Synchronization of random bit generators based on coupled chaotic lasers and application to cryptography

Ido Kanter; Maria Butkovski; Yitzhak Peleg; Meital Zigzag; Yaara Aviad; Igor Reidler; Michael Rosenbluh; Wolfgang Kinzel

doi:10.1364/OE.18.018292

Synchronization of random bit generators based on coupled chaotic lasers and application to cryptography

Ido Kanter, Maria Butkovski, Yitzhak Peleg, Meital Zigzag, Yaara Aviad, Igor Reidler, Michael Rosenbluh, and Wolfgang Kinzel

Optics Express
Vol. 18,
Issue 17,
pp. 18292-18302
(2010)
•https://doi.org/10.1364/OE.18.018292

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Ido Kanter, Maria Butkovski, Yitzhak Peleg, Meital Zigzag, Yaara Aviad, Igor Reidler, Michael Rosenbluh, and Wolfgang Kinzel, "Synchronization of random bit generators based on coupled chaotic lasers and application to cryptography," Opt. Express 18, 18292-18302 (2010)

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Abstract

Random bit generators (RBGs) constitute an important tool in cryptography, stochastic simulations and secure communications. The later in particular has some difficult requirements: high generation rate of unpredictable bit strings and secure key-exchange protocols over public channels. Deterministic algorithms generate pseudo-random number sequences at high rates, however, their unpredictability is limited by the very nature of their deterministic origin. Recently, physical RBGs based on chaotic semiconductor lasers were shown to exceed Gbit/s rates. Whether secure synchronization of two high rate physical RBGs is possible remains an open question. Here we propose a method, whereby two fast RBGs based on mutually coupled chaotic lasers, are synchronized. Using information theoretic analysis we demonstrate security against a powerful computational eavesdropper, capable of noiseless amplification, where all parameters are publicly known. The method is also extended to secure synchronization of a small network of three RBGs.

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References

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Year
|
Author
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Publication

M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information. (Cambridge University Press, Cambridge, 2000).
D. R. Stinson, Cryptography: Theory and Practice. (CRC Press, Boca Raton, 1995).
R. G. Gallager, Principles of Digital Communication. (Cambridge University Press, Cambridge, 2008).
C. H. Bennett, C. Hand, and G. Brassard, “Quantum Cryptography: Public key distribution and coin tossing,” Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, p. 175 (1984).
A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[Crossref] [PubMed]
V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]
N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44(247), 335–341 (1949).
[Crossref] [PubMed]
S. Asmussen, and P. W. Glynn, Stochastic Simulation: Algorithms and Analysis. (Springer-Verlag, New York, 2007).
T. E. Murphy and R. Roy, “Chaotic lasers: The world's fastest dice,” Nat. Photonics 2(12), 714–715 (2008).
[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(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]
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]
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]
E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(6), 066214 (2006).
[Crossref] [PubMed]
I. Kanter, E. Kopelowitz, and W. Kinzel, “Public channel cryptography: chaos synchronization and Hilbert’s tenth problem,” Phys. Rev. Lett. 101(8), 084102 (2008).
[Crossref] [PubMed]
M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. 85(6), 60005 (2009).
[Crossref]
R. J. Jones, P. S. Spencer, J. Lawrence, and D. M. Kane, “Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback,” IEEE Proc. Optoelectron. 148(1), 7–12 (2001).
[Crossref]
R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]
V. Ahlers, U. Parlitz, and W. Lauterborn, “Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(6), 7208–7213 (1998).
[Crossref]
E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006).
[Crossref] [PubMed]
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]
T. M. Cover, and J. A. Thomas, Elements of Information Theory (John Wiley and Sons, New York, 1991).
V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]
J. Kestler, W. Kinzel, and I. Kanter, “Sublattice synchronization of chaotic networks with delayed couplings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(3), 035202 (2007).
[Crossref] [PubMed]

2010 (2)

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]

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]

2009 (4)

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]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009).
[Crossref]

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. 85(6), 60005 (2009).
[Crossref]

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

2008 (3)

I. Kanter, E. Kopelowitz, and W. Kinzel, “Public channel cryptography: chaos synchronization and Hilbert’s tenth problem,” Phys. Rev. Lett. 101(8), 084102 (2008).
[Crossref] [PubMed]

T. E. Murphy and R. Roy, “Chaotic lasers: The world's fastest dice,” Nat. Photonics 2(12), 714–715 (2008).
[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(12), 728–732 (2008).
[Crossref]

2007 (1)

J. Kestler, W. Kinzel, and I. Kanter, “Sublattice synchronization of chaotic networks with delayed couplings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(3), 035202 (2007).
[Crossref] [PubMed]

2006 (2)

E. Klein, N. Gross, E. Kopelowitz, M. Rosenbluh, L. Khaykovich, W. Kinzel, and I. Kanter, “Public-channel cryptography based on mutual chaos pass filters,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(4), 046201 (2006).
[Crossref] [PubMed]

E. Klein, N. Gross, M. Rosenbluh, W. Kinzel, L. Khaykovich, and I. Kanter, “Stable isochronal synchronization of mutually coupled chaotic lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(6), 066214 (2006).
[Crossref] [PubMed]

2005 (1)

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]

2001 (1)

R. J. Jones, P. S. Spencer, J. Lawrence, and D. M. Kane, “Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback,” IEEE Proc. Optoelectron. 148(1), 7–12 (2001).
[Crossref]

1998 (1)

V. Ahlers, U. Parlitz, and W. Lauterborn, “Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(6), 7208–7213 (1998).
[Crossref]

1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[Crossref] [PubMed]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

1949 (1)

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

Ahlers, V.

Aida, H.

Amano, K.

Annovazzi-Lodi, V.

Argyris, A.

Aviad, Y.

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]

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]

Bechmann-Pasquinucci, H.

Butkovski, M.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. 85(6), 60005 (2009).
[Crossref]

Cerf, N. J.

Cohen, E.

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]

Colet, P.

D’Huys, O.

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

Danckaert, J.

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

Davis, P.

Dušek, M.

Ekert, A. K.

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[Crossref] [PubMed]

Englert, A.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. 85(6), 60005 (2009).
[Crossref]

Fischer, I.

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

Flunkert, V.

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

García-Ojalvo, J.

Gross, N.

Harayama, T.

Hirano, K.

Inoue, M.

Jones, R. J.

Kane, D. M.

Kanter, I.

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]

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]

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. 85(6), 60005 (2009).
[Crossref]

I. Kanter, E. Kopelowitz, and W. Kinzel, “Public channel cryptography: chaos synchronization and Hilbert’s tenth problem,” Phys. Rev. Lett. 101(8), 084102 (2008).
[Crossref] [PubMed]

Kestler, J.

Khaykovich, L.

Kinzel, W.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. 85(6), 60005 (2009).
[Crossref]

I. Kanter, E. Kopelowitz, and W. Kinzel, “Public channel cryptography: chaos synchronization and Hilbert’s tenth problem,” Phys. Rev. Lett. 101(8), 084102 (2008).
[Crossref] [PubMed]

Klein, E.

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Kopelowitz, E.

I. Kanter, E. Kopelowitz, and W. Kinzel, “Public channel cryptography: chaos synchronization and Hilbert’s tenth problem,” Phys. Rev. Lett. 101(8), 084102 (2008).
[Crossref] [PubMed]

Kurashige, T.

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Larger, L.

Lauterborn, W.

Lawrence, J.

Lütkenhaus, N.

Metropolis, N.

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

Mirasso, C. R.

Morikatsu, S.

Murphy, T. E.

T. E. Murphy and R. Roy, “Chaotic lasers: The world's fastest dice,” Nat. Photonics 2(12), 714–715 (2008).
[Crossref]

Naito, S.

Okumura, H.

Oowada, I.

Parlitz, U.

Peev, M.

Pesquera, L.

Reidler, I.

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]

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]

Rosenbluh, M.

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]

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]

Roy, R.

T. E. Murphy and R. Roy, “Chaotic lasers: The world's fastest dice,” Nat. Photonics 2(12), 714–715 (2008).
[Crossref]

Scarani, V.

Schöll, E.

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

Shiki, M.

Shore, K. A.

Someya, H.

Spencer, P. S.

Syvridis, D.

Uchida, A.

Ulam, S.

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

Yamazaki, T.

Yoshimori, S.

Yoshimura, K.

Zigzag, M.

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. 85(6), 60005 (2009).
[Crossref]

Europhys. Lett. (1)

M. Zigzag, M. Butkovski, A. Englert, W. Kinzel, and I. Kanter, “Zero lag synchronization of chaotic units with time-delayed couplings,” Europhys. Lett. 85(6), 60005 (2009).
[Crossref]

IEEE J. Quantum Electron. (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

IEEE Proc. Optoelectron. (1)

J. Am. Stat. Assoc. (1)

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

Nat. Photonics (3)

T. E. Murphy and R. Roy, “Chaotic lasers: The world's fastest dice,” Nat. Photonics 2(12), 714–715 (2008).
[Crossref]

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]

Nature (1)

Opt. Express (1)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (4)

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

Phys. Rev. Lett. (3)

I. Kanter, E. Kopelowitz, and W. Kinzel, “Public channel cryptography: chaos synchronization and Hilbert’s tenth problem,” Phys. Rev. Lett. 101(8), 084102 (2008).
[Crossref] [PubMed]

I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009).
[Crossref] [PubMed]

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661–663 (1991).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

Other (6)

S. Asmussen, and P. W. Glynn, Stochastic Simulation: Algorithms and Analysis. (Springer-Verlag, New York, 2007).

M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information. (Cambridge University Press, Cambridge, 2000).

D. R. Stinson, Cryptography: Theory and Practice. (CRC Press, Boca Raton, 1995).

R. G. Gallager, Principles of Digital Communication. (Cambridge University Press, Cambridge, 2008).

C. H. Bennett, C. Hand, and G. Brassard, “Quantum Cryptography: Public key distribution and coin tossing,” Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, p. 175 (1984).

T. M. Cover, and J. A. Thomas, Elements of Information Theory (John Wiley and Sons, New York, 1991).

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Fig. 1

Zero lag synchronization scheme for two and three lasers and an attacker. (a) Two mutually coupled SLs, A and B, where a third SL, C, is unidirectionally coupled to the mutually transmitted signals. The self-coupling delays for A and B are τ_A and τ_B, mutual coupling delay is τ, κ and σ are the strengths of self and mutual couplings. and similarly κ_C and σ_C for C. (b) A small network of three symmetrically mutually coupled SLs, A, B and C, where all coupling delays are equal to τ and a fourth laser, D, is coupled unidirectionally to each of the three transmitted signals. All delay times are equal.

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Fig. 2

Cross correlation for mutual and unidirectional coupling. (a) Cross correlation at zero time lag is calculated for two mutually coupled SLs (Fig. 1(a)) for a range of parameter values: κ, feedback strength and σ, coupling strength. (b) Cross correlation at zero time lag between a third SL coupled unidirectionally to one of the parties (Fig. 1(a)), using identical κ and σ as the parties.

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Fig. 3

Bit error rate for the attacker and the parties. (a) BER is calculated for laser C in the setup of Fig. 1(a) as a function of (κ, σ), when A and B are operating with κ=90 ns⁻¹ and σ=40 ns⁻¹ (indicated by the arrow). (b) BER among the parties in the setup of Fig. 1(a) as a function of (κ, σ). The BER for each (κ, σ) is averaged over 1 μs and the modulation bandwidth is 1 Gbit/s.

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Fig. 4

Secure regions for two and three synchronized RBGs. (a) Two mutually coupled lasers as in the setup of Fig. 1(a). where all delays are equal. (b) Three mutually coupled lasers as in schematic Fig. 1(b). The BER of the MCPF procedure between a pair of parties, p, and between a party and the attacker, q, is calculated assuming uncorrelated decoded bits by the parties and by the attacker. The colored regions (colored in either blue or red) indicate a necessary condition for the failure of an attacker before the reconciliation procedure, P_AC<P_AB . The region where an attacker cannot succeed in recovering the key, even when using the leakage of information of the reconciliation procedure, is indicated in red.

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Fig. 5

Histogram of averaged chaotic intensity and BER as a function of bit length. (a) 1 ns bit length for a run time of 25 μs. (b) 50 ns bit length for a run time of 25 μs. (c) BER of the parties for a run time of 16 μs. Parameters: κ=90 ns⁻¹, σ=40 ns⁻¹.

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Fig. 6

Scheme of two mutually coupled lasers via a semi-permeable mirror. Two mutually coupled SLs, A and B, where a 50% semi-transparent mirror (M) is placed at a distance of τ_Α/2 and τ_Β/2 from A and B, respectively. The scheme is equivalent to Fig. 1(a) with τ=(τ_A+τ_B)/2 and κ = σ.

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Fig. 7

Venn diagram for the mutual information of three communicating parties. The entropy of a transmitted bit of each party is represented by a circle normalized to 1. T₃ – common information for all parties, T₂ – common information for each pair of parties and 1-T₁ stands for the independent information of each party.

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Tables (1)

Table 1 Probabilities for all possible scenarios of transmitted and received bits by the communicating parties.

View Table

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

P_{A B} = 1 - p + 0.5 p^{2}

\frac{σ M^{2}}{I (σ + κ)}

\frac{σ_{C} M^{2}}{I (σ_{C} + κ_{C})}

P_{A C} = 1 - 0.5 p - q (1 - 1.5 p) + q^{2} (0.5 - p)

{I(S}_{A} {,S}_{B}) {>I(S}_{C} {,S}_{A} {)+I(S}_{C} {,S}_{B} {|S}_{A})

{I(S}_{D} {,S}_{A} {)+I(S}_{D} {,S}_{B} {|S}_{C} {)+I(S}_{D} {,S}_{C} {|S}_{A} {,S}_{B} {)-2I(S}_{A} {,S}_{B} {)-I(S}_{A} {,S}_{C} {|S}_{B})+1>0

0 < p < \frac{1}{2} (1 + 3 q - 2 q^{2} - \sqrt{{(1 + 3 q - 2 q^{2})}^{2} - 4 q (2 - q)})

0 < p < \sqrt{2 Z^{4} - 6 Z^{3} + (6 + 5 q - 8 q^{2} + 4 q^{3}) Z^{2} - (2 + 8 q - 10 q^{2} + 4 q^{3}) Z + (3 q - 3 q^{2} + q^{3}) = 0}

M = (\begin{array}{l} 0.899 0.035 \\ 0.035 0.031 \end{array}), N = (\begin{array}{l} 0.714 0.122 \\ 0.122 0.042 \end{array})

M_{} = (\begin{array}{l} 0.872 0.062 \\ 0.062 0.004 \end{array}), N = (\begin{array}{l} 0.697 0.138 \\ 0.138 0.027 \end{array})

A	R_A	B	R_B	Act. A	Act. B	Prob. of configuration
1	1	1	1	1	1	(1-p)²/4
1	1	1	−1	1	0	p(1-p)/4
1	1	−1	1	1	0	p(1-p)/4
1	1	−1	−1	1	−1	P²/4
1	−1	1	1	0	1	p(1-p)/4
1	−1	1	−1	0	0	P²/4
1	−1	−1	1	0	0	(1-p)²/4
1	−1	−1	−1	0	−1	p(1-p)/4
−1	1	1	1	0	1	p(1-p)/4
−1	1	1	−1	0	0	(1-p)²/4
−1	1	−1	1	0	0	P²/4
−1	1	−1	−1	0	−1	p(1-p)/4
−1	−1	1	1	−1	1	P²/4
−1	−1	1	−1	−1	0	p(1-p)/4
−1	−1	−1	1	−1	0	p(1-p)/4
−1	−1	−1	−1	−1	−1	(1-p)²/4