## A robust random number generator based on differential comparison of chaotic laser signals |

Optics Express, Vol. 20, Issue 7, pp. 7496-7506 (2012)

http://dx.doi.org/10.1364/OE.20.007496

Acrobat PDF (2260 KB)

### Abstract

We experimentally realize a robust real-time random number generator by differentially comparing the signal from a chaotic semiconductor laser and its delayed signal through a 1-bit analog-to-digital converter. The probability density distribution of the output chaotic signal based on the differential comparison method possesses an extremely small coefficient of Pearson’s median skewness (1.5 × 10^{−6}), which can yield a balanced random sequence much easily than the previously reported method that compares the signal from the chaotic laser with a certain threshold value. Moveover, we experimently demonstrate that our method can stably generate good random numbers at rates of 1.44 Gbit/s with excellent immunity from external perturbations while the previously reported method fails.

© 2012 OSA

## 1. Introduction

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

2. R. L. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory of spread-spectrum communications-a tutorial,” IEEE Trans. Commun. **30**(5), 855–884 (1982). [CrossRef]

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

4. M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna Jr, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. **7**(1), 24–38 (1989). [CrossRef]

5. C. Petrie and J. Connelly, “A noise-based IC random number generator for applications in cryptography,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. **47**(5), 615–621 (2000). [CrossRef]

6. 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**(4), 403–409 (2003). [CrossRef]

7. T. Stojanovski and L. Kocarev, “Chaos-based random number generators-Part I: analysis,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. **48**(3), 281–288 (2001). [CrossRef]

8. T. Stojanovski, J. Pihl, and 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]

*et al.*[9

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. 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]

*et al.*constructed a high-speed stable RNG by combining together multi-bits from an 8-bit ADC [10

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]

12. 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]

13. 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]

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

11. 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]

15. S. Sunada, T. Harayama, K. Arai, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Chaos laser chips with delayed optical feedback using a passive ring waveguide,” Opt. Express **19**(7), 5713–5724 (2011). [CrossRef] [PubMed]

16. 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]

## 2. Experimental setup

## 3. Experimental results

### 3.1 Characteristics of chaotic laser signals

### 3.2 Performances of differential comparison

18. 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 **18**(23), 23584–23597 (2010). [CrossRef] [PubMed]

*b*denotes the

_{i}*i*-th sampling value of the chaotic signal,

*k*denotes the delayed sampling dot number, and

*n*is the length of chaotic time series. Chaotic time series for a 500 ns long data stream, i.e.,

*n*= 10000, is chosen to calculate the correlation trace of the chaotic signal with the maximum delay time of 200 ns, as shown in Fig. 3 . It can be seen that when the delay time is larger than 4 ns, the correlation coefficient is decreased to less than 0.01. Here, the delay time is selected to be 5 ns (the delay line length is about 1 m mentioned as before), and the corresponding correlation coefficient is 0.004. Under the condition, the chaotic signal and its delayed signal are considered to be almost independent of each other. Thus, we select the length of about 1 m as the delay line length between two differential chaotic signals into the comparator. From the inset of Fig. 3, we can see that the secondary peak of the overall correlation trace appears at 84 ns corresponding to the external cavity round-trip time, also showing the periodicity induced by the external cavity.

*V*

_{1}(

*t*) and its delayed signal

*V*

_{2}(

*t*) injected into the comparator. If

*V*

_{1}(

*t*) -

*V*

_{2}(

*t*) > 0, the random bit 1 is obtained; otherwise, the random bit 0 is obtained. Thus, a good distribution of the differentiated chaotic signal helps to extract high-quality random number sequences. From the above analysis, we find when the delay time between

*V*

_{1}(

*t*) and

*V*

_{2}(

*t*) is set to an appropriate value (for example, 5 ns), they are suggested to be mutually independent. Moreover, they obey the same amplitude distribution. Here, we suppose that their probability density functions are

*f*(

*x*) and

*f*(

*y*), respectively, and their joint probability density function is

*f*(

*x*,

*y*), where

*f*(

*x*,

*y*) =

*f*(

*x*)

*f*(

*y*) due to mutual independence of

*V*

_{1}(

*t*) and

*V*

_{2}(

*t*). For the differentiated chaotic signal, a variable

*z*is introduced to represent it, where

*z*=

*V*

_{1}(

*t*) -

*V*

_{2}(

*t*). Its amplitude distribution function

*F*(

*z*) is calculated by the following equation.

*f*(

_{z}*z*) is obtained by the derivation of the distribution function.Then,

*v*= -

*z*+

*y*, thus,

19. Pearson's skewness coefficients. http://en.wikipedia.org/wiki/Skewness#cite_note-4.

*x*,

_{i}*M*denote the sampling amplitude value, the mean and the median of the chaotic signal, respectively. Closer to zero the skewness coefficient is, the more symmetric the distribution. Specially,

*γ*= 0 indicates that the distribution is completely symmetric. For the chaotic signal without the differential comparison, its skewness coefficient is

*γ*= 0.384 by calculation. However, for the chaotic signal with the differential comparison, its skewness is decreased to

*γ*= 1.5 × 10

^{−6}. The decrease of the skewness coefficient is in agreement with the statistical histograms as shown in Fig. 4.

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]

### 3.3 Random number generation

20. 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,” http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html

21. G. Marsaglia, “Diehard: A battery of tests of randomness,” http://www.stat.fsu.edu/pub/diehard/

## 4. Analysis of system robustness

## 5. Conclusions

^{−6}. Since the whole experimental system is very robust to the external perturbations and can be maintained for a long time during continuous operation of the devices, it is very useful for practical applications.

## Acknowledgments

## References and links

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

2. | R. L. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory of spread-spectrum communications-a tutorial,” IEEE Trans. Commun. |

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

4. | M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna Jr, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol. |

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

6. | 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. |

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

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

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. | 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 |

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

13. | 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 |

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

15. | S. Sunada, T. Harayama, K. Arai, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Chaos laser chips with delayed optical feedback using a passive ring waveguide,” Opt. Express |

16. | 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 |

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

18. | 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 |

19. | Pearson's skewness coefficients. http://en.wikipedia.org/wiki/Skewness#cite_note-4. |

20. | 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,” http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html |

21. | G. Marsaglia, “Diehard: A battery of tests of randomness,” http://www.stat.fsu.edu/pub/diehard/ |

**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

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 2, 2011

Revised Manuscript: February 2, 2012

Manuscript Accepted: March 5, 2012

Published: March 19, 2012

**Citation**

Jianzhong Zhang, Yuncai Wang, Ming Liu, Lugang Xue, Pu Li, Anbang Wang, and Mingjiang Zhang, "A robust random number generator based on differential comparison of chaotic laser signals," Opt. Express **20**, 7496-7506 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7496

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### References

- Security requirements for cryptographic modules. FIPS 140–2 (2001). http://csrc.nist.gov/publications/fips/fips140-2/fips1402.pdf
- R. L. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory of spread-spectrum communications-a tutorial,” IEEE Trans. Commun.30(5), 855–884 (1982). [CrossRef]
- N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc.44(247), 335–341 (1949). [CrossRef] [PubMed]
- M. Nazarathy, S. A. Newton, R. P. Giffard, D. S. Moberly, F. Sischka, W. R. Trutna, and S. Foster, “Real-time long range complementary correlation optical time domain reflectometer,” J. Lightwave Technol.7(1), 24–38 (1989). [CrossRef]
- C. Petrie and J. Connelly, “A noise-based IC random number generator for applications in cryptography,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl.47(5), 615–621 (2000). [CrossRef]
- 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(4), 403–409 (2003). [CrossRef]
- T. Stojanovski and L. Kocarev, “Chaos-based random number generators-Part I: analysis,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl.48(3), 281–288 (2001). [CrossRef]
- T. Stojanovski, J. Pihl, and 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]
- 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. Photonics2(12), 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(2), 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. Express18(18), 18763–18768 (2010). [CrossRef] [PubMed]
- I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4(1), 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. Express18(6), 5512–5524 (2010). [CrossRef] [PubMed]
- P. Li, Y. C. Wang, and J. Z. Zhang, “All-optical fast random number generator,” Opt. Express18(19), 20360–20369 (2010). [CrossRef] [PubMed]
- S. Sunada, T. Harayama, K. Arai, K. Yoshimura, P. Davis, K. Tsuzuki, and A. Uchida, “Chaos laser chips with delayed optical feedback using a passive ring waveguide,” Opt. Express19(7), 5713–5724 (2011). [CrossRef] [PubMed]
- 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. A83(3), 031803 (2011). [CrossRef]
- A. B. Wang, Y. C. Wang, and H. C. 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]
- 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. Express18(23), 23584–23597 (2010). [CrossRef] [PubMed]
- Pearson's skewness coefficients. http://en.wikipedia.org/wiki/Skewness#cite_note-4 .
- 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,” http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html
- G. Marsaglia, “Diehard: A battery of tests of randomness,” http://www.stat.fsu.edu/pub/diehard/

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