## Two approaches for ultrafast random bit generation based on the chaotic dynamics of a semiconductor laser |

Optics Express, Vol. 22, Issue 6, pp. 6634-6646 (2014)

http://dx.doi.org/10.1364/OE.22.006634

Acrobat PDF (1502 KB)

### Abstract

This paper reports the experimental investigation of two different approaches to random bit generation based on the chaotic dynamics of a semiconductor laser with optical feedback. By computing high-order finite differences of the chaotic laser intensity time series, we obtain time series with symmetric statistical distributions that are more conducive to ultrafast random bit generation. The first approach is guided by information-theoretic considerations and could potentially reach random bit generation rates as high as 160 Gb/s by extracting 4 bits per sample. The second approach is based on pragmatic considerations and could lead to rates of 2.2 Tb/s by extracting 55 bits per sample. The randomness of the bit sequences obtained from the two approaches is tested against three standard randomness tests (ENT, Diehard, and NIST tests), as well as by calculating the statistical bias and the serial correlation coefficients on longer sequences of random bits than those used in the standard tests.

© 2014 Optical Society of America

## 1. Introduction

1. C. Masoller and N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external cavity semiconductor laser,” Phys. Rev. A **57**(2), 1313–1322 (1998). [CrossRef]

2. J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. **65**(16), 1999–2002 (1990). [CrossRef] [PubMed]

4. M. C. Soriano, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductor lasers,” Rev. Mod. Phys. **85**(1), 421–470 (2013). [CrossRef]

*coherence collapse*, has been exploited for several applications, such as chaos-based communications [5

5. 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 **438**(7066), 343–346 (2005). [CrossRef] [PubMed]

6. R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. **107**(3), 034103 (2011). [CrossRef] [PubMed]

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

8. L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express **20**(3), 3241–3249 (2012). [CrossRef] [PubMed]

9. 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]

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

*pseudo*RBG [12]; unfortunately, the quality and generation rate may be inadequate, for applications such as cryptography and large-scale Monte Carlo numerical computations in which it is crucial to have nearly unpredictable bits [13

13. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. **74**(1), 145–195 (2002). [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]

*physical*RBG [15

15. T. Durt, C. Belmont, L. P. Lamoureux, K. Panajotov, F. Van den Berghe, and H. Thienpont, “Fast quantum-optical random-number generators,” Phys. Rev. A **87**(2), 022339 (2013). [CrossRef]

25. X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, and H. Lo, “Postprocessing for quantum random-numbers generators: Entropy evaluation and randomness extraction,” Phys. Rev. A **87**(6), 062327 (2013). [CrossRef]

26. V. N. Chizhevsky, “Fast generation of random bits based on polarization noises in a semiconductor vertical-cavity laser,” Opt. Spectrosc. **111**(5), 689–694 (2011). [CrossRef]

9. 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]

10. 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), 024102 (2009). [CrossRef] [PubMed]

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

27. T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A **83**(3), 031803 (2011). [CrossRef]

33. N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Fast random bit generation using a chaotic laser: approaching the information theoretic limit,” IEEE J. Quantum Electron. **49**(11), 910–918 (2013). [CrossRef]

34. T. Harayama, S. Sunada, K. Yoshimura, J. Muramatsu, K. Arai, A. Uchida, and P. Davis, “Theory of fast nondeterministic physical random-bit generation with chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **85**(4), 046215 (2012). [CrossRef] [PubMed]

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

37. 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**(6), 5512–5524 (2010). [CrossRef] [PubMed]

40. X. Z. Li and S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. **49**(10), 829–838 (2013). [CrossRef]

41. Y. Akizawa, T. Yamazaki, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast random number generation with bandwidth-enhanced chaotic semiconductor lasers at**24**(12), 1042–1044 (2012). [CrossRef]

27. T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A **83**(3), 031803 (2011). [CrossRef]

28. 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**(18), 18763–18768 (2010). [CrossRef] [PubMed]

30. P. Li, Y. C. Wang, and J. Z. Zhang, “All-optical fast random number generator,” Opt. Express **18**(19), 20360–20369 (2010). [CrossRef] [PubMed]

34. T. Harayama, S. Sunada, K. Yoshimura, J. Muramatsu, K. Arai, A. Uchida, and P. Davis, “Theory of fast nondeterministic physical random-bit generation with chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **85**(4), 046215 (2012). [CrossRef] [PubMed]

*et al.*have demonstrated ultrafast chaos-based RBG with a rate of 300 Gb/s based on the use of the high-order derivatives using a sampling rate of 20 GHz and extracting 15 random bits per sample, although the raw data were digitized with 8-bit resolution [11

**4**(1), 58–61 (2010). [CrossRef]

*et al.*have highlighted that the fundamental limits of maximum RBG rate imposed by information theory depend on the analog bandwidth of the chaotic laser and cannot be improved by simply increasing the sampling rate or by creating additional bits through post-processing [33

33. N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Fast random bit generation using a chaotic laser: approaching the information theoretic limit,” IEEE J. Quantum Electron. **49**(11), 910–918 (2013). [CrossRef]

31. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bits generation based on a single chaotic semiconductor ring laser,” Opt. Express **20**(27), 28603–28613 (2012). [CrossRef] [PubMed]

33. N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Fast random bit generation using a chaotic laser: approaching the information theoretic limit,” IEEE J. Quantum Electron. **49**(11), 910–918 (2013). [CrossRef]

37. 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**(6), 5512–5524 (2010). [CrossRef] [PubMed]

**49**(11), 910–918 (2013). [CrossRef]

44. M. A. Wayne and P. G. Kwiat, “Low-bias high-speed quantum random number generator via shaped optical pulses,” Opt. Express **18**(9), 9351–9357 (2010). [CrossRef] [PubMed]

9. 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]

29. J. Z. Zhang, Y. C. Wang, M. Liu, L. G. Xue, P. Li, A. B. Wang, and M. J. Zhang, “A robust random number generator based on differential comparison of chaotic laser signals,” Opt. Express **20**(7), 7496–7506 (2012). [CrossRef] [PubMed]

31. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bits generation based on a single chaotic semiconductor ring laser,” Opt. Express **20**(27), 28603–28613 (2012). [CrossRef] [PubMed]

38. J. G. Wu, X. Tang, Z. M. Wu, G. Q. Xia, and G. Y. Feng, “Parallel generation of 10 Gbits/s physical random number streams using chaotic semiconductor lasers,” Laser Phys. **22**(10), 1476–1480 (2012). [CrossRef]

41. Y. Akizawa, T. Yamazaki, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast random number generation with bandwidth-enhanced chaotic semiconductor lasers at**24**(12), 1042–1044 (2012). [CrossRef]

45. 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]

*physical RBG*, and the second

*physical-based pseudo RBG*. Proving that our conservative first approach does indeed ensure information-theoretic RNG requires an in-depth analysis of our post-processing, which is beyond the scope of the present paper and will be the subject of a subsequent work. Some previous works show that the interplay of dynamical properties, acquisition conditions, and post-processing plays a critical role in the performance of RBG [33

**49**(11), 910–918 (2013). [CrossRef]

45. 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]

46. T. Yamazaki and A. Uchida, “Performance of random number generators using noise-based superluminescent diode and chaos-based semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. **19**(4), 0600309 (2013). [CrossRef]

47. V. N. Chizhevsky, “Symmetrization of single-sided or nonsymmetrical distributions: The way to enhance a generation rate of random bits from a physical source of randomness,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **82**(5), 050101 (2010). [CrossRef] [PubMed]

**4**(1), 58–61 (2010). [CrossRef]

47. V. N. Chizhevsky, “Symmetrization of single-sided or nonsymmetrical distributions: The way to enhance a generation rate of random bits from a physical source of randomness,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **82**(5), 050101 (2010). [CrossRef] [PubMed]

**4**(1), 58–61 (2010). [CrossRef]

**2**(12), 728–732 (2008). [CrossRef]

**4**(1), 58–61 (2010). [CrossRef]

27. T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A **83**(3), 031803 (2011). [CrossRef]

**49**(11), 910–918 (2013). [CrossRef]

37. 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**(6), 5512–5524 (2010). [CrossRef] [PubMed]

41. Y. Akizawa, T. Yamazaki, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast random number generation with bandwidth-enhanced chaotic semiconductor lasers at**24**(12), 1042–1044 (2012). [CrossRef]

## 2. Experimental setup

31. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bits generation based on a single chaotic semiconductor ring laser,” Opt. Express **20**(27), 28603–28613 (2012). [CrossRef] [PubMed]

**49**(11), 910–918 (2013). [CrossRef]

**18**(6), 5512–5524 (2010). [CrossRef] [PubMed]

38. J. G. Wu, X. Tang, Z. M. Wu, G. Q. Xia, and G. Y. Feng, “Parallel generation of 10 Gbits/s physical random number streams using chaotic semiconductor lasers,” Laser Phys. **22**(10), 1476–1480 (2012). [CrossRef]

### 2.1 Chaos generation

### 2.2 Post-processing

## 3. Experimental results

### 3.1 Statistical properties of laser chaos

*e.g.,*a pronounced peak with a correlation coefficient about 0.34 is located at a delay time of 57.68 ns, which corresponds to the roundtrip time in the external-cavity. Therefore, to extract random bits from the chaotic sources, post-processing techniques should be employed.

**20**(27), 28603–28613 (2012). [CrossRef] [PubMed]

**49**(11), 910–918 (2013). [CrossRef]

**18**(6), 5512–5524 (2010). [CrossRef] [PubMed]

40. X. Z. Li and S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. **49**(10), 829–838 (2013). [CrossRef]

51. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “Towards the generation of random bits at terahertz rates based on chaotic semiconductor laser,” J. Phys. Conf. Ser. **233**, 012002 (2010). [CrossRef]

### 3.2 High-quality physical RBG

*n*th-order finite differences (HFD) procedure described in [47

47. V. N. Chizhevsky, “Symmetrization of single-sided or nonsymmetrical distributions: The way to enhance a generation rate of random bits from a physical source of randomness,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **82**(5), 050101 (2010). [CrossRef] [PubMed]

10. 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), 024102 (2009). [CrossRef] [PubMed]

**4**(1), 58–61 (2010). [CrossRef]

*M*-bit resolution (

*n*th-order differences of the floating-point data are calculated. It is worth noting that one cannot aim to extract all significant bits from each obtained sample since strong correlations appear in the bit stream after the calculation of HFD. Therefore, one should discard certain MSBs to eliminate these correlations. To this end, in the second step, one just retains

25. X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, and H. Lo, “Postprocessing for quantum random-numbers generators: Entropy evaluation and randomness extraction,” Phys. Rev. A **87**(6), 062327 (2013). [CrossRef]

44. M. A. Wayne and P. G. Kwiat, “Low-bias high-speed quantum random number generator via shaped optical pulses,” Opt. Express **18**(9), 9351–9357 (2010). [CrossRef] [PubMed]

54. J. Walker, Ent-a pseudorandom sequence test program, http://www. fourmilab.ch/random.

55. G. Marsaglia, The diehard test suite (2003), http:// www. Csis. hku. hk/diehard.

56. A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dary, and S. Vo, “A statistical test suite for the validation of random number generators and pseudo random number generators for cryptographic applications”, http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html.

55. G. Marsaglia, The diehard test suite (2003), http:// www. Csis. hku. hk/diehard.

**82**(5), 050101 (2010). [CrossRef] [PubMed]

### 3.3 Ultrafast physical-based pseudo RBG

**4**(1), 58–61 (2010). [CrossRef]

26. V. N. Chizhevsky, “Fast generation of random bits based on polarization noises in a semiconductor vertical-cavity laser,” Opt. Spectrosc. **111**(5), 689–694 (2011). [CrossRef]

**82**(5), 050101 (2010). [CrossRef] [PubMed]

**82**(5), 050101 (2010). [CrossRef] [PubMed]

57. T. Granlund., GMP, the GNU multiple precision arithmetic library.http://gmplib.org (2013).

38. J. G. Wu, X. Tang, Z. M. Wu, G. Q. Xia, and G. Y. Feng, “Parallel generation of 10 Gbits/s physical random number streams using chaotic semiconductor lasers,” Laser Phys. **22**(10), 1476–1480 (2012). [CrossRef]

40. X. Z. Li and S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. **49**(10), 829–838 (2013). [CrossRef]

45. 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]

**22**(10), 1476–1480 (2012). [CrossRef]

49. J. Z. Zhang, Y. C. Wang, L. G. Xue, J. Y. Hou, B. B. Zhang, A. B. Wang, and M. J. Zhang, “Delay line length selection in generating fast random numbers with a chaotic laser,” Appl. Opt. **51**(11), 1709–1714 (2012). [CrossRef] [PubMed]

50. A. B. Wang, P. Li, J. G. Zhang, J. Z. Zhang, L. Li, and Y. C. Wang, “4.5 Gbps high-speed real-time physical random bit generator,” Opt. Express **21**(17), 20452–20462 (2013). [CrossRef] [PubMed]

**83**(3), 031803 (2011). [CrossRef]

28. 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**(18), 18763–18768 (2010). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgment

## References and links

1. | C. Masoller and N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external cavity semiconductor laser,” Phys. Rev. A |

2. | J. Mørk, J. Mark, and B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. |

3. | J. Ohtsubo, |

4. | M. C. Soriano, J. Garcia-Ojalvo, C. R. Mirasso, and I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductor lasers,” Rev. Mod. Phys. |

5. | 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 |

6. | R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. |

7. | F. Y. Lin and J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. |

8. | L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, and I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express |

9. | 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 |

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

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

12. | D. Knuth, |

13. | N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. |

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. | T. Durt, C. Belmont, L. P. Lamoureux, K. Panajotov, F. Van den Berghe, and H. Thienpont, “Fast quantum-optical random-number generators,” Phys. Rev. A |

16. | C. R. S. Williams, J. C. Salevan, X. W. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express |

17. | 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. |

18. | 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. |

19. | X. Li, A. B. Cohen, T. E. Murphy, and R. Roy, “Scalable parallel physical random number generator based on a superluminescent LED,” Opt. Lett. |

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

21. | J. Walker, Hotbits: Genuine Random Numbers, Generated by Radioactive Decay, http://www.fourmilab.ch/hotbits. |

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

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

24. | D. P. Rosin, D. Rontani, and D. J. Gauthier, “Ultrafast physical generation of random numbers using hybrid Boolean networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

25. | X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, and H. Lo, “Postprocessing for quantum random-numbers generators: Entropy evaluation and randomness extraction,” Phys. Rev. A |

26. | V. N. Chizhevsky, “Fast generation of random bits based on polarization noises in a semiconductor vertical-cavity laser,” Opt. Spectrosc. |

27. | T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A |

28. | 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 |

29. | J. Z. Zhang, Y. C. Wang, M. Liu, L. G. Xue, P. Li, A. B. Wang, and M. J. Zhang, “A robust random number generator based on differential comparison of chaotic laser signals,” Opt. Express |

30. | P. Li, Y. C. Wang, and J. Z. Zhang, “All-optical fast random number generator,” Opt. Express |

31. | R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, and G. Van der Sande, “Fast random bits generation based on a single chaotic semiconductor ring laser,” Opt. Express |

32. | N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. |

33. | N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Fast random bit generation using a chaotic laser: approaching the information theoretic limit,” IEEE J. Quantum Electron. |

34. | T. Harayama, S. Sunada, K. Yoshimura, J. Muramatsu, K. Arai, A. Uchida, and P. Davis, “Theory of fast nondeterministic physical random-bit generation with chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

35. | T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, and P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

36. | X. Fang, B. Wetzel, J. Merolla, J. M. Dudley, L. Larger, C. Guyeux, and J. M. Bahi, “Noise and chaos contributions in fast random bit sequence generated from broadband optoelectronic entropy sources,” IEEE Trans. Circuits Syst. I |

37. | 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 |

38. | J. G. Wu, X. Tang, Z. M. Wu, G. Q. Xia, and G. Y. Feng, “Parallel generation of 10 Gbits/s physical random number streams using chaotic semiconductor lasers,” Laser Phys. |

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

40. | X. Z. Li and S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. |

41. | Y. Akizawa, T. Yamazaki, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, and P. Davis, “Fast random number generation with bandwidth-enhanced chaotic semiconductor lasers at |

42. | |

43. | C. E. Shannon, “A mathematical theory of communication,” Bell System Technical Journal |

44. | M. A. Wayne and P. G. Kwiat, “Low-bias high-speed quantum random number generator via shaped optical pulses,” Opt. Express |

45. | 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. |

46. | T. Yamazaki and A. Uchida, “Performance of random number generators using noise-based superluminescent diode and chaos-based semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. |

47. | V. N. Chizhevsky, “Symmetrization of single-sided or nonsymmetrical distributions: The way to enhance a generation rate of random bits from a physical source of randomness,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

48. | T. E. Murphy and R. Roy, “Chaotic lasers: The world’s fastest dice,” Nat. Photonics |

49. | J. Z. Zhang, Y. C. Wang, L. G. Xue, J. Y. Hou, B. B. Zhang, A. B. Wang, and M. J. Zhang, “Delay line length selection in generating fast random numbers with a chaotic laser,” Appl. Opt. |

50. | A. B. Wang, P. Li, J. G. Zhang, J. Z. Zhang, L. Li, and Y. C. Wang, “4.5 Gbps high-speed real-time physical random bit generator,” Opt. Express |

51. | I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “Towards the generation of random bits at terahertz rates based on chaotic semiconductor laser,” J. Phys. Conf. Ser. |

52. | D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. |

53. | S. Priyadarshi, Y. Hong, I. Pierce, and K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external-cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. |

54. | J. Walker, Ent-a pseudorandom sequence test program, http://www. fourmilab.ch/random. |

55. | G. Marsaglia, The diehard test suite (2003), http:// www. Csis. hku. hk/diehard. |

56. | A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dary, and S. Vo, “A statistical test suite for the validation of random number generators and pseudo random number generators for cryptographic applications”, http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html. |

57. | T. Granlund., GMP, the GNU multiple precision arithmetic library.http://gmplib.org (2013). |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and 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: October 22, 2013

Revised Manuscript: February 14, 2014

Manuscript Accepted: March 4, 2014

Published: March 14, 2014

**Citation**

Nianqiang Li, Byungchil Kim, V. N. Chizhevsky, A. Locquet, M. Bloch, D. S. Citrin, and Wei Pan, "Two approaches for ultrafast random bit generation based on the chaotic dynamics of a semiconductor laser," Opt. Express **22**, 6634-6646 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6634

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### References

- C. Masoller, N. B. Abraham, “Stability and dynamical properties of the coexisting attractors of an external cavity semiconductor laser,” Phys. Rev. A 57(2), 1313–1322 (1998). [CrossRef]
- J. Mørk, J. Mark, B. Tromborg, “Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback,” Phys. Rev. Lett. 65(16), 1999–2002 (1990). [CrossRef] [PubMed]
- J. Ohtsubo, Semiconductor Laser: Stability, Instability and Chaos (Springer, 2008).
- M. C. Soriano, J. Garcia-Ojalvo, C. R. Mirasso, I. Fischer, “Complex photonics: Dynamics and applications of delay-coupled semiconductor lasers,” Rev. Mod. Phys. 85(1), 421–470 (2013). [CrossRef]
- A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005). [CrossRef] [PubMed]
- R. M. Nguimdo, P. Colet, L. Larger, L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. 107(3), 034103 (2011). [CrossRef] [PubMed]
- F. Y. Lin, J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004). [CrossRef]
- L. Larger, M. C. Soriano, D. Brunner, L. Appeltant, J. M. Gutierrez, L. Pesquera, C. R. Mirasso, I. Fischer, “Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing,” Opt. Express 20(3), 3241–3249 (2012). [CrossRef] [PubMed]
- A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008). [CrossRef]
- I. Reidler, Y. Aviad, M. Rosenbluh, I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009). [CrossRef] [PubMed]
- I. Kanter, Y. Aviad, I. Reidler, E. Cohen, M. Rosenbluh, “An optical ultrafast random number generator,” Nat. Photonics 4(1), 58–61 (2010). [CrossRef]
- D. Knuth, The Art of Computer Programming, 3rd ed. (Addison Wesley Longman, 1998), Vol. 2.
- N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]
- T. Honjo, A. Uchida, K. Amano, K. Hirano, H. Someya, H. Okumura, K. Yoshimura, P. Davis, 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]
- T. Durt, C. Belmont, L. P. Lamoureux, K. Panajotov, F. Van den Berghe, H. Thienpont, “Fast quantum-optical random-number generators,” Phys. Rev. A 87(2), 022339 (2013). [CrossRef]
- C. R. S. Williams, J. C. Salevan, X. W. Li, R. Roy, T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express 18(23), 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 Stat. Nonlin. Soft Matter Phys. 81(5), 051137 (2010). [CrossRef] [PubMed]
- B. Qi, Y.-M. Chi, H.-K. Lo, 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]
- X. Li, A. B. Cohen, T. E. Murphy, R. Roy, “Scalable parallel physical random number generator based on a superluminescent LED,” Opt. Lett. 36(6), 1020–1022 (2011). [CrossRef] [PubMed]
- T. Stojanovski, J. Pihl, L. Kocarev, “Chaos-based random number generators–Part II: practical realization,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48(3), 382–385 (2001). [CrossRef]
- J. Walker, Hotbits: Genuine Random Numbers, Generated by Radioactive Decay, http://www.fourmilab.ch/hotbits .
- W. T. Holman, J. A. Connelly, 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]
- D. P. Rosin, D. Rontani, D. J. Gauthier, “Ultrafast physical generation of random numbers using hybrid Boolean networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 87(4), 040902 (2013). [CrossRef] [PubMed]
- X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, H. Lo, “Postprocessing for quantum random-numbers generators: Entropy evaluation and randomness extraction,” Phys. Rev. A 87(6), 062327 (2013). [CrossRef]
- V. N. Chizhevsky, “Fast generation of random bits based on polarization noises in a semiconductor vertical-cavity laser,” Opt. Spectrosc. 111(5), 689–694 (2011). [CrossRef]
- T. Harayama, S. Sunada, K. Yoshimura, P. Davis, K. Tsuzuki, A. Uchida, “Fast nondeterministic random-bit generation using on-chip chaos lasers,” Phys. Rev. A 83(3), 031803 (2011). [CrossRef]
- A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express 18(18), 18763–18768 (2010). [CrossRef] [PubMed]
- J. Z. Zhang, Y. C. Wang, M. Liu, L. G. Xue, P. Li, A. B. Wang, M. J. Zhang, “A robust random number generator based on differential comparison of chaotic laser signals,” Opt. Express 20(7), 7496–7506 (2012). [CrossRef] [PubMed]
- P. Li, Y. C. Wang, J. Z. Zhang, “All-optical fast random number generator,” Opt. Express 18(19), 20360–20369 (2010). [CrossRef] [PubMed]
- R. M. Nguimdo, G. Verschaffelt, J. Danckaert, X. Leijtens, J. Bolk, G. Van der Sande, “Fast random bits generation based on a single chaotic semiconductor ring laser,” Opt. Express 20(27), 28603–28613 (2012). [CrossRef] [PubMed]
- N. Oliver, M. C. Soriano, D. W. Sukow, I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36(23), 4632–4634 (2011). [CrossRef] [PubMed]
- N. Oliver, M. C. Soriano, D. W. Sukow, I. Fischer, “Fast random bit generation using a chaotic laser: approaching the information theoretic limit,” IEEE J. Quantum Electron. 49(11), 910–918 (2013). [CrossRef]
- T. Harayama, S. Sunada, K. Yoshimura, J. Muramatsu, K. Arai, A. Uchida, P. Davis, “Theory of fast nondeterministic physical random-bit generation with chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(4), 046215 (2012). [CrossRef] [PubMed]
- T. Mikami, K. Kanno, K. Aoyama, A. Uchida, T. Ikeguchi, T. Harayama, S. Sunada, K.-i. Arai, K. Yoshimura, P. Davis, “Estimation of entropy rate in a fast physical random-bit generator using a chaotic semiconductor laser with intrinsic noise,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(1 Pt 2), 016211 (2012). [CrossRef] [PubMed]
- X. Fang, B. Wetzel, J. Merolla, J. M. Dudley, L. Larger, C. Guyeux, J. M. Bahi, “Noise and chaos contributions in fast random bit sequence generated from broadband optoelectronic entropy sources,” IEEE Trans. Circuits Syst. I 99, 1–14 (2013).
- K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express 18(6), 5512–5524 (2010). [CrossRef] [PubMed]
- J. G. Wu, X. Tang, Z. M. Wu, G. Q. Xia, G. Y. Feng, “Parallel generation of 10 Gbits/s physical random number streams using chaotic semiconductor lasers,” Laser Phys. 22(10), 1476–1480 (2012). [CrossRef]
- X. Z. Li, S. C. Chan, “Random bit generation using an optically injected semiconductor laser in chaos with oversampling,” Opt. Lett. 37(11), 2163–2165 (2012). [CrossRef] [PubMed]
- X. Z. Li, S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. 49(10), 829–838 (2013). [CrossRef]
- Y. Akizawa, T. Yamazaki, A. Uchida, T. Harayama, S. Sunada, K. Arai, K. Yoshimura, P. Davis, “Fast random number generation with bandwidth-enhanced chaotic semiconductor lasers at8×50Gb/s, ” IEEE Photon. Technol. Lett. 24(12), 1042–1044 (2012). [CrossRef]
- http://people.seas.harvard.edu/~salil/pseudorandomness .
- C. E. Shannon, “A mathematical theory of communication,” Bell System Technical Journal 27, 379–423 and 623–656 (1948).
- M. A. Wayne, P. G. Kwiat, “Low-bias high-speed quantum random number generator via shaped optical pulses,” Opt. Express 18(9), 9351–9357 (2010). [CrossRef] [PubMed]
- K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45(11), 1367–1379 (2009). [CrossRef]
- T. Yamazaki, A. Uchida, “Performance of random number generators using noise-based superluminescent diode and chaos-based semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. 19(4), 0600309 (2013). [CrossRef]
- V. N. Chizhevsky, “Symmetrization of single-sided or nonsymmetrical distributions: The way to enhance a generation rate of random bits from a physical source of randomness,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 82(5), 050101 (2010). [CrossRef] [PubMed]
- T. E. Murphy, R. Roy, “Chaotic lasers: The world’s fastest dice,” Nat. Photonics 2(12), 714–715 (2008). [CrossRef]
- J. Z. Zhang, Y. C. Wang, L. G. Xue, J. Y. Hou, B. B. Zhang, A. B. Wang, M. J. Zhang, “Delay line length selection in generating fast random numbers with a chaotic laser,” Appl. Opt. 51(11), 1709–1714 (2012). [CrossRef] [PubMed]
- A. B. Wang, P. Li, J. G. Zhang, J. Z. Zhang, L. Li, Y. C. Wang, “4.5 Gbps high-speed real-time physical random bit generator,” Opt. Express 21(17), 20452–20462 (2013). [CrossRef] [PubMed]
- I. Kanter, Y. Aviad, I. Reidler, E. Cohen, M. Rosenbluh, “Towards the generation of random bits at terahertz rates based on chaotic semiconductor laser,” J. Phys. Conf. Ser. 233, 012002 (2010). [CrossRef]
- D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. 32(20), 2960–2962 (2007). [CrossRef] [PubMed]
- S. Priyadarshi, Y. Hong, I. Pierce, K. A. Shore, “Experimental investigations of time-delay signature concealment in chaotic external-cavity VCSELs subject to variable optical polarization angle of feedback,” IEEE J. Sel. Top. Quantum Electron. 19(4), 1700707 (2013). [CrossRef]
- J. Walker, Ent-a pseudorandom sequence test program, http://www. fourmilab.ch/random .
- G. Marsaglia, The diehard test suite (2003), http:// www. Csis. hku. hk/diehard .
- A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dary, S. Vo, “A statistical test suite for the validation of random number generators and pseudo random number generators for cryptographic applications”, http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html .
- T. Granlund et al.., GMP, the GNU multiple precision arithmetic library. http://gmplib.org (2013).

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