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]

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]

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]

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]

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]

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).

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]

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]

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).

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]

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).

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

[CrossRef]

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).

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]

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]

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]

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]

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

[CrossRef]

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]

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]

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

[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, “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]

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]

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]

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]

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]

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

[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, “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]

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

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]

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).

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]

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]

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]

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).

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]

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

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

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]

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]

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]

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]

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

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]

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]

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]

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]

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]

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

[CrossRef]

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]

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).

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]

R. Raz, O. Reingold, and S. Vadhan, in Proc. of the 31st Annual ACM Symposium on Theory of Computing, 149–158 (1999).

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]

R. Raz, O. Reingold, and S. Vadhan, in Proc. of the 31st Annual ACM Symposium on Theory of Computing, 149–158 (1999).

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]

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]

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]

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]

B. Schneier and P. Sutherland, Applied Cryptography: Protocols, Algorithms, and Source Code in C (John Wiley & Sons, 1995).

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

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]

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, “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]

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

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]

B. Schneier and P. Sutherland, Applied Cryptography: Protocols, Algorithms, and Source Code in C (John Wiley & Sons, 1995).

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]

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]

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]

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

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]

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).

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

[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, “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]

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

R. Raz, O. Reingold, and S. Vadhan, in Proc. of the 31st Annual ACM Symposium on Theory of Computing, 149–158 (1999).

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

[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]

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]

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]

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]

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]

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]

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

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

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).

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]

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

[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, “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]

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]

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]

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]

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]

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

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

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]

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]

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]

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

[CrossRef]

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

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

[CrossRef]

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

[CrossRef]

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

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]

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]

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]

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

[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]

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

[CrossRef]

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]

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]

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]

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]

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]

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]

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]

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]

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).

B. Schneier and P. Sutherland, Applied Cryptography: Protocols, Algorithms, and Source Code in C (John Wiley & Sons, 1995).

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).

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.

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

[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]).

R. Raz, O. Reingold, and S. Vadhan, in Proc. of the 31st Annual ACM Symposium on Theory of Computing, 149–158 (1999).

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).