## Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode |

Optics Express, Vol. 22, Issue 2, pp. 1645-1654 (2014)

http://dx.doi.org/10.1364/OE.22.001645

Acrobat PDF (1020 KB)

### 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

## 1. Introduction

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

2. X. Cai and 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]

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

4. C. Petrie and 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]

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

6. A. Argyris, E. Pikasis, S. Deligiannidis, and D. Syvridis, “Sub-tb/s physical random bit generators based on direct detection of amplified spontaneous emission signals,” J. Lightwave Technol. **30**, 1329–1334 (2012). [CrossRef]

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

9. O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Quantum random number generator using photon-number path entanglement,” Appl. Opt. **48**, 1774–1778 (2009). [CrossRef] [PubMed]

10. 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. Express **18**, 23584–23597 (2010). [CrossRef] [PubMed]

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

13. F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Opt. Express **20**, 12366–12377 (2012). [CrossRef] [PubMed]

14. M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, 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]

15. A. Marandi, N. C. Leindecker, K. L. Vodopyanov, and R. L. Byer, “All-optical quantum random bit generation from intrinsically binary phase of parametric oscillators,” Opt. Express **20**, 19322–19330 (2012). [CrossRef] [PubMed]

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

6. A. Argyris, E. Pikasis, S. Deligiannidis, and D. Syvridis, “Sub-tb/s physical random bit generators based on direct detection of amplified spontaneous emission signals,” J. Lightwave Technol. **30**, 1329–1334 (2012). [CrossRef]

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

17. 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**, 20452–20462 (2013). [CrossRef] [PubMed]

12. 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. Express **19**, 20665–20672 (2011). [CrossRef] [PubMed]

14. M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, 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]

18. 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, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature **464**, 1021–1024 (2010). [CrossRef] [PubMed]

12. 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. Express **19**, 20665–20672 (2011). [CrossRef] [PubMed]

13. F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Opt. Express **20**, 12366–12377 (2012). [CrossRef] [PubMed]

12. 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. Express **19**, 20665–20672 (2011). [CrossRef] [PubMed]

13. F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Opt. Express **20**, 12366–12377 (2012). [CrossRef] [PubMed]

## 2. System configuration

## 3. Principle of operation

*k*th PMC from input port

*i*to output port

*j*,

*ℰ*

^{(laser)}is the field emitted by the laser and

*t*

_{1,2}are the delays caused by arms 1,2 of the U-MZI, respectively. For convenience, we write

*ℰ*

^{(laser)}(

*t*) =

*A*(

*t*)exp[−

*iωt*]exp[

*iθ*] where

_{j}*ω*is the central frequency of the laser emission and

*θ*is the phase of the

_{j}*j*th pulse. Defining the powers

*u*

^{(out)}(

*t*) ≡ |

**

^{(out)}(

*t*)|

^{2},

*ϕ*≡

*ω*(

*t*

_{1}−

*t*

_{2}) is the relative phase delay of the two arms,

*u*

_{noise}is noise contribution from the detection and digitization electronics. Fluctuation of Δ

*ϕ*due to changes in

*t*

_{1}−

*t*

_{2}, e.g. stretching of the fiber, were measured by above-threshold continous-wave operation of the laser, which showed RMS phase fluctuations, between points separated by 1/PRF, of ≤ 2 × 10

^{−7}rad, which is negligible on the scale of the quantum fluctuations described below.

## 4. Phase diffusion

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

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

*θ*

^{2}(

*t*)〉 is derived under various conditions from the Langevin equation where

*F*is a Langevin force responsible for the phase diffusion process,

_{θ}*α*= 5.4 is the linewidth enhancement factor first derived by C. Henry [19

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

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

*t*. Here

*R*is the spontaneous emission rate, which depends (see next paragraph) on the number of carriers

_{sp}*n*, and

*s*is the number of photons in the cavity [19

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

*R*and

_{sp}*s*are described by differential equations, given in [21

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

*s*to

*n*. where

*G*is the gain per carrier,

_{N}*n*

_{0}is the number of carriers at transparency,

*n*is the number of carriers at threshold,

_{th}*s*is the saturation photon number,

_{sat}*γ*is the carrier recombination lifetime = 1/

_{e}*τ*≈ 1 × 10

_{e}^{9},

*I*the injection current and

*q*is the electronic charge. To give values to these parameters, we note that

*G*=

_{N}*γ*/(

*n*−

_{th}*n*

_{0}), where

*γ*=

*c*(

*α*+

_{m}*α*)/

_{s}*n̄*is the cavity decay rate where

*c*is the speed of light in vacuum,

*n̄*= 4.33 is the effective refractive index,

*α*= 4.5

_{s}*cm*

^{−1}describes losses due to scattering and absorption and

*α*≈ 1.4/

_{m}*L*is the cavity escape loss, where

*L*is the cavity length. Because

*s*,

_{sat}*L*and

*n*

_{0}are unknown to us, we choose them by fitting to the observed pulse-shape (see below).

*R*can be written

_{sp}*R*=

_{sp}*nγ*

_{e}R_{0}, where

*R*

_{0}=

*K*

_{tot}Γ

_{conf}

*β*

_{SE}is a constant containing Γ

_{conf}the “confinement factor”,

*β*

_{SE}the fraction of spontaneous emission coupled to the lasing mode, and

*K*

_{tot}the “total enhancement factor” in a DFB LD, i.e. the enhancement in the fraction of spontaneous emission actually coupled to the lasing mode because of the geometry. We also do not have access to these parameters, but their combined effect is measurable using the steady-state power (see below).

*s*(

*t*) and

*n*(

*t*), permits the calculation of

*R*and therefore the dynamical evolution of the average phase diffusion 〈Δ

_{sp}*θ*

^{2}〉 in Eq. (4). In order to fit a solution of these equations to the measurements on our DFB LD, we first need to estimate the parameters

*s*,

_{sat}*L*,

*n*,

_{th}*G*,

_{N}*n*

_{0}and

*R*

_{0}(these are not all independent). Also, the solution is low pass filtered with a single-pole recursive filter with a time constant of

*τ*= 0.35/

_{f}*BW*, where

_{osc}*BW*= 12.5 GHz is the bandwidth of the oscilloscope.

_{osc}- We set
*s*to a reasonable value, and_{sat}*L*to one of 100, 200, 500 or 1000*μ*m. - We solve the steady-state solution of the differential equations, Eq. (7) and Eq. (8), near threshold, i.e., with
*n*≈*n*. The DFB LD emits_{th}*s*= 0.3 mW when it is biased with a constant current of_{th′}*I*= 10 mA. (The subscript_{th′}*th*refers to steady-state solution slightly above threshold). Eqs. (5) and (6) then take the form from which we extract_{′}*n*and_{th}*R*_{0}. - We choose the maximum
*G*, which mainly controls the speed of the dynamics, such that the predicted detected output power vs. time is always larger than the observed output power vs. time (from Fig. 1). Note that setting_{N}*G*directly specifies_{N}*n*_{0}, via*G*=_{N}*γ*/(*n*−_{th}*n*_{0}).

*s*= 7.7 × 10

_{sat}^{5},

*L*= 500

*μ*m,

*n*= 5.62 × 10

_{th}^{7},

*R*

_{0}= 8.8 × 10

^{−4},

*G*= 2.3 × 10

_{N}^{4}and

*n*

_{0}= 3.46 × 10

^{7}. The result is shown in Fig. 1. Note that phase diffusion is larger for low optical intensities. Hence, the fitting is conservative because the simulated optical power is always larger than the measurement.

*θ*

^{2}〉 > (9.45 rad)

^{2}. For all practical purposes this value describes a full randomization. We note that cos

*θ*is unchanged under

*θ*→

*θ*+ 2

*πn*(

*n*integer) and under

*θ*→ −

*θ*. If

*θ*is described by a Gaussian PDF

*G*(

*θ*) with width 〈Δ

*θ*

^{2}〉, then an equivalent distribution, for the purposes of the cosine, is

*θ*<

*π*and

*G*(

_{π}*θ*) ≡ 0 otherwise. Already with 〈Δ

*θ*

^{2}〉 = (2

*π*)

^{2},

*G*approximates a uniform distribution on [0,

_{π}*π*) with a fractional error below 10

^{−8}.

## 5. Raw data analysis

^{6}pulses were recorded on a fast 14-bit sampling oscilloscope. From each pulse a single sample was recorded, taken ∼ 13 ps after the pulse peak, to produce 120 × 10

^{6}14-bit numbers. This delay was chosen experimentally to give a broad distribution in Fig. 3(a). We observe experimentally that the distribution takes on the expected shape after power saturation occurs, i.e. after the pulse peak, presumably due to damping of transients associated with the rapid turn-on. These data were taken over a five day period as a test of stability and repeatability.

*μW*. This deviation around the mean arises from background noise and spontaneous emission events, producing narrow distributions for the interfering pulses. In contrast, due to interference of equal power optical pulses with random phases, the output power distribution is expected to broaden. The fact that the optical pulses have a predictable power and true random phase implies that when interfering these fields, random input phases of quantum origin are converted into macroscopically measurable random output intensities. This behaviour is shown in Fig. 3(a).

## 6. Randomness extraction and post-processing analysis

*X*distributed according to the distribution

*P*[

*X*=

*x*], the

_{i}*min-entropy*quantifies the amount of extractable randomness, in the sense that from a sequence {

*X*

_{1},...,

*X*} with

_{N}*N*≫ 1, a uniform random bit sequence of length

*NH*

_{∞}(

*X*) can be extracted. For our digitized input with

*b*bits of resolution, we define the reduction factor RF ≡

*b/H*

_{∞}(

*X*).

*ϕ*is the random phase due to phase diffusion. If this were the only fluctuating term, the observed signal would obey an arcsine distribution, the distribution of the cosine of a uniformly-distributed phase. Other fluctuating contributions to the signal, including photo-detection noise, laser current fluctuations, and digitization errors, broaden and smooth this ideal distribution to give the observed distribution shown in Fig. 3. These other noise sources are not guaranteed to be random. Fortunately, the randomness extraction step will remove any non-random effects of these other noise sources, provided that the reduction factor is chosen using the min-entropy of the quantum contribution. In what follows, we show that this quantum contribution can be determined from second order statistics of the measured observable and noise sources, plus our knowledge about the distribution of the phase.

### 6.1. Procedure

*u*

_{1},

*u*

_{2}and

*u*

^{(out)}, the variance at both sides of Eq. (2) is calculated. Assuming uncorrelation: From laser physics, we know that the intra-cavity field describes a diffusion process and therefore

*θ*is gaussianly distributed. Also, using measures and well-established models of semiconductor DFB lasers, an standard deviation larger than

*π*was estimated in Section 4. As described above, this justifies considering

*θ*to be uniformly distributed on [0,

*π*), and thus that cos

*θ*is described by an arcsine distribution. From Eq. (10) and Eq. (11), it then follows that where

*E*[·] indicates the mean value.

*u*

^{(out)}) = 1.4 mW

^{2}, var(

*u*

_{1}) = 2.0 × 10

^{−3}mW

^{2}, var(

*u*

_{2}) = 2.1 × 10

^{−3}mW

^{2},

*u*

_{noise}) = 1.45 × 10

^{−4}mW

^{2}. Applying Eq. (12) we find |

*g*(

*t*)| = 0.90.

_{loop}### 6.2. Min-entropy estimation

*u*

_{min}and

*u*

_{max}, where Because the arcsine distribution is peaked at

*x*=

*u*

_{min}, and because the min-entropy only depends on the weight of the most probable event, the min-entropy of the digitized arcsine distribution only depends on the probability of the first bin. If

*A*

_{ADC}is the dynamic range of the analog-to-digital converter (ADC),

*b*its resolution in bits so that Δ

*u*=

*A*

_{ADC}/2

*is the bin size, the probability mass of the first bin is Using the fact that arcsin*

^{b}*x*≈

*x*for

*x*small, it can easily be seen that the probability of the first bin decreases exponentially with

*b*and therefore the min-entropy, given by Eq. (16), increases linearly with the resolution of the digitization. In the experiment, where

*A*

_{ADC}= 5 mW,

*b*= 14 bits,

^{6}14-bit numbers to 125 × 10

^{6}7-bit numbers. The hashing is performed by the Whirlpool hash function [23

23. V. Rijmen and P. S. L. M. Barreto, “The WHIRLPOOL hash function,” http://www.larc.usp.br/~pbarreto/WhirlpoolPage.html (2008).

*δ*=

_{eu}*P*− 1/128, where

_{i}*P*is the probability of the

_{i}*i*-th bin. No statistically significant deviation from uniformity is observed.

### 6.3. Statistical testing

*α*

_{SL}) is set at 0.01. It means that 1 in a 100 sequences is expected to be rejected even if it is produced by a fair random generator. Using suggestions in [24], the proportion of accepted/rejected sequences and the uniformity of the

*P*-values are computed. As shown in Fig. 5, all tests are passed with a sequence of 1.5 Gbits.

*m*= 1500 is the number of 1 Mbit sequences tested, see Fig. 5(a). Second, the

*ε*-uniformity of the

*P*-values is examined. The idea is to compute

*P*

_{valueT}, a

*‘P-value of P-values’*. The procedure is as follows: for each test, (i) calculate a 10-bit histogram of

*P*-values, (ii) compute the

*χ*

^{2}defined in Eq. (17) and (iii) calculate the incomplete gamma statistical function Γ(9/2,

*χ*

^{2}/2), which must be larger than 10

^{−4}, see Fig. 5(b). where

*s*is the number of

*P*-values per test and

*F*is the number of

_{i}*P*-values in the

*i*-th bin.

## 7. Conclusions

## Acknowledgments

## References and links

1. | A. Tajima, A. Tanaka, W. Maeda, S. Takahashi, and A. Tomita, “Practical quantum cryptosystem for metro area applications,” IEEE J. Sel. Top. Quantum Electron. |

2. | X. Cai and 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. |

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

4. | C. Petrie and J. Connelly, “A noise-based ic random number generator for applications in cryptography,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. |

5. | I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics |

6. | A. Argyris, E. Pikasis, S. Deligiannidis, and D. Syvridis, “Sub-tb/s physical random bit generators based on direct detection of amplified spontaneous emission signals,” J. Lightwave Technol. |

7. | Y. Yoshizawa, H. Kimura, H. Inoue, K. Fujita, M. Toyama, and O. Miyatake, “Physical random numbers generated by radioactivity,” J. Jpn. Soc. Comput. Stat. |

8. | T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Intrum. |

9. | O. Kwon, Y.-W. Cho, and Y.-H. Kim, “Quantum random number generator using photon-number path entanglement,” Appl. Opt. |

10. | 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. Express |

11. | H. Guo, W. Tang, Y. Liu, and W. Wei, “Truly random number generation based on measurement of phase noise of a laser,” Phys. Rev. E |

12. | 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. Express |

13. | F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Opt. Express |

14. | M. Wahl, M. Leifgen, M. Berlin, T. Röhlicke, 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. |

15. | A. Marandi, N. C. Leindecker, K. L. Vodopyanov, and R. L. Byer, “All-optical quantum random bit generation from intrinsically binary phase of parametric oscillators,” Opt. Express |

16. | X.-Z. Li and S.-C. Chan, “Random bit generation using an optically injected semiconductor laser in chaos with oversampling,” Opt. Lett. |

17. | 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 |

18. | 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, and C. Monroe, “Random numbers certified by Bell’s theorem,” Nature |

19. | C. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. |

20. | C. Henry, “Phase noise in semiconductor lasers,” J. Lightwave Technol. |

21. | G. Agrawal, “Effect of gain and index nonlinearities on single-mode dynamics in semiconductor lasers,” IEEE J. Quantum Electron. |

22. | N. Nisan and A. Ta-Shma, “Extracting randomness: A survey and new constructions,” J. Comput. Sci. Tech. |

23. | V. Rijmen and P. S. L. M. Barreto, “The WHIRLPOOL hash function,” http://www.larc.usp.br/~pbarreto/WhirlpoolPage.html (2008). |

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

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(230.0230) Optical devices : Optical devices

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: November 6, 2013

Revised Manuscript: December 30, 2013

Manuscript Accepted: December 31, 2013

Published: January 16, 2014

**Citation**

C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acín, J. Capmany, V. Pruneri, and M. W. Mitchell, "Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode," Opt. Express **22**, 1645-1654 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-2-1645

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### References

- 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]
- C. Hall, B. Schneier, “Remote electronic gambling,” in “13th Annual Computer Security Applications Conference” (1997), pp. 232–238. [CrossRef]
- 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]
- I. Kanter, Y. Aviad, I. Reidler, E. Cohen, M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2010). [CrossRef]
- A. Argyris, E. Pikasis, S. Deligiannidis, D. Syvridis, “Sub-tb/s physical random bit generators based on direct detection of amplified spontaneous emission signals,” J. Lightwave Technol. 30, 1329–1334 (2012). [CrossRef]
- 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).
- 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]
- O. Kwon, Y.-W. Cho, Y.-H. Kim, “Quantum random number generator using photon-number path entanglement,” Appl. Opt. 48, 1774–1778 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Opt. Express 20, 12366–12377 (2012). [CrossRef] [PubMed]
- 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]
- A. Marandi, N. C. Leindecker, K. L. Vodopyanov, R. L. Byer, “All-optical quantum random bit generation from intrinsically binary phase of parametric oscillators,” Opt. Express 20, 19322–19330 (2012). [CrossRef] [PubMed]
- X.-Z. Li, S.-C. Chan, “Random bit generation using an optically injected semiconductor laser in chaos with oversampling,” Opt. Lett. 37, 2163–2165 (2012). [CrossRef] [PubMed]
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