Abstract

We demonstrate the physical generation of random bits at high bit rates (> 100 Gb/s) using optical chaos from a solitary laser diode and therefore without the complex addition of either external optical feedback or injection. This striking result is obtained despite the low dimension and relatively small bandwidth of the laser chaos, i.e. two characteristics that have been so far considered as limiting the performances of optical chaos-based applications. We unambiguously attribute the successful randomness at high speed to the physics of the laser chaotic polarization dynamics and the resulting growth rate of the dynamical entropy.

© 2014 Optical Society of America

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  3. T. Symul, S. M. Assad, and P. K. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett. 98, 231103 (2011).
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  4. W. Li, I. Reidler, Y. Aviad, Y. Huang, H. Song, Y. Zhang, M. Rosenbluh, and I. Kanter, “Fast physical random-number generation based on room-temperature chaotic oscillations in weakly coupled superlattices,” Phys. Rev. Lett. 111, 044102 (2013).
    [CrossRef] [PubMed]
  5. Y. Liu, M.-Y. Zhu, B. Luo, J.-W. Zhang, and H. Guo, “Implementation of 1.6 Tb s 1 truly random number generation based on a super-luminescent emitting diode,” Laser Phys. Lett. 10, 045001 (2013).
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  6. C. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express 18, 23584–23597 (2010).
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  7. 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).
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    [CrossRef] [PubMed]
  13. T. Yamazaki and A. Uchida, “Performance of Random Number Generators Using Noise-based Super-Luminescent Diode and Chaos-based Semiconductor Lasers,” IEEE J. Sel. Top. Quantum Electron. 19, 0600309 (2013).
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  14. X. Li and S. Chan, “Heterodyne Random Bit Generation Using an Optically Injected Semiconductor Laser in Chaos,” IEEE J. Quantum Electron. 49, 829–838 (2013).
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  17. 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, 1367–1379 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  27. C. Bracikowski, R. Fox, and R. Roy, “Amplification of intrinsic noise in a chaotic multimode laser system,” Phys. Rev. A 45, 403 (1992).
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    [CrossRef]
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2013 (6)

W. Li, I. Reidler, Y. Aviad, Y. Huang, H. Song, Y. Zhang, M. Rosenbluh, and I. Kanter, “Fast physical random-number generation based on room-temperature chaotic oscillations in weakly coupled superlattices,” Phys. Rev. Lett. 111, 044102 (2013).
[CrossRef] [PubMed]

Y. Liu, M.-Y. Zhu, B. Luo, J.-W. Zhang, and H. Guo, “Implementation of 1.6 Tb s 1 truly random number generation based on a super-luminescent emitting diode,” Laser Phys. Lett. 10, 045001 (2013).
[CrossRef]

T. Yamazaki and A. Uchida, “Performance of Random Number Generators Using Noise-based Super-Luminescent Diode and Chaos-based Semiconductor Lasers,” IEEE J. Sel. Top. Quantum Electron. 19, 0600309 (2013).
[CrossRef]

X. Li and S. Chan, “Heterodyne Random Bit Generation Using an Optically Injected Semiconductor Laser in Chaos,” IEEE J. Quantum Electron. 49, 829–838 (2013).
[CrossRef]

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

M. Virte, K. Panajotov, and M. Sciamanna, “Bifurcation to nonlinear polarization dynamics and chaos in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87, 013834 (2013).
[CrossRef]

2012 (3)

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 85, 016211 (2012).
[CrossRef]

M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics 7, 60–65 (2012).
[CrossRef]

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

2011 (3)

2010 (5)

S. Pironio, A. Acín, and S. Massar, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

C. Gabriel, C. Wittmann, D. Sych, and R. Dong, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4, 711–715 (2010).
[CrossRef]

C. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express 18, 23584–23597 (2010).
[CrossRef] [PubMed]

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, 1367–1379 (2010).
[CrossRef]

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).
[CrossRef] [PubMed]

2009 (4)

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2009).
[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, 1367–1379 (2009).
[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]

L. Olejniczak, M. Sciamanna, H. Thienpont, K. Panajotov, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching in quantum-dot vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 21, 1008 (2009).
[CrossRef]

2008 (1)

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]

2006 (1)

F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. a. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, a. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. 89, 141106 (2006).
[CrossRef]

2005 (1)

R. Vicente, J. Dauden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41, 541–548 (2005).
[CrossRef]

1997 (1)

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

1995 (1)

M. San Miguel, Q. Feng, and J.V. Moloney, “Light-polarization dynamics in surface emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[CrossRef] [PubMed]

1994 (1)

I. Fischer, O. Hess, W. Elspher, and E. Göbel, “High-Dimensional Chaotic Dynamics of an External Cavity Semiconductor Laser,” Phys. Rev. Lett. 73, 2188–2191 (1994).
[CrossRef] [PubMed]

1992 (1)

C. Bracikowski, R. Fox, and R. Roy, “Amplification of intrinsic noise in a chaotic multimode laser system,” Phys. Rev. A 45, 403 (1992).
[CrossRef] [PubMed]

1991 (1)

R. Fox and J. Keizer, “Amplification of intrinsic fluctuations by chaotic dynamics in physical systems,” Phys. Rev. A 43, 1709–1720 (1991).
[CrossRef] [PubMed]

1979 (1)

I. Shimada and T. Nagashima, “A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems,” Prog. Theor. Phys. 61, 1605–1616 (1979).
[CrossRef]

Abraham, N. B.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

Acín, A.

S. Pironio, A. Acín, and S. Massar, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

Aida, H.

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, 1367–1379 (2010).
[CrossRef]

Amano, K.

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, 1367–1379 (2009).
[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]

Aoyama, K.

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 85, 016211 (2012).
[CrossRef]

Arai, K.-i.

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 85, 016211 (2012).
[CrossRef]

Argyris, A.

Assad, S. M.

T. Symul, S. M. Assad, and P. K. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett. 98, 231103 (2011).
[CrossRef]

Aviad, Y.

W. Li, I. Reidler, Y. Aviad, Y. Huang, H. Song, Y. Zhang, M. Rosenbluh, and I. Kanter, “Fast physical random-number generation based on room-temperature chaotic oscillations in weakly coupled superlattices,” Phys. Rev. Lett. 111, 044102 (2013).
[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, 024102 (2009).
[CrossRef] [PubMed]

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2009).
[CrossRef]

Bimberg, D.

L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching and polarization mode hopping in quantum dot vertical-cavity surface-emitting lasers,” Opt. Express 19, 2476–2484 (2011).
[CrossRef] [PubMed]

L. Olejniczak, M. Sciamanna, H. Thienpont, K. Panajotov, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching in quantum-dot vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 21, 1008 (2009).
[CrossRef]

F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. a. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, a. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. 89, 141106 (2006).
[CrossRef]

Bogris, A.

Bracikowski, C.

C. Bracikowski, R. Fox, and R. Roy, “Amplification of intrinsic noise in a chaotic multimode laser system,” Phys. Rev. A 45, 403 (1992).
[CrossRef] [PubMed]

Chan, S.

X. Li and S. Chan, “Heterodyne Random Bit Generation Using an Optically Injected Semiconductor Laser in Chaos,” IEEE J. Quantum Electron. 49, 829–838 (2013).
[CrossRef]

Chan, S.-C.

Cohen, E.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2009).
[CrossRef]

Colet, P.

R. Vicente, J. Dauden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41, 541–548 (2005).
[CrossRef]

Dauden, J.

R. Vicente, J. Dauden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41, 541–548 (2005).
[CrossRef]

Davis, P.

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 85, 016211 (2012).
[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, 1367–1379 (2010).
[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, 1367–1379 (2009).
[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]

Deligiannidis, S.

Dong, R.

C. Gabriel, C. Wittmann, D. Sych, and R. Dong, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4, 711–715 (2010).
[CrossRef]

Elspher, W.

I. Fischer, O. Hess, W. Elspher, and E. Göbel, “High-Dimensional Chaotic Dynamics of an External Cavity Semiconductor Laser,” Phys. Rev. Lett. 73, 2188–2191 (1994).
[CrossRef] [PubMed]

Feng, Q.

M. San Miguel, Q. Feng, and J.V. Moloney, “Light-polarization dynamics in surface emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[CrossRef] [PubMed]

Fiol, G.

F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. a. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, a. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. 89, 141106 (2006).
[CrossRef]

Fischer, I.

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

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. 36, 4632–4634 (2011).
[CrossRef] [PubMed]

I. Fischer, O. Hess, W. Elspher, and E. Göbel, “High-Dimensional Chaotic Dynamics of an External Cavity Semiconductor Laser,” Phys. Rev. Lett. 73, 2188–2191 (1994).
[CrossRef] [PubMed]

Fox, R.

C. Bracikowski, R. Fox, and R. Roy, “Amplification of intrinsic noise in a chaotic multimode laser system,” Phys. Rev. A 45, 403 (1992).
[CrossRef] [PubMed]

R. Fox and J. Keizer, “Amplification of intrinsic fluctuations by chaotic dynamics in physical systems,” Phys. Rev. A 43, 1709–1720 (1991).
[CrossRef] [PubMed]

Gabriel, C.

C. Gabriel, C. Wittmann, D. Sych, and R. Dong, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4, 711–715 (2010).
[CrossRef]

Göbel, E.

I. Fischer, O. Hess, W. Elspher, and E. Göbel, “High-Dimensional Chaotic Dynamics of an External Cavity Semiconductor Laser,” Phys. Rev. Lett. 73, 2188–2191 (1994).
[CrossRef] [PubMed]

Guo, H.

Y. Liu, M.-Y. Zhu, B. Luo, J.-W. Zhang, and H. Guo, “Implementation of 1.6 Tb s 1 truly random number generation based on a super-luminescent emitting diode,” Laser Phys. Lett. 10, 045001 (2013).
[CrossRef]

Harayama, T.

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 85, 016211 (2012).
[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, 1367–1379 (2010).
[CrossRef]

Hess, O.

I. Fischer, O. Hess, W. Elspher, and E. Göbel, “High-Dimensional Chaotic Dynamics of an External Cavity Semiconductor Laser,” Phys. Rev. Lett. 73, 2188–2191 (1994).
[CrossRef] [PubMed]

Hirano, K.

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, 1367–1379 (2010).
[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, 1367–1379 (2009).
[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]

Hopfer, F.

L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching and polarization mode hopping in quantum dot vertical-cavity surface-emitting lasers,” Opt. Express 19, 2476–2484 (2011).
[CrossRef] [PubMed]

L. Olejniczak, M. Sciamanna, H. Thienpont, K. Panajotov, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching in quantum-dot vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 21, 1008 (2009).
[CrossRef]

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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, 1367–1379 (2009).
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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 85, 016211 (2012).
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W. Li, I. Reidler, Y. Aviad, Y. Huang, H. Song, Y. Zhang, M. Rosenbluh, and I. Kanter, “Fast physical random-number generation based on room-temperature chaotic oscillations in weakly coupled superlattices,” Phys. Rev. Lett. 111, 044102 (2013).
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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).
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I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2009).
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F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. a. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, a. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. 89, 141106 (2006).
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T. Symul, S. M. Assad, and P. K. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett. 98, 231103 (2011).
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F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. a. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, a. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. 89, 141106 (2006).
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W. Li, I. Reidler, Y. Aviad, Y. Huang, H. Song, Y. Zhang, M. Rosenbluh, and I. Kanter, “Fast physical random-number generation based on room-temperature chaotic oscillations in weakly coupled superlattices,” Phys. Rev. Lett. 111, 044102 (2013).
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S. Pironio, A. Acín, and S. Massar, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
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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 85, 016211 (2012).
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F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. a. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, a. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. 89, 141106 (2006).
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Mutig, A.

L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching and polarization mode hopping in quantum dot vertical-cavity surface-emitting lasers,” Opt. Express 19, 2476–2484 (2011).
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L. Olejniczak, M. Sciamanna, H. Thienpont, K. Panajotov, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching in quantum-dot vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 21, 1008 (2009).
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F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. a. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, a. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. 89, 141106 (2006).
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I. Shimada and T. Nagashima, “A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems,” Prog. Theor. Phys. 61, 1605–1616 (1979).
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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, 1367–1379 (2009).
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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).
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Okumura, H.

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, 1367–1379 (2010).
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L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching and polarization mode hopping in quantum dot vertical-cavity surface-emitting lasers,” Opt. Express 19, 2476–2484 (2011).
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L. Olejniczak, M. Sciamanna, H. Thienpont, K. Panajotov, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching in quantum-dot vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 21, 1008 (2009).
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N. Oliver, M. Soriano, D. Sukow, and I. Fischer, “Fast random bit generation using a chaotic laser: approaching the information theoretic limit,” IEEE J. Quant. Electron. 49, 910–918 (2013).
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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. 36, 4632–4634 (2011).
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[CrossRef]

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M. Virte, K. Panajotov, and M. Sciamanna, “Bifurcation to nonlinear polarization dynamics and chaos in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87, 013834 (2013).
[CrossRef]

M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics 7, 60–65 (2012).
[CrossRef]

L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching and polarization mode hopping in quantum dot vertical-cavity surface-emitting lasers,” Opt. Express 19, 2476–2484 (2011).
[CrossRef] [PubMed]

L. Olejniczak, M. Sciamanna, H. Thienpont, K. Panajotov, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching in quantum-dot vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 21, 1008 (2009).
[CrossRef]

Pikasis, E.

Pironio, S.

S. Pironio, A. Acín, and S. Massar, “Random numbers certified by Bell’s theorem,” Nature 464, 1021–1024 (2010).
[CrossRef] [PubMed]

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J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

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W. Li, I. Reidler, Y. Aviad, Y. Huang, H. Song, Y. Zhang, M. Rosenbluh, and I. Kanter, “Fast physical random-number generation based on room-temperature chaotic oscillations in weakly coupled superlattices,” Phys. Rev. Lett. 111, 044102 (2013).
[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, 024102 (2009).
[CrossRef] [PubMed]

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2009).
[CrossRef]

Rosenbluh, M.

W. Li, I. Reidler, Y. Aviad, Y. Huang, H. Song, Y. Zhang, M. Rosenbluh, and I. Kanter, “Fast physical random-number generation based on room-temperature chaotic oscillations in weakly coupled superlattices,” Phys. Rev. Lett. 111, 044102 (2013).
[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, 024102 (2009).
[CrossRef] [PubMed]

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4, 58–61 (2009).
[CrossRef]

Roy, R.

Rukhin, A.

A. Rukhin, J. Soto, and J. Nechvatal, “A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications - Special Publication 800-22 Rev1a,” Tech. Rep. April, National Institute of Standards and Technology (2010).

Salevan, J. C.

San Miguel, M.

J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765–783 (1997).
[CrossRef]

M. San Miguel, Q. Feng, and J.V. Moloney, “Light-polarization dynamics in surface emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995).
[CrossRef] [PubMed]

Sciamanna, M.

M. Virte, K. Panajotov, and M. Sciamanna, “Bifurcation to nonlinear polarization dynamics and chaos in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87, 013834 (2013).
[CrossRef]

M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics 7, 60–65 (2012).
[CrossRef]

L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching and polarization mode hopping in quantum dot vertical-cavity surface-emitting lasers,” Opt. Express 19, 2476–2484 (2011).
[CrossRef] [PubMed]

L. Olejniczak, M. Sciamanna, H. Thienpont, K. Panajotov, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching in quantum-dot vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 21, 1008 (2009).
[CrossRef]

Shchukin, V. a.

F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. a. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, a. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. 89, 141106 (2006).
[CrossRef]

Shiki, M.

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]

Shimada, I.

I. Shimada and T. Nagashima, “A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems,” Prog. Theor. Phys. 61, 1605–1616 (1979).
[CrossRef]

Someya, H.

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]

Song, H.

W. Li, I. Reidler, Y. Aviad, Y. Huang, H. Song, Y. Zhang, M. Rosenbluh, and I. Kanter, “Fast physical random-number generation based on room-temperature chaotic oscillations in weakly coupled superlattices,” Phys. Rev. Lett. 111, 044102 (2013).
[CrossRef] [PubMed]

Soriano, M.

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

Soriano, M. C.

Soto, J.

A. Rukhin, J. Soto, and J. Nechvatal, “A Statistical Test Suite for Random and Pseudorandom Number Generators for Cryptographic Applications - Special Publication 800-22 Rev1a,” Tech. Rep. April, National Institute of Standards and Technology (2010).

Sukow, D.

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

Sukow, D. W.

Sunada, S.

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 85, 016211 (2012).
[CrossRef]

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C. Gabriel, C. Wittmann, D. Sych, and R. Dong, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4, 711–715 (2010).
[CrossRef]

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T. Symul, S. M. Assad, and P. K. Lam, “Real time demonstration of high bitrate quantum random number generation with coherent laser light,” Appl. Phys. Lett. 98, 231103 (2011).
[CrossRef]

Syvridis, D.

Thienpont, H.

M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics 7, 60–65 (2012).
[CrossRef]

L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching and polarization mode hopping in quantum dot vertical-cavity surface-emitting lasers,” Opt. Express 19, 2476–2484 (2011).
[CrossRef] [PubMed]

L. Olejniczak, M. Sciamanna, H. Thienpont, K. Panajotov, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching in quantum-dot vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 21, 1008 (2009).
[CrossRef]

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R. Vicente, J. Dauden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41, 541–548 (2005).
[CrossRef]

Uchida, A.

T. Yamazaki and A. Uchida, “Performance of Random Number Generators Using Noise-based Super-Luminescent Diode and Chaos-based Semiconductor Lasers,” IEEE J. Sel. Top. Quantum Electron. 19, 0600309 (2013).
[CrossRef]

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 85, 016211 (2012).
[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, 1367–1379 (2010).
[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, 1367–1379 (2009).
[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]

Vicente, R.

R. Vicente, J. Dauden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. 41, 541–548 (2005).
[CrossRef]

Virte, M.

M. Virte, K. Panajotov, and M. Sciamanna, “Bifurcation to nonlinear polarization dynamics and chaos in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87, 013834 (2013).
[CrossRef]

M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics 7, 60–65 (2012).
[CrossRef]

Werner, P.

F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. a. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, a. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. 89, 141106 (2006).
[CrossRef]

Williams, C. R. S.

Wittmann, C.

C. Gabriel, C. Wittmann, D. Sych, and R. Dong, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics 4, 711–715 (2010).
[CrossRef]

Yamazaki, T.

T. Yamazaki and A. Uchida, “Performance of Random Number Generators Using Noise-based Super-Luminescent Diode and Chaos-based Semiconductor Lasers,” IEEE J. Sel. Top. Quantum Electron. 19, 0600309 (2013).
[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, 1367–1379 (2010).
[CrossRef]

Yoshimori, S.

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, 1367–1379 (2010).
[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, 1367–1379 (2009).
[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).
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Figures (3)

Fig. 1
Fig. 1

Experimental setup, from polarization chaos to chaotic time series. The VCSEL generates polarization chaos: using a polarizer at 45° with respect to the linear polarization direction at threshold, polarization chaos can be transformed into chaotic intensity fluctuations. This polarizer combined with a rotator and a second polarizer form an optical isolator which prevents reflections from the fibre. RF spectra are acquired with a fast photo detector, an electronic amplifier and a RF spectrum analyser, whereas time-series are recorded with the small-bandwidth photodiode of the oscilloscope optical channel. Electronic bandwidths are indicated for each device.

Fig. 2
Fig. 2

Characterization of Polarization Chaos Time-Series. (a) Typical intensity fluctuations recorded by the oscilloscope 8-bit ADC and the small-bandwidth photodiode with the corresponding distribution given by the black curve in (b); the green curve shows the resulting histogram after a comparison of the time-series with a time-shifted (250 ps) version of itself. (c) RF spectra of the chaotic fluctuations, laser noise and amplification noise in red, black and blue respectively. Vertical dotted lines give the electronic bandwidth of the small-bandwidth photodiode of the oscilloscope optical channel and of the oscilloscope (2.4 and 4 GHz resp.). (d) Autocorrelation of the time-series.

Fig. 3
Fig. 3

Entropy evolution for polarization chaos. (a) Black, red, blue and green give the entropy evolution for noise levels of −30, −40, −50 and −60 dB respectively. Dash-doted, solid and dashed lines give the evolution for unfiltered, filtered, and filtered + processed cases respectively. (b) is the same plot as (a) but in logarithmic scale for the entropy. (c) and (d) gives an example of the separation between 3 different chaotic time-series with a unfiltered (c) and filtered output power (d) and noise level of −60 dB.

Tables (2)

Tables Icon

Table 1 Results of the NIST Statistical Test. The tests were performed with 1000 sequences of 1 million bits each. We use a significance level of α = 0.01, hence the tests are successful if the P-value (uniformity of the p-values) is larger than 0.0001 and the proportion is in the range 0.99 ± 0.0094392. For tests generating multiple outputs we give the worst case scenario. In case 1, we use a single comparison and keep the 5 LSBs of the waveform. In case 2, we perform 23 comparisons and keep 23 LSBs of the waveform. In both case the time-shift is 250 ps.

Tables Icon

Table 2 Results of the Diehard Statistical Test. The tests were performed with a sequence of 80 millions bits. For tests with multiple P-value worst case was selected. The tests are considered successful if the P-value is between 0.01 and 0.99. Description of case 1 and 2 is similar to the one given for Table 1. [KS] means that a Kolmogorov-Sinai test was applied.

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