## Fast physical random number generator using amplified spontaneous emission |

Optics Express, Vol. 18, Issue 23, pp. 23584-23597 (2010)

http://dx.doi.org/10.1364/OE.18.023584

Acrobat PDF (1392 KB)

### Abstract

We report a 12.5 Gb/s physical random number generator (RNG) that uses high-speed threshold detection of the spectrally-sliced incoherent light produced by a fiber amplifier. The system generates a large-amplitude, easily measured, fluctuating signal with bandwidth that is constrained only by the optical filter and electrical detector used. The underlying physical process (spontaneous emission) is inherently quantum mechanical in origin, and therefore cannot be described deterministically. Unlike competing optical RNG approaches that require photon counting electronics, chaotic laser cavities, or state-of-the-art analog-to-digital converters, the system employs only commonly available telecommunications-grade fiber optic components and can be scaled to higher speeds or multiplexed into parallel channels. The quality of the resulting random bitstream is verified using industry-standard statistical tests.

© 2010 Optical Society of America

## 1. Introduction

1. A. M. Ferrenberg, D. P. Landau, and Y. J. Wong, “Monte Carlo simulations: hidden errors from ‘good’ random number generators,” Phys. Rev. Lett. **69**, 3382–3384 (1992). [CrossRef] [PubMed]

2. M. Isida and H. Ikeda, “Random number generator,” Ann. Inst. Stat. Math. **8**, 119–126 (1956). [CrossRef]

3. J. Walker, “HotBits: Genuine random numbers, generated by radioactive decay,” Online: http://www.fourmilab.ch/hotbits/.

4. W. T. Holman, J. A. Connelly, and A. B. Dowlatabadi, “An integrated analog/digital random noise source,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. **44**, 521–528 (1997). [CrossRef]

5. P. Xu, Y. Wong, T. Horiuchi, and P. Abshire, “Compact floating-gate true random number generator,” Electron. Lett. **42**, 1346 –1347 (2006). [CrossRef]

6. 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]

8. M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanonuovo, “A high-speed oscillator-based truly random number source for cryptographic applications on a smart card IC,” IEEE Trans. Comput. **52**, 403–409 (2003). [CrossRef]

9. G. Bernstein and M. Lieberman, “Secure random number generation using chaotic circuits,” IEEE Trans. Circuits Syst. **37**, 1157–1164 (1990). [CrossRef]

11. T. Stojanovski, J. Pihl, and L. Kocarev, “Chaos-based random number generators – Part II: practical realization,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. **48**, 382–385 (2001). [CrossRef]

12. M. Haahr, “Random.org: True Random Number Service,” Online: http://www.random.org/.

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

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

15. C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nature Photon. **4**, 711–715 (2010). [CrossRef]

16. L. C. Noll and S. Cooper, “What is LavaRnd?” Online: http://www.lavarnd.org/.

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

18. 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]

19. 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,” Nature Photon. **2**, 728–732 (2008). [CrossRef]

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

23. K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express18, 5512–5524 (2010). http://www.opticsexpress.org/abstract.cfm?URI=oe-18-6-5512. [CrossRef] [PubMed]

## 2. Theory

*u*(

*t*) is taken to be white noise generated by amplified spontaneous emission with a power spectral density of

*S*

_{0}. We assume that the noise is polarized, both to simplify the analysis and also because that is how our experimental system is constructed. The noise passes through an optical bandpass filter that has a (dimensionless) complex transfer function

*H*

_{BP}(

*f*), so that the power spectral density of the emerging optical signal is

*S*

_{0}|

*H*

_{BP}(

*f*)|

^{2}. The photodiode produces an electrical current proportional to the squared magnitude of the optical field, and the resulting photocurrent is passed through a low-pass filter with transfer function

*H*

_{LP}(

*f*).

*B*

_{BP}and

*B*

_{LP}represent the 3 dB bandwidths of the bandpass and lowpass filters, respectively.

*H*

_{LP}(0) is the DC gain of lowpass filter, and Eq. (2b) gives the specific result for the case of Gaussian filters. Because the responsivity ℛ is typically measured at DC frequencies, one typically takes

*H*

_{LP}(0) = 1 with the assumption that any DC filter attenuation has been factored into ℛ.

24. N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. **7**, 1071–1082 (1989). [CrossRef]

25. R. C. Steele, G. R. Walker, and N. G. Walker, “Sensitivity of optically preamplified receivers with optical filtering,” IEEE Photon. Technol. Lett. **3**, 545–547 (1991). [CrossRef]

*S*(

_{i}*f*), which would appear as a term proportional to 〈

*i*〉

^{2}

*δ*(

*f*). Thus, Eq. (3a) represents the power spectral density of the zero-mean process

*i*(

*t*) − 〈

*i*〉.

26. M. S. Leeson, “Performance analysis of direct detection spectrally sliced receivers using Fabry-Perot filters,” J. Lightwave Technol. **18**, 13–25 (2000). [CrossRef]

29. A. J. Keating and D. D. Sampson, “Reduction of excess intensity noise in spectrum-sliced incoherent light for WDM applications,” J. Lightwave Technol. **15**, 53–61 (1997). [CrossRef]

*a*describes the signal to noise ratio [30

30. J.-S. Lee, “Signal-to-noise ratio of spectrum-sliced incoherent light sources including optical modulation effects,” J. Lightwave Technol. **14**, 2197–2201 (1996). [CrossRef]

*a*) depends only on the shapes of the optical and electrical filters employed.

*i*〉 cannot be too large, or else the photoreciever will saturate, producing only a DC output with no noise. This saturation will occur even if the output signal is AC-coupled. Therefore, in order to produce a strong electrical noise signal at the output without saturating the photoreceiver, one seeks to minimize the signal-to-noise ratio. From Eq. (7b), this can only be achieved by choosing bandpass and lowpass filters that have comparable bandwidths.

## 3. Experimental System

*λ*

_{0}= 1552.5 nm. The resulting filtered noise signal is then amplified in a low-noise erbium-doped fiber amplifier (MPB EFA-R35W). A fiber polarization splitter divides the noise into independent, identically distributed, orthogonally polarized noise signals that are separately detected in a pair of matched photoreceivers (Discovery DSC-R402). Each photoreceiver consists of a photodiode with responsivity of ℛ = 0.8 A/W followed by a transimpedance amplifier with a gain of 500 V/A. The photoreceivers have an electrical bandwidth of 11 GHz, and the transimpedance amplifiers are AC coupled with a cut-on frequency of 30 kHz. Variable optical attenuators were used to adjust the total noise power, and also to balance the noise power in the two orthogonal polarization arms. Because amplified spontaneous emission is generated in both polarization states with equal intensity, we do not require precise polarization control or tracking in order to maintain an acceptable balance between the two arms of the system. The DC photocurrent in each photodiode was adjusted to be 0.77 mA.

*v*

_{1}(

*t*) and

*v*

_{2}(

*t*) were connected to the differential logic inputs (

*X*and

*X̄*) of a bit error rate tester (BERT). In this configuration, the BERT may be thought of as performing a clocked comparison of the two input signals, producing a logical one when

*v*

_{1}(

*t*) >

*v*

_{2}(

*t*) and a logical zero otherwise. An external 12.5 GHz clock signal supplied to the BERT determines the sampling frequency and bit generation rate. A DC bias voltage may be optionally added to either of the input signals, to control the comparison threshold.

## 4. Noise Characterization

*H*

_{BP}(

*f*)|

^{2}* |

*H*

_{BP}(–

*f*)|

^{2}[25

25. R. C. Steele, G. R. Walker, and N. G. Walker, “Sensitivity of optically preamplified receivers with optical filtering,” IEEE Photon. Technol. Lett. **3**, 545–547 (1991). [CrossRef]

*H*

_{LP}(

*f*)|

^{2}. The photoreceiver spectral response was measured by exciting the detector with a 200 fs pulses from an 80 MHz mode-locked laser system, and observing the resulting 80 MHz comb of spectral lines on an RF spectrum analyzer. The spectra shown in Figs. 4a–b are both normalized to a DC value of 0 dB. Finally, in Fig. 4c, we show the electrical spectrum of the ASE noise from one detector, measured with a resolution bandwidth of 3 MHz. For comparison, we also show the computed noise spectrum obtained by multiplying the two traces from (a) and (b), as described in Eq. (2a), which closely matches the measured spectrum. The computed spectrum was scaled in order to match the DC value observed in the measurement. The final noise spectrum has a bandwidth of 7.5 GHz, which agrees with the result calculated from Eq. (4) using

*B*

_{BP}= 14.5 GHz and

*B*

_{LP}= 11 GHz. The dotted black line in Fig. 4c shows the background electrical noise spectrum obtained by completely extinguishing the optical signal. Over the frequency range of interest, the electrical noise is more than 40 dB smaller than the optical noise produced by ASE.

*a*= 1.44, which is in reasonable agreement with the result of 1.37 predicted from Eq. (5b).

*v*

_{1}(

*t*) and

*v*

_{2}(

*t*) are detected differentially by the bit error rate tester, which assigns a one or zero based on the difference signal

*v*

_{1}(

*t*) –

*v*

_{2}(

*t*). Fig. 5c shows the calculated difference between the two channels and the corresponding statistical distribution of voltages. Unlike the single channels shown in Fig. 5a–b, the differential voltage has a symmetric distribution, with a mean and median of 0. The theoretical distribution was numerically calculated by performing a self-correlation of the gamma distribution shown in Figs. 5a–b. The balanced detection scheme is insensitive to common-mode interference and drift – even if the source power changes, the decision threshold does not need to be adjusted in order to produce an unbiased bit sequence. Although the fluctuations produced here are macroscopic and unpredictable, we note that for cryptographic applications the security of the resulting bit sequence assumes that a would-be adversary does have access to the physical system or intermediate optical or electrical signals.

^{9}bit sequence used in subsequent statistical testing.

## 5. Statistical Testing

*k*(or time delay

*τ*) for a 10

^{9}-bit random sequence produced by our system. The normalized correlation at lag

*k*was calculated in the following way where 〈•〉 denotes a statistical average over the

*N*bits of the binary sequence

*b*[

*n*]. When computing the average 〈

*b*[

*n*]

*b*[

*n*+

*k*]〉, the

*N*-bit sequence

*b*[

*n*] is assumed to repeat with a period of

*N*, e.g.,

*b*[

*N*+

*k*] =

*b*[

*k*]. The correlation

*ρ*defined in Eq. (8) is a symmetric function of the lag

_{k}*k*, with

*ρ*

_{0}= 1. For a finite length sequence of

*N*ideal, independent, unbiased bits, the correlation calculated by Eq. (8) has an expected value that decreases as (−1/

*N*) and a standard deviation that decreases as

*N*= 10

^{9}, we therefore expect the correlation for

*k*≠ 0 to be statistically centered about 0 with a standard deviation of 3.16 × 10

^{−5}.

*k*, even for large lags. Without the XOR processing, the small but statistically significant correlation seen in Fig. 6a would cause the raw bit sequence to fail several of the statistical tests.

5. P. Xu, Y. Wong, T. Horiuchi, and P. Abshire, “Compact floating-gate true random number generator,” Electron. Lett. **42**, 1346 –1347 (2006). [CrossRef]

6. 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]

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

19. 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,” Nature Photon. **2**, 728–732 (2008). [CrossRef]

*p*and correlation of

*ρ*, the binary sequence obtained by computing the XOR will have a mark ratio and correlation of If the original sequences are unbiased, then the XOR process will produce an unbiased sequence with new correlation

_{k}*b*[

*n*] ⊕

*b*[

*n*– 20]. The resulting sequence exhibits a correlation near the statistical noise level, with no discernible pattern or trend. Although we computed the XOR using off-line postprocessing, it could easily be implemented in real-time using simple high-speed logic operations. The lagged XOR process does not require more than 20 bits of delay, and does not reduce the generation rate.

32. A. Rukhin, J. Soto, J. Nechvatal, M. Smid, 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)*, National Institute of Standards and Technology (2010).

*p*-value” that, for a truly random bit sequence, would be uniformly distributed between 0 and 1. The NIST test suite applies each test to 1000 sequences (a total of 10

^{9}bits) and then computes a single composite

*p*-value to assess whether the constituent

*p*-values are uniformly distributed. For a truly random sequence, the composite

*p*-value should also be uniformly distributed between 0 and 1. The composite

*p*-values must all exceed 10

^{−4}in order to pass the NIST test. Furthermore, of the 1000 individual

*p*-values obtained for each test, no fewer than 1 nor more than 19 may fall below the threshold of

*α*= 0.01. Fig. 7 plots the results of the NIST tests applied to the 10

^{9}bit XORed data sequence. For tests that produce multiple composite

*p*-values, all are shown in Fig. 7a. The number of tests (out of 1000) with

*p*< 0.01 is plotted in Fig. 7b. For tests that produce multiple results, the numbers are shown as a grayscale histogram. The XORed data set passes all of the NIST statistical tests.

33. G. Marsaglia, “DIEHARD: A battery of tests of randomness,” Online: http://www.stat.fsu.edu/pub/diehard/ (1996).

*p*-value that, for a random sequence, would be uniformly distributed between 0 and 1. For some tests, the Diehard suite computes a composite

*p*-value using the Kolmogorov-Smirnov (K-S) test to asses the degree of uniformity. In Fig. 8 we plot the results of the Diehard tests.

*p*-values obtained from the K-S test are indicated by thick red lines. Where available, the individual

*p*-values from which the composite was calculated are shown by the thin blue lines. In order to pass each test, the computed

*p*-values (or, where available, the K-S

*p*-value) must all exceed 10

^{−4}.

34. 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]

## 6. Improving Generation Rate with Analog-to-Digital Conversion

21. A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express18, 18763–18768 (2010). http://www.opticsexpress.org/abstract.cfm?URI=oe-18-18-18763. [CrossRef] [PubMed]

23. K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express18, 5512–5524 (2010). http://www.opticsexpress.org/abstract.cfm?URI=oe-18-6-5512. [CrossRef] [PubMed]

23. K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express18, 5512–5524 (2010). http://www.opticsexpress.org/abstract.cfm?URI=oe-18-6-5512. [CrossRef] [PubMed]

35. R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. Sel. Areas Commun. **17**, 539–550 (1999). [CrossRef]

*x*[

*n*] (in two’s-complement format) taken from these records, we computed a 9-th order discrete derivative (using 32-bit, two’s-complement arithmetic), and retained only the 8 least significant bits of the resulting sequence [22

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

*y*[

*n*] at a rate of 50 GHz, for a cumulative random generation rate of 400 Gb/s (or 800 Gb/s if one considers both orthogonal polarization channels.) The resulting sequence was confirmed to pass all of the standard NIST and Diehard tests for randomness. Next, we completely extinguished the optical signal and performed the same process using only the background electrical noise present in our system. The resulting sequence

*also*passed all of the NIST and Diehard statistical tests.

## 7. Conclusion

## 8. Acknowledgements

## References and links

1. | A. M. Ferrenberg, D. P. Landau, and Y. J. Wong, “Monte Carlo simulations: hidden errors from ‘good’ random number generators,” Phys. Rev. Lett. |

2. | M. Isida and H. Ikeda, “Random number generator,” Ann. Inst. Stat. Math. |

3. | J. Walker, “HotBits: Genuine random numbers, generated by radioactive decay,” Online: http://www.fourmilab.ch/hotbits/. |

4. | W. T. Holman, J. A. Connelly, and A. B. Dowlatabadi, “An integrated analog/digital random noise source,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. |

5. | P. Xu, Y. Wong, T. Horiuchi, and P. Abshire, “Compact floating-gate true random number generator,” Electron. Lett. |

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

7. | B. Jun and P. Kocher, “The Intel Random Number Generator,” Cryptography Research Inc., white paper prepared for Inter Corp. (1999). |

8. | M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanonuovo, “A high-speed oscillator-based truly random number source for cryptographic applications on a smart card IC,” IEEE Trans. Comput. |

9. | G. Bernstein and M. Lieberman, “Secure random number generation using chaotic circuits,” IEEE Trans. Circuits Syst. |

10. | T. Stojanovski and L. Kocarev, “Chaos-based random number generators – Part I: analysis,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. |

11. | T. Stojanovski, J. Pihl, and L. Kocarev, “Chaos-based random number generators – Part II: practical realization,” IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. |

12. | M. Haahr, “Random.org: True Random Number Service,” Online: http://www.random.org/. |

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

14. | J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. |

15. | C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nature Photon. |

16. | L. C. Noll and S. Cooper, “What is LavaRnd?” Online: http://www.lavarnd.org/. |

17. | B. Qi, Y.-M. Chi, H.-K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. Lett. |

18. | 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 |

19. | 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,” Nature Photon. |

20. | I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. |

21. | A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express18, 18763–18768 (2010). http://www.opticsexpress.org/abstract.cfm?URI=oe-18-18-18763. [CrossRef] [PubMed] |

22. | I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nature Photon. |

23. | K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express18, 5512–5524 (2010). http://www.opticsexpress.org/abstract.cfm?URI=oe-18-6-5512. [CrossRef] [PubMed] |

24. | N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. |

25. | R. C. Steele, G. R. Walker, and N. G. Walker, “Sensitivity of optically preamplified receivers with optical filtering,” IEEE Photon. Technol. Lett. |

26. | M. S. Leeson, “Performance analysis of direct detection spectrally sliced receivers using Fabry-Perot filters,” J. Lightwave Technol. |

27. | J. W. Goodman, |

28. | P. A. Humblet and M. Azizog̃lu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. |

29. | A. J. Keating and D. D. Sampson, “Reduction of excess intensity noise in spectrum-sliced incoherent light for WDM applications,” J. Lightwave Technol. |

30. | J.-S. Lee, “Signal-to-noise ratio of spectrum-sliced incoherent light sources including optical modulation effects,” J. Lightwave Technol. |

31. | D. Knuth, |

32. | A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, |

33. | G. Marsaglia, “DIEHARD: A battery of tests of randomness,” Online: http://www.stat.fsu.edu/pub/diehard/ (1996). |

34. | 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 |

35. | R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. Sel. Areas Commun. |

**OCIS Codes**

(030.6600) Coherence and statistical optics : Statistical optics

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

(230.2285) Optical devices : Fiber devices and optical amplifiers

**ToC Category:**

Optical Devices

**History**

Original Manuscript: September 9, 2010

Revised Manuscript: October 19, 2010

Manuscript Accepted: October 20, 2010

Published: October 26, 2010

**Citation**

Caitlin R. S. Williams, Julia C. Salevan, Xiaowen Li, Rajarshi Roy, and Thomas E. Murphy, "Fast physical random number generator using amplified spontaneous emission," Opt. Express **18**, 23584-23597 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-23584

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### References

- A. M. Ferrenberg, D. P. Landau, and Y. J. Wong, “Monte Carlo simulations: hidden errors from ‘good’ random number generators,” Phys. Rev. Lett. 69, 3382–3384 (1992). [CrossRef] [PubMed]
- M. Isida, and H. Ikeda, “Random number generator,” Ann. Inst. Stat. Math. 8, 119–126 (1956). [CrossRef]
- J. Walker, “HotBits: Genuine random numbers, generated by radioactive decay,” Online: http://www.fourmilab.ch/hotbits/.
- W. T. Holman, J. A. Connelly, and A. B. Dowlatabadi, “An integrated analog/digital random noise source,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 44, 521–528 (1997). [CrossRef]
- P. Xu, Y. Wong, T. Horiuchi, and P. Abshire, “Compact floating-gate true random number generator,” Electron. Lett. 42, 1346–1347 (2006). [CrossRef]
- 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]
- B. Jun, and P. Kocher, “The Intel Random Number Generator,” Cryptography Research Inc., white paper prepared for Inter Corp. (1999).
- M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanonuovo, “A high-speed oscillator-based truly random number source for cryptographic applications on a smart card IC,” IEEE Trans. Comput. 52, 403–409 (2003). [CrossRef]
- G. Bernstein, and M. Lieberman, “Secure random number generation using chaotic circuits,” IEEE Trans. Circ. Syst. 37, 1157–1164 (1990). [CrossRef]
- T. Stojanovski, and L. Kocarev, “Chaos-based random number generators – Part I: analysis,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 48, 281–288 (2001). [CrossRef]
- T. Stojanovski, J. Pihl, and L. Kocarev, “Chaos-based random number generators – Part II: practical realization,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 48, 382–385 (2001). [CrossRef]
- M. Haahr, “Random.org: True Random Number Service,” Online: http://www.random.org/.
- T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum. 71, 1675–1680 (2000). [CrossRef]
- J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, postprocessing free, quantum random number generator,” Appl. Phys. Lett. 93, 031109 (2008). [CrossRef]
- C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4, 711–715 (2010). [CrossRef]
- L. C. Noll, and S. Cooper, “What is LavaRnd?” Online: http://www.lavarnd.org/.
- B. Qi, Y.-M. Chi, H.-K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. Lett. 35, 312–314 (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. E Stat. Nonlin. Soft Matter Phys. 81, 051137 (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. Photonics 2, 728–732 (2008). [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, 024102 (2009). [CrossRef] [PubMed]
- A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express 18, 18763–18768 (2010), http://www.opticsexpress.org/abstract.cfm?URI=oe-18-18-18763. [CrossRef] [PubMed]
- I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2010). [CrossRef]
- K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18, 5512–5524 (2010), http://www.opticsexpress.org/abstract.cfm?URI=oe-18-6-5512. [CrossRef] [PubMed]
- N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7, 1071–1082 (1989). [CrossRef]
- R. C. Steele, G. R. Walker, and N. G. Walker, “Sensitivity of optically preamplified receivers with optical filtering,” IEEE Photon. Technol. Lett. 3, 545–547 (1991). [CrossRef]
- M. S. Leeson, “Performance analysis of direct detection spectrally sliced receivers using Fabry-Perot filters,” J. Lightwave Technol. 18, 13–25 (2000). [CrossRef]
- J. W. Goodman, Statistical Optics (Wiley, 1985). p. 246.
- P. A. Humblet and M. Azizõglu, “On the bit error rate of lightwave systems with optical amplifiers,” J. Lightwave Technol. 9, 1576–1582 (1991). [CrossRef]
- A. J. Keating, and D. D. Sampson, “Reduction of excess intensity noise in spectrum-sliced incoherent light for WDM applications,” J. Lightwave Technol. 15, 53–61 (1997). [CrossRef]
- J.-S. Lee, “Signal-to-noise ratio of spectrum-sliced incoherent light sources including optical modulation effects,” J. Lightwave Technol. 14, 2197–2201 (1996). [CrossRef]
- D. Knuth, The Art of Computer Programming, Volume 2: Seminumerical Algorithms (3rd ed.) (Addison-Wesley, 1996). pp. 64–65.
- A. Rukhin, J. Soto, J. Nechvatal, M. Smid, 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), National Institute of Standards and Technology (2010).
- G. Marsaglia, “DIEHARD: A battery of tests of randomness,” Online: http://www.stat.fsu.edu/pub/diehard/ (1996).
- 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]
- R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. Sel. Areas Comm. 17, 539–550 (1999). [CrossRef]

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