Abstract

We demonstrate a high bit-rate quantum random number generator by interferometric detection of phase diffusion in a gain-switched DFB laser diode. Gain switching at few-GHz frequencies produces a train of bright pulses with nearly equal amplitudes and random phases. An unbalanced Mach-Zehnder interferometer is used to interfere subsequent pulses and thereby generate strong random-amplitude pulses, which are detected and digitized to produce a high-rate random bit string. Using established models of semiconductor laser field dynamics, we predict a regime of high visibility interference and nearly complete vacuum-fluctuation-induced phase diffusion between pulses. These are confirmed by measurement of pulse power statistics at the output of the interferometer. Using a 5.825 GHz excitation rate and 14-bit digitization, we observe 43 Gbps quantum randomness generation.

© 2014 Optical Society of America

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2013

2012

2011

M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, H.-J. Rahn, 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. Jofre, M. Curty, F. Steinlechner, G. Anzolin, J. P. Torres, M. W. Mitchell, V. Pruneri, “True random numbers from amplified quantum vacuum,” Opt. Express 19, 20665–20672 (2011).
[CrossRef] [PubMed]

2010

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

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (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 81, 051137 (2010).
[CrossRef]

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

2009

2007

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[CrossRef]

X. Cai, X. Wang, “Stochastic modeling and simulation of gene networks - a review of the state-of-the-art research on stochastic simulations,” IEEE Signal Process. Mag. 24, 27–36 (2007).
[CrossRef]

2000

C. Petrie, J. Connelly, “A noise-based ic random number generator for applications in cryptography,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 615–621 (2000).
[CrossRef]

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[CrossRef]

1999

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

N. Nisan, A. Ta-Shma, “Extracting randomness: A survey and new constructions,” J. Comput. Sci. Tech. 58, 148–173 (1999).

1990

G. Agrawal, “Effect of gain and index nonlinearities on single-mode dynamics in semiconductor lasers,” IEEE J. Quantum Electron. 26, 1901–1909 (1990).
[CrossRef]

1986

C. Henry, “Phase noise in semiconductor lasers,” J. Lightwave Technol. 4, 298–311 (1986).
[CrossRef]

1982

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

Achleitner, U.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[CrossRef]

Acín, A.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

Agrawal, G.

G. Agrawal, “Effect of gain and index nonlinearities on single-mode dynamics in semiconductor lasers,” IEEE J. Quantum Electron. 26, 1901–1909 (1990).
[CrossRef]

Anzolin, G.

Argyris, A.

Aviad, Y.

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

Banks, D.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Barker, E.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Benson, O.

M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, H.-J. Rahn, 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. Röhlicke, H.-J. Rahn, 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]

Byer, R. L.

Cai, X.

X. Cai, X. Wang, “Stochastic modeling and simulation of gene networks - a review of the state-of-the-art research on stochastic simulations,” IEEE Signal Process. Mag. 24, 27–36 (2007).
[CrossRef]

Chan, S.-C.

Cho, Y.-W.

Cohen, E.

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

Connelly, J.

C. Petrie, J. Connelly, “A noise-based ic random number generator for applications in cryptography,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 615–621 (2000).
[CrossRef]

Curty, M.

de la Giroday, A. B.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

Deligiannidis, S.

Dray, J.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Fujita, K.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

Guo, H.

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

Hall, C.

C. Hall, B. Schneier, “Remote electronic gambling,” in “13th Annual Computer Security Applications Conference” (1997), pp. 232–238.
[CrossRef]

Hayes, D.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

Heckert, A.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Henry, C.

C. Henry, “Phase noise in semiconductor lasers,” J. Lightwave Technol. 4, 298–311 (1986).
[CrossRef]

C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

Inoue, H.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

Jennewein, T.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[CrossRef]

Jofre, M.

Kanter, I.

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

Kim, Y.-H.

Kimura, H.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

Kwon, O.

Leifgen, M.

M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, H.-J. Rahn, 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]

Leigh, S.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Leindecker, N. C.

Levenson, M.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Li, L.

Li, P.

Li, X.

Li, X.-Z.

Liu, Y.

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

Lo, H.-K.

Luo, L.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

Ma, X.

Maeda, W.

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[CrossRef]

Manning, T. A.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

Marandi, A.

Massar, S.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

Matsukevich, D. N.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

Maunz, P.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

Mitchell, M. W.

Miyatake, O.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

Monroe, C.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

Murphy, T. E.

Nechvatal, J.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Nisan, N.

N. Nisan, A. Ta-Shma, “Extracting randomness: A survey and new constructions,” J. Comput. Sci. Tech. 58, 148–173 (1999).

Olmschenk, S.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

Petrie, C.

C. Petrie, J. Connelly, “A noise-based ic random number generator for applications in cryptography,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 615–621 (2000).
[CrossRef]

Pikasis, E.

Pironio, S.

S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, C. Monroe, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

Pruneri, V.

Qi, B.

Rahn, H.-J.

M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, H.-J. Rahn, 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]

Reidler, I.

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

Röhlicke, T.

M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, H.-J. Rahn, 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]

Rosenbluh, M.

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

Roy, R.

Rukhin, A.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Salevan, J. C.

Schneier, B.

C. Hall, B. Schneier, “Remote electronic gambling,” in “13th Annual Computer Security Applications Conference” (1997), pp. 232–238.
[CrossRef]

Soto, J.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Steinlechner, F.

Syvridis, D.

Tajima, A.

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[CrossRef]

Takahashi, S.

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[CrossRef]

Tanaka, A.

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[CrossRef]

Tang, W.

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

Ta-Shma, A.

N. Nisan, A. Ta-Shma, “Extracting randomness: A survey and new constructions,” J. Comput. Sci. Tech. 58, 148–173 (1999).

Tomita, A.

A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. 13, 1031–1038 (2007).
[CrossRef]

Torres, J. P.

Toyama, M.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

Vangel, M.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Vo, S.

A. Rukhin, J. Soto, J. Nechvatal, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” NIST Special Publication 800-22 revision 1a (2010).

Vodopyanov, K. L.

Wahl, M.

M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, H.-J. Rahn, 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]

Wang, A.

Wang, X.

X. Cai, X. Wang, “Stochastic modeling and simulation of gene networks - a review of the state-of-the-art research on stochastic simulations,” IEEE Signal Process. Mag. 24, 27–36 (2007).
[CrossRef]

Wang, Y.

Wei, W.

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

Weihs, G.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[CrossRef]

Weinfurter, H.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[CrossRef]

Williams, C. R. S.

Xu, F.

Xu, H.

Yoshizawa, Y.

Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. 12, 67–81 (1999).

Zeilinger, A.

T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. 71, 1675–1680 (2000).
[CrossRef]

Zhang, J.

Zheng, H.

Appl. Opt.

Appl. Phys. Lett.

M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, H.-J. Rahn, 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]

IEEE J. Quantum Electron.

G. Agrawal, “Effect of gain and index nonlinearities on single-mode dynamics in semiconductor lasers,” IEEE J. Quantum Electron. 26, 1901–1909 (1990).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Electrical and optical pulse trains. Magenta, (upper trace): electrical current drive applied to the laser, with PRF of 172 ps. Blue, (lower trace): photo-detected optical pulses of 85 ps time width and 7.65 mW peak power and (black, dashed line) 9 mA LD current threshold. Simulation is a conservative fitting of the rate equations such that the predicted detected output power vs. time is always larger than the observed output power vs. time.

Fig. 2
Fig. 2

Unbalanced Mach-Zehnder interferometer (U-MZI). Phase-randomized coherent optical pulses interfering at the output of the U-MZI produce random intensities. (Pulse driver) denotes the electrical pulse generator that directly modulates the laser, (LD) laser diode, (PMC) polarization maintaining coupler, (PMF) polarization maintaining fiber, (θ0–3) optical phases of different consecutive pulses, (θloop) phase introduced by the delay line and (PD) a fast photodetector.

Fig. 3
Fig. 3

In Fig. 3(a), input power distributions (left y axis) and the resultant output power distribution (right y axis). The visibility achieved for the interferometer is 0.9. The power distribution has clearly widened due to the random phase generated by amplified spontaneous emission (ASE). In Fig. 3(b), normalized correlation of 50 subsequent sampled pulses. The autocorrelation has been evaluated with 120 × 106 14-bit samples, but just the first 50 terms are shown. As expected, it follows a delta-function like behaviour indicating the random nature of the process.

Fig. 4
Fig. 4

Statistical characterization of 125 × 106 7-bit numbers produced by hashing the experimental data. (a) autocorrelation of the hashed data. Autocorrelation at the −40 dB level is seen, corresponding to the expected statistical variation. (b) deviation from the ideal 7-bit uniform distribution. Dashed lines show plus/minus 1 sigma expected variation.

Fig. 5
Fig. 5

Summary of the results of the NIST test suite to assess randomness. Dashed lines in Fig. 5(a) represent the confidence interval where the proportion of sequences accepted/rejected per test must fall in. In Fig. 5(b) is plot PvalueT = Γ(9/2, χ2/2) for each test. All PvalueT > 10−4. The smallest one is for the non-periodic template matching test, giving a value of PvalueT = 0.0926.

Equations (17)

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( out ) ( t ) = ε 11 ( 1 ) ε 11 ( 2 ) ( laser ) ( t t 1 ) + ε 12 ( 1 ) ε 21 ( 2 ) ( laser ) ( t t 2 ) ,
u ( out ) ( t ) = u 1 ( t ) + u 2 ( t ) + 2 | g ( 1 ) ( t ) | u 1 ( t ) u 2 ( t ) cos ( θ j θ j 1 + Δ ϕ ) + u noise ,
θ ˙ = α 2 G N ( n n th ) β SE 2 G N ( n n 0 ) p 1 + 1 + p + F θ ( t ) ,
d d t Δ θ ( t ) 2 = R sp 2 s ( 1 + α 2 ) ,
s ˙ = G N ( n n 0 1 + s / s sat ( n th n 0 ) ) s + R sp ,
n ˙ = I / q γ e n G N n n 0 1 + s / s sat s ,
γ ( 1 1 + s th / s sat 1 ) s th + R 0 γ e n th = 0 ,
I q γ e n th γ s th 1 1 + s th / s sat = 0 ,
H ( X ) log 2 ( max x i P [ X = x i ] ) ,
PDF ( x ) = 1 π ( x a ) ( b x ) ; μ x = a + b 2 ; var [ x ] = 1 8 ( b a ) 2 .
var ( u ( out ) ) = var ( u 1 ) + var ( u 2 ) + var ( u noise ) + 4 | g ( t loop ) | 2 var ( u 1 u 2 cos θ ) .
| g ( t loop ) | 2 var ( u ( out ) ) var ( u 1 ) var ( u 2 ) var ( u noise ) 2 E [ u 1 ] 2 E [ u 2 ] 2 ,
u min E [ u 1 + u 2 2 | g ( t loop ) | u 1 u 2 ] ,
u max E [ u 1 + u 2 + 2 | g ( t loop ) | u 1 u 2 ] .
P [ X = x 1 ] = 1 π u min u min + Δ u 1 ( u u min ) ( u max u ) d u = 2 π arcsin A ADC 2 b ( u max u min ) .
H ( X ) b 2 1 2 log 2 ( 4 A ADC π 2 ( u max u min ) ) .
χ 2 = s 10 i = 1 10 ( F i s 10 ) 2 ,

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