## Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers

Optics Express, Vol. 18, Issue 6, pp. 5512-5524 (2010)

http://dx.doi.org/10.1364/OE.18.005512

Acrobat PDF (528 KB)

### Abstract

We experimentally demonstrate random bit generation using multi-bit samples of bandwidth-enhanced chaos in semiconductor lasers. Chaotic fluctuation of laser output is generated in a semiconductor laser with optical feedback and the chaotic output is injected into a second semiconductor laser to obtain a chaotic intensity signal with bandwidth enhanced up to 16 GHz. The chaotic signal is converted to an 8-bit digital signal by sampling with a digital oscilloscope at 12.5 Giga samples per second (GS/s). Random bits are generated by bitwise exclusive-OR operation on corresponding bits in samples of the chaotic signal and its time-delayed signal. Statistical tests verify the randomness of bit sequences obtained using 1 to 6 bits per sample, corresponding to fast random bit generation rates from 12.5 to 75 Gigabit per second (Gb/s) ( = 6 bit × 12.5 GS/s).

© 2010 OSA

## 1. Introduction

1. D. Eastlake, J. Schiller, and S. Crocker, “Randomness requirements for security,” RFC4086 (2005) http://tools.ietf.org/html/rfc4086

2. Security requirements for cryptographic modules. FIPS 140–2 (2001) http://csrc.nist.gov/publications/fips/fips140-2/fips1402.pdf

3. N. Gisin, G. Robordy, W. Tittel, and H. Zbinden, “Quantum Cryptography,” Rev. Mod. Phys. **74**(1), 145–195 (2002). [CrossRef]

4. N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. **44**(247), 335–341 (1949). [CrossRef] [PubMed]

6. W. T. Holman, J. A. Connelly, and A. B. Dowlatabadi, “An integrated analog/digital random noise source,” IEEE Trans. Circuits Syst. I **44**(6), 521–528 (1997). [CrossRef]

10. 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**(3), 312–314 (2010). [CrossRef] [PubMed]

8. M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanouvo, “A high-speed oscillator-based truly random number source for cryptographic applications on a Smart Card IC,” IEEE Trans. Comput. **52**(4), 403–409 (2003). [CrossRef]

10. 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**(3), 312–314 (2010). [CrossRef] [PubMed]

*in real time*by directly sampling the output of two chaotic semiconductor lasers with one-bit analog-digital converters [11

11. 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**(12), 728–732 (2008). [CrossRef]

13. K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. **45**(11), 1367–1379 (2009). [CrossRef]

14. T. Honjo, A. Uchida, K. Amano, K. Hirano, H. Someya, H. Okumura, K. Yoshimura, P. Davis, and Y. Tokura, “Differential-phase-shift quantum key distribution experiment using fast physical random bit generator with chaotic semiconductor lasers,” Opt. Express **17**(11), 9053–9061 (2009). [CrossRef] [PubMed]

15. I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. **103**(2), 1–4 (2009). [CrossRef]

16. Y. Takiguchi, K. Ohyagi, and J. Ohtsubo, “Bandwidth-enhanced chaos synchronization in strongly injection-locked semiconductor lasers with optical feedback,” Opt. Lett. **28**(5), 319–321 (2003). [CrossRef] [PubMed]

20. H. Someya, I. Oowada, H. Okumura, T. Kida, and A. Uchida, “Synchronization of bandwidth-enhanced chaos in semiconductor lasers with optical feedback and injection,” Opt. Express **17**(22), 19536–19543 (2009). [CrossRef] [PubMed]

16. Y. Takiguchi, K. Ohyagi, and J. Ohtsubo, “Bandwidth-enhanced chaos synchronization in strongly injection-locked semiconductor lasers with optical feedback,” Opt. Lett. **28**(5), 319–321 (2003). [CrossRef] [PubMed]

20. H. Someya, I. Oowada, H. Okumura, T. Kida, and A. Uchida, “Synchronization of bandwidth-enhanced chaos in semiconductor lasers with optical feedback and injection,” Opt. Express **17**(22), 19536–19543 (2009). [CrossRef] [PubMed]

21. G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science **279**(5354), 1198–1200 (1998). [CrossRef] [PubMed]

23. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature **437**(7066), 343–346 (2005). [CrossRef]

24. F.-Y. Lin and J.-M. Liu, “Chaotic lider,” IEEE J. Sel. Top. Quantum Electron. **10**(5), 991–997 (2004). [CrossRef]

## 2. Experimental setup

*I*for solitary Laser 1 and 2 were 9.43 and 9.31 mA, respectively. Both Laser 1 and 2 were prepared without standard optical isolators, to allow optical feedback and injection. Laser 1 was connected to a fiber coupler and a variable fiber reflector which reflects a fraction of the light back into the laser, inducing high-frequency chaotic oscillations of the optical intensity. The amount of the optical feedback light was adjusted by the variable fiber reflector. The fiber length between Laser 1 and the variable fiber reflector was 4.55 m, corresponding to a feedback delay time (round-trip) of 43.8 ns. On the other hand, there was no optical feedback for Laser 2. Polarization maintaining fibers were used for all the optical fiber components.

_{th}## 3. Experimental results

### 3.1 Bandwidth enhancement of chaos

*I*) and 59.00 mA (6.34

_{th}*I*), respectively. To enhance the bandwidth of chaos, we detuned the optical wavelength of Laser 2 to the positive direction with respect to that of Laser 1, i.e., we set the optical wavelength to be 1547.585 nm for Laser 1 and 1547.677 nm for Laser 2, by controlling the temperature of the two lasers. The optical wavelength for Laser 2 was shifted to 1547.718 nm due to the presence of the optical injection from Laser 1. The optical wavelength detuning was defined as Δλ = λ

_{th}_{2}- λ

_{1}, where λ

_{1}and λ

_{2}indicate the optical wavelength of Laser 1 and 2 in the presence of the optical injection, respectively. Δλ was set to 0.133 nm (16.6 GHz in frequency), which corresponds to the positive detuning condition in the literature [25]. At this condition, no injection locking was achieved between Laser 1 and 2, where injection locking was defined as the matching of optical wavelengths between the two lasers due to the coherent unidirectional coupling. The existence of the frequency component corresponding to the optical wavelength detuning is crucial for the bandwidth enhancement of the laser chaos [20

20. H. Someya, I. Oowada, H. Okumura, T. Kida, and A. Uchida, “Synchronization of bandwidth-enhanced chaos in semiconductor lasers with optical feedback and injection,” Opt. Express **17**(22), 19536–19543 (2009). [CrossRef] [PubMed]

**17**(22), 19536–19543 (2009). [CrossRef] [PubMed]

18. F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. **221**, 173–180 (2003). [CrossRef]

**17**(22), 19536–19543 (2009). [CrossRef] [PubMed]

13. K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. **45**(11), 1367–1379 (2009). [CrossRef]

11. 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**(12), 728–732 (2008). [CrossRef]

15. I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. **103**(2), 1–4 (2009). [CrossRef]

### 3.2 Generation of random bits

*m*least significant bits (LSBs) from each sample are then selected and interleaved to generate a single bit sequence [15

15. I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. **103**(2), 1–4 (2009). [CrossRef]

*s*or

_{k}= 0*1 (k = 1, 2, ..., m)*is the

*k*-th LSB of the selected

*m*LSBs. The method is illustrated in Fig. 3 .

26. A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, and M. Levenson, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” National Institute of Standards and Technology, Special Publication 800–22, 2001; Revision 1, August 2008. http://csrc.nist.gov/publications/nistpubs/800-22-rev1/SP800-22rev1.pdf

26. A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, and M. Levenson, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” National Institute of Standards and Technology, Special Publication 800–22, 2001; Revision 1, August 2008. http://csrc.nist.gov/publications/nistpubs/800-22-rev1/SP800-22rev1.pdf

28. G. Marsaglia, DIEHARD: A battery of tests of randomness. http://stat.fsu.edu/ geo, 1996.

## 4. Statistical characteristics of random bits

### 4.1 Probability density function

*m*LSBs (see Sec. 4.2).

### 4.2 NIST tests of sequences generated using multiple significant bits

^{−4}to pass the test. The P-value gradually decreases as the number of LSBs increases. Both the runs and block-frequency tests are passed when the length of LSBs is set to be up to 6 LSBs.

### 4.3 NIST tests of sequences generated using single significant bits

^{−1}and 10

^{0}for the significant bits 1 to 7. Both the runs and block-frequency tests are passed by all significant bits up to 7. Figure 9(b) shows

*V(obs)*as a function of the significant bit.

*V(obs)*is close to 50% for the significant bits 1 to 7, and decreases significantly for the significant bit 8 (i.e., MSB). The 0/1 ratio of the MSB is more sensitively dependent on the match between the signal amplitudes and the range of the 8-bit detectors, compared with the other bits. We do not exclude the possibility of improving the test performance of MSB sequences by adjusting the signal offset or amplitude with respect to the MSB detection threshold [11

11. 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**(12), 728–732 (2008). [CrossRef]

13. K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. **45**(11), 1367–1379 (2009). [CrossRef]

29. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics **4**(1), 58–61 (2010). [CrossRef]

## 5. Conclusion

## Acknowledgments

## References and links

1. | D. Eastlake, J. Schiller, and S. Crocker, “Randomness requirements for security,” RFC4086 (2005) http://tools.ietf.org/html/rfc4086 |

2. | Security requirements for cryptographic modules. FIPS 140–2 (2001) http://csrc.nist.gov/publications/fips/fips140-2/fips1402.pdf |

3. | N. Gisin, G. Robordy, W. Tittel, and H. Zbinden, “Quantum Cryptography,” Rev. Mod. Phys. |

4. | N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. |

5. | D. Knuth, “ |

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

7. | J. T. Gleeson, “Truly random number generator based on turbulent electroconvection,” Appl. Phys. Lett. |

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

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

10. | 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. |

11. | 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 |

12. | T. E. Murphy and R. Roy, “The world’s fastest dice,” Nat. Photonics |

13. | K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. |

14. | T. Honjo, A. Uchida, K. Amano, K. Hirano, H. Someya, H. Okumura, K. Yoshimura, P. Davis, and Y. Tokura, “Differential-phase-shift quantum key distribution experiment using fast physical random bit generator with chaotic semiconductor lasers,” Opt. Express |

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

16. | Y. Takiguchi, K. Ohyagi, and J. Ohtsubo, “Bandwidth-enhanced chaos synchronization in strongly injection-locked semiconductor lasers with optical feedback,” Opt. Lett. |

17. | A. Uchida, T. Heil, Y. Liu, P. Davis, and T. Aida, “High-frequency broad-band signal generation using a semiconductor laser with a chaotic optical injection,” IEEE J. Quantum Electron. |

18. | F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. |

19. | A. Wang, Y. Wang, and H. He, “Enhancing the bandwidth of the optical chaotic signal generated by a semiconductor laser with optical feedback,” IEEE Photon. Technol. Lett. |

20. | H. Someya, I. Oowada, H. Okumura, T. Kida, and A. Uchida, “Synchronization of bandwidth-enhanced chaos in semiconductor lasers with optical feedback and injection,” Opt. Express |

21. | G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science |

22. | J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. |

23. | A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature |

24. | F.-Y. Lin and J.-M. Liu, “Chaotic lider,” IEEE J. Sel. Top. Quantum Electron. |

25. | J. Ohtsubo, “ |

26. | A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, and M. Levenson, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” National Institute of Standards and Technology, Special Publication 800–22, 2001; Revision 1, August 2008. http://csrc.nist.gov/publications/nistpubs/800-22-rev1/SP800-22rev1.pdf |

27. | S. J. Kim, K. Umeno, and A. Hasegawa, “Corrections of the NIST statistical test suite for randomness,” arXiv:nlin.CD/0401040v1, 2004. |

28. | G. Marsaglia, DIEHARD: A battery of tests of randomness. http://stat.fsu.edu/ geo, 1996. |

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

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(140.1540) Lasers and laser optics : Chaos

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.3100) Nonlinear optics : Instabilities and chaos

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 7, 2009

Revised Manuscript: February 12, 2010

Manuscript Accepted: February 14, 2010

Published: March 3, 2010

**Citation**

Kunihito Hirano, Taiki Yamazaki, Shinichiro Morikatsu, Haruka Okumura, Hiroki Aida, Atsushi Uchida, Shigeru Yoshimori, Kazuyuki Yoshimura, Takahisa Harayama, and Peter Davis, "Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers," Opt. Express **18**, 5512-5524 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-5512

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### References

- D. Eastlake, J. Schiller, and S. Crocker, “Randomness requirements for security,” RFC4086 (2005) http://tools.ietf.org/html/rfc4086
- Security requirements for cryptographic modules. FIPS 140–2 (2001) http://csrc.nist.gov/publications/fips/fips140-2/fips1402.pdf
- N. Gisin, G. Robordy, W. Tittel, and H. Zbinden, “Quantum Cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]
- N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44(247), 335–341 (1949). [CrossRef] [PubMed]
- D. Knuth, “The Art of Computer Programming,” Volume 2: Seminumerical Algorithms (3rd Edition), Addison-Wesley Professional (1996).
- W. T. Holman, J. A. Connelly, and A. B. Dowlatabadi, “An integrated analog/digital random noise source,” IEEE Trans. Circuits Syst. I 44(6), 521–528 (1997). [CrossRef]
- J. T. Gleeson, “Truly random number generator based on turbulent electroconvection,” Appl. Phys. Lett. 81(11), 1949 (2002). [CrossRef]
- M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanouvo, “A high-speed oscillator-based truly random number source for cryptographic applications on a Smart Card IC,” IEEE Trans. Comput. 52(4), 403–409 (2003). [CrossRef]
- J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, post-processing free, quantum random number generator,” Appl. Phys. Lett. 93(3), 1–3 (2008). [CrossRef]
- 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(3), 312–314 (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, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008). [CrossRef]
- T. E. Murphy and R. Roy, “The world’s fastest dice,” Nat. Photonics 2(12), 714–715 (2008). [CrossRef]
- K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009). [CrossRef]
- T. Honjo, A. Uchida, K. Amano, K. Hirano, H. Someya, H. Okumura, K. Yoshimura, P. Davis, and Y. Tokura, “Differential-phase-shift quantum key distribution experiment using fast physical random bit generator with chaotic semiconductor lasers,” Opt. Express 17(11), 9053–9061 (2009). [CrossRef] [PubMed]
- I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 1–4 (2009). [CrossRef]
- Y. Takiguchi, K. Ohyagi, and J. Ohtsubo, “Bandwidth-enhanced chaos synchronization in strongly injection-locked semiconductor lasers with optical feedback,” Opt. Lett. 28(5), 319–321 (2003). [CrossRef] [PubMed]
- A. Uchida, T. Heil, Y. Liu, P. Davis, and T. Aida, “High-frequency broad-band signal generation using a semiconductor laser with a chaotic optical injection,” IEEE J. Quantum Electron. 39(11), 1462–1467 (2003). [CrossRef]
- F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221, 173–180 (2003). [CrossRef]
- A. Wang, Y. Wang, and H. He, “Enhancing the bandwidth of the optical chaotic signal generated by a semiconductor laser with optical feedback,” IEEE Photon. Technol. Lett. 20(19), 1633–1635 (2008). [CrossRef]
- H. Someya, I. Oowada, H. Okumura, T. Kida, and A. Uchida, “Synchronization of bandwidth-enhanced chaos in semiconductor lasers with optical feedback and injection,” Opt. Express 17(22), 19536–19543 (2009). [CrossRef] [PubMed]
- G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science 279(5354), 1198–1200 (1998). [CrossRef] [PubMed]
- J.-P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80(10), 2249–2252 (1998). [CrossRef]
- A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 437(7066), 343–346 (2005). [CrossRef]
- F.-Y. Lin and J.-M. Liu, “Chaotic lider,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004). [CrossRef]
- J. Ohtsubo, “Semiconductor Lasers, -Stability, Instability and Chaos-,” Second Ed., Springer-Verlag, Berlin Heidelberg (2005).
- A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, and M. Levenson, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” National Institute of Standards and Technology, Special Publication 800–22, 2001; Revision 1, August 2008. http://csrc.nist.gov/publications/nistpubs/800-22-rev1/SP800-22rev1.pdf
- S. J. Kim, K. Umeno, and A. Hasegawa, “Corrections of the NIST statistical test suite for randomness,” arXiv:nlin.CD/0401040v1, 2004.
- G. Marsaglia, DIEHARD: A battery of tests of randomness. http://stat.fsu.edu/ geo, 1996.
- I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 58–61 (2010). [CrossRef]

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