with provably bounded output bias based on photon arrival time measurements,” Appl. Phys. Lett.98, 171105 (2011).
[CrossRef]

Berlin, M.

M. Wahl, M. Leifgen, M. Berlin, T. Rhlicke, H.-J. Rahn, and O. Benson, “An ultrafast quantum random number generator with provably bounded output bias based on photon arrival time measurements,” Appl. Phys. Lett.98, 171105 (2011).
[CrossRef]

Brassard, G.

C. Bennett and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in Proc. of IEEE Inter. Conf. on Computer Systems and Signal Processing, 175–179 (IEEE Press, 1984).

Carter, J. L.

M. N. Wegman and J. L. Carter, “New hash functions and their use in authentication and set equality,” J. Comput. Syst. Sci.22, 265–279 (1981).
[CrossRef]

Chi, Y.

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

B. Qi, Y. Chi, H.-K. Lo, and Q. Li, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” in Proc. of the 9th Asian Conf. on Quant. Info. Sci.64–65 (2009).

Cohen, A.

Cohen, E.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4(1), 58–61 (2010).
[CrossRef]

Cover, Thomas M.

In information theory, the channel capacity of a given channel is the limiting information rate that can be achieved with arbitrarily small error probability by the noisy-channel coding theorem. For a more detailed discussion, see Thomas M. Cover and Joy A. Thomas, Elements of Information Theory (John Wiley & Sons, 2006).

Curty, M.

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, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[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. Photonics2, 728–732 (2008).
[CrossRef]

de la Giroday, A. B.

S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature464, 1021–1024 (2010).
[CrossRef] [PubMed]

Dynes, J.

J. Dynes, Z. Yuan, A. Sharpe, and A. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett.93, 031109 (2008).
[CrossRef]

Fürst, M.

Guo, H.

H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. Rev. E81, 051137 (2010).
[CrossRef]

Hadeld, R. H.

R. H. Hadeld, “Single-photon detectors for optical quantum information applications,” Nat. Photonics3, 696–705 (2009).
[CrossRef]

Hadfield, R.

H. Takesue, S. Nam, Q. Zhang, R. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over a 40-dB channel loss using superconducting single-photon detectors,” Nat. Photonics1, 343–348 (2007).
[CrossRef]

Hayes, D.

S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature464, 1021–1024 (2010).
[CrossRef] [PubMed]

Henry, C.

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron.18259–264, (1982).
[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, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[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. Photonics2, 728–732 (2008).
[CrossRef]

Honjo, T.

H. Takesue, S. Nam, Q. Zhang, R. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over a 40-dB channel loss using superconducting single-photon detectors,” Nat. Photonics1, 343–348 (2007).
[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, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[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. Photonics2, 728–732 (2008).
[CrossRef]

Jennewein, T.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum.71, 1675–1679 (2000).
[CrossRef]

Jofre, M.

Kanter, I.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4(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, 24102 (2009)
[CrossRef]

Krawczyk, H.

H. Krawczyk, in Advances in Cryptology - CRYPTO’94, Lecture Notes in Computer Science, 893, 129–139 (Springer-Verlag, 1994).
[CrossRef]

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, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[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. Photonics2, 728–732 (2008).
[CrossRef]

Kurtsiefer, C.

Kwiat, P.

L’Ecuyer, P.

P. L’Ecuyer and R. Simard “TestU01: AC library for empirical testing of random number generators,” ACM Trans. Math. Softw.33, 22 (2007).

Lam, P.

T. Symul, S. Assad, and P. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett.98, 231103 (2011).
[CrossRef]

Leifgen, M.

M. Wahl, M. Leifgen, M. Berlin, T. Rhlicke, H.-J. Rahn, and O. Benson, “An ultrafast quantum random number generator with provably bounded output bias based on photon arrival time measurements,” Appl. Phys. Lett.98, 171105 (2011).
[CrossRef]

Li, Q.

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

B. Qi, Y. Chi, H.-K. Lo, and Q. Li, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” in Proc. of the 9th Asian Conf. on Quant. Info. Sci.64–65 (2009).

Li, X.

Liu, Y.

H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. Rev. E81, 051137 (2010).
[CrossRef]

Lo, H.-K.

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

F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, arXiv:1109.0643 (2011).

X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, and H.-K. Lo, under preparation (2011).

B. Qi, Y. Chi, H.-K. Lo, and Q. Li, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” in Proc. of the 9th Asian Conf. on Quant. Info. Sci.64–65 (2009).

Luo, L.

S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature464, 1021–1024 (2010).
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S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature464, 1021–1024 (2010).
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S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature464, 1021–1024 (2010).
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S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature464, 1021–1024 (2010).
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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, and P. Davis, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
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H. Takesue, S. Nam, Q. Zhang, R. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over a 40-dB channel loss using superconducting single-photon detectors,” Nat. Photonics1, 343–348 (2007).
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H. Takesue, S. Nam, Q. Zhang, R. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over a 40-dB channel loss using superconducting single-photon detectors,” Nat. Photonics1, 343–348 (2007).
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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, and P. Davis, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
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J. Dynes, Z. Yuan, A. Sharpe, and A. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett.93, 031109 (2008).
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T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum.71, 1675–1679 (2000).
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F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, arXiv:1109.0643 (2011).

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T. Symul, S. Assad, and P. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett.98, 231103 (2011).
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J. Dynes, Z. Yuan, A. Sharpe, and A. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett.93, 031109 (2008).
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N. Meteopolis and S. Ulam, “The monte carlo method,” J. Am. Stat. Assoc.44, 335–341 (1949).

J. Comput. Syst. Sci.

M. N. Wegman and J. L. Carter, “New hash functions and their use in authentication and set equality,” J. Comput. Syst. Sci.22, 265–279 (1981).
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Nat. Photonics

H. Takesue, S. Nam, Q. Zhang, R. Hadfield, T. Honjo, K. Tamaki, and Y. Yamamoto, “Quantum key distribution over a 40-dB channel loss using superconducting single-photon detectors,” Nat. Photonics1, 343–348 (2007).
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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, and P. Davis, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
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I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4(1), 58–61 (2010).
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Nature

S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature464, 1021–1024 (2010).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. E

H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. Rev. E81, 051137 (2010).
[CrossRef]

Phys. Rev. Lett.

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett.103, 24102 (2009)
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Other

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http://www.idquantique.com

The measured accuracy of the temperature controller is 0.01°C. The fluctuations of the setpoint temperature of the PLC-MZI are smaller than 0.01°C during a few hours.

K. Petermann, Laser Diode Modulation and Noise (Springer, 1988).
[CrossRef]

A practical laser presents some classical noises, such as occupation fluctuations [26] and 1/f noise (see Electron. Lett., 19, 812, 1983). These classical noises are power independent [26].

To experimentally determine γ, the key idea is that when the laser is operated at a significant high power level, the classical noise part (C in Eq. (3)) will dominate over the quantum fluctuations part (QP in Eq. (3)). It consists of three steps: a) at an optical power level Po, we measured the variance of Vpr(t) as σ12. b) the laser was operated to its maximal power (around 25 mW for our DFB laser diode) and an optical attenuator (JDS Uniphase HA1) was applied right after the laser to attenuate the output power down to Po, in which the variance of Vpr(t) was measured as σ22. From σ12 and σ22, we could derive the experimental value γ=σ12−σ22σ22 at power Po. c) the process was repeated at different power levels and the experimental results were shown in Fig. 3.

X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, and H.-K. Lo, under preparation (2011).

There are mainly five spikes around 0, 100, 200, 500, and 650 MHz. These frequencies are all within practical broadcast radio bands (see http://www.fcc.gov/oet/spectrum ).

To reduce the correlations and ensure the independence between adjacent samples, the sampling time (1 ns) has been chosen to be larger than the sum of PLC-MZI time difference (500 ps) and detector response time (200 ps). For details, see Ref. [19].

We remark that in a practical system, it will be interesting for future research to investigate how to determine an optimal ADC range, which can maximize the extractable randomness.

In information theory, the channel capacity of a given channel is the limiting information rate that can be achieved with arbitrarily small error probability by the noisy-channel coding theorem. For a more detailed discussion, see Thomas M. Cover and Joy A. Thomas, Elements of Information Theory (John Wiley & Sons, 2006).

The final security parameter of randomness extractor (i.e. statistical distance between output distribution and a perfect-random distribution) is a function of input data size n. In the infinite key limit, the output of randomness extractor is determined by the min-entropy. In general, randomness extractors are quite efficient (close to 100% for a reasonable input data size, such as 100Mbits). See [31] for a rigorious discussion.

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[CrossRef]

For demonstration purpose, we use pseudo-random number generator of Matlab to generate the seed constructing Toeplitz matrix. In the future, we plan to generate the seed from either some well-developed QRNGs (such as Ref. [16]) or pre-stored random bits generated by our own QRNG system. Note that Toeplitz-hashing allows the re-use of the seed in subsequent applications (see details in [31]).

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http://www.stat.fsu.edu/pub/diehard/

http://csrc.nist.gov/groups/ST/toolkit/rng/

F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, arXiv:1109.0643 (2011).

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Figures (6)

Fig. 1
Fig. 1

Experimental setup. Laser, 1550nm cw DFB laser diode (ILX Lightwave); PLC-MZI, planar lightwave circuit Mach-Zehnder interferometer with a 500ps delay difference (manufactured by NTT); TC, temperature controller (PTC 5K from Wavelength Electronics Inc.); PD, 5GHz InGaAs photodetector (Thorlabs SIR5-FC); ADC, 8-bit analog-to-digital convertor inside an oscilloscope (Agilent DSO81204A).

Fig. 2
Fig. 2

Experimental voltage variance. The variance of the output a.c. voltage (Vpr(t)) from the photodetector (see Fig. 1) is measured by an oscilloscope. Here, the error bars are smaller than the symbol size. The experimental data is fitted by a quadratic polynomial function.

Fig. 3
Fig. 3

Quantum signal to classical noise ratio. The theoretical curve of signal-to-noise ratio is obtained from Egn. 5 and the parameters given in Table 1. The experimental results are measured with an oscilloscope at different laser powers [30] (the corresponding error bars are smaller than the symbol size). At low and high power range, either the background noise F or the classical phase noise ACP2 will dominate over the quantum signal. The optimal ratio γ = 21 is achieved at P = 0.95 mW.

Fig. 4
Fig. 4

Noise spectra. The spectral power density of total phase fluctuations (blue), intensity noise (green), and background noise (red).

Fig. 5
Fig. 5

A simple illustration of min-entropy evaluation (toy model). The raw-data follows a Gaussian distribution (μ = 0 and σ = 7) and is digitized by a 3-bit ADC (sampling range is defined as [−a, a] with a=15 here). From Eq. (6), the min-entropy is determined by the sample point x with the maximal probability Pmax. Here, x equals to the bin of ‘100’ (or ‘011’) and Pmax can be calculated from its bin area.

Fig. 6
Fig. 6

(a) Autocorrelation of the raw-data. The raw-data is obtained by sampling the output a.c. voltage (Vpr(t) from the photodetector) with an ADC (see Fig. 1). Each sample consists of 8 bits and the correlation between samples cannot reach zero for a practical detector with finite bandwidth. (b) Autocorrelation of the Toeplitz-hashing output. Data size is 1 × 107 bits and the average value within 100 bit-delay is −1.0 × 10−5. In theory, for a truly random 1×107 bit string, the average normalized correlation is 0 and the standard deviation is 3 × 10−4. In practice, due to the inevitable presence of bias and finite data size, the autocorrelation of data sequence can never reach 0. (c) NIST results of the Toeplitz-hashing output. Data size is 3.25 Gbits (500 sequences with each sequence around 6.5 Mbits). To pass the test, P-value should be larger than the lowest significant level α = 0.01, and the proportion of sequences satisfying P > α should be greater than 0.976. Where the test has multiple P-values, the worst case is selected. (d) Diehard results of the Trevisan’s extractor output. Data size is 240Mbits. A Kolmogorov-Smirnov (KS) test is used to obtain a final P-value from the case of multiple P-values. Successful P-value is 0.01 ≤ P ≤ 0.99.

Tables (1)

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Table 1 Experimental results (with 0.99 confidence intervals) of parameters in Eq. (4).

Equations (8)

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E ( t ) = E 0 exp [ i ( ω t + θ ( t ) ) ]
V ( t ) 2 E ( t ) E ( t + τ ) sin ( Δ θ ( t ) ) P Δ θ ( t )
Δ θ ( t ) 2 = Q P + C
V p r ( t ) 2 = A Q P + A C P 2 + F
γ = A Q P A C P 2 + F
H ( X ) = log 2 ( max x { 0 , 1 } n Pr [ X = x ] )
{ 0 , 1 } n × { 0 , 1 } d { 0 , 1 } m
R ( j ) = E [ ( X i μ ) ( X i + j μ ) ] σ 2

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