Abstract

We propose and experimentally demonstrate an approach to generate and distribute secret keys over optical fiber communication infrastructure. Mach-Zehnder interferometers (MZIs) are adopted for key generation by transferring the environmental noise to random optical signals. A novel combination of wideband optical noise and an asymmetric MZI structure enables the secret keys to be securely transmitted and exchanged over public fiber links without being detected. We experimentally demonstrate this system and show reliable performance: keys are generated at the rate of 502 bit/s, and are successfully exchanged between two parties over a 10 km optical fiber with a bit error of $\sim$ 0.3%. System security analysis is performed by corroborating our experimental findings with simulations. The results show that our system can protect the key distribution under different attacks, attributed to wideband optical noise and asymmetric MZI structures. Compared to the previous schemes based on distributed MZIs, our scheme exploits localized MZI which provides twofold advantages. Firstly, the key generation rate can be increased by a factor of 5.7 at a negligible additional cost. Secondly, the system becomes robust to, in particular, active intrusion attack. The proposed system is a reliable and cost-effective solution for key establishment, and is compatible with the existing optical fiber communication infrastructure.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

C. Huang, P. Y. Ma, B. J. Shastri, P. Mittal, and P. R. Prucnal, “Robustness of optical steganographic communication under coherent detection attack,” IEEE Photonics Technol. Lett. 31(4), 327–330 (2019).
[Crossref]

2018 (2)

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate–distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

A. A. E. Hajomer, X. Yang, A. Sultan, and W. Hu, “Key Distribution Based on Phase Fluctuation Between Polarization Modes in Optical Channel,” IEEE Photonics Technol. Lett. 30(8), 704–707 (2018).
[Crossref]

2017 (1)

B. Colombier, L. Bossuet, V. Fischer, and D. Hély, “Key reconciliation protocols for error correction of silicon puf responses,” IEEE Trans. Inform. Forensic Secur. 12(8), 1988–2002 (2017).
[Crossref]

2016 (3)

E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” npj Quantum Inf. 2(1), 16025 (2016).
[Crossref]

J. Zhang, T. Q. Duong, A. Marshall, and R. Woods, “Key Generation From Wireless Channels: A Review,” IEEE Access 4, 614–626 (2016).
[Crossref]

N. Skorin-Kapov, M. Furdek, S. Zsigmond, and L. Wosinska, “Physical-layer security in evolving optical networks,” IEEE Commun. Mag. 54(8), 110–117 (2016).
[Crossref]

2015 (2)

W. Trappe, “The challenges facing physical layer security,” IEEE Commun. Mag. 53(6), 16–20 (2015).
[Crossref]

B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015).
[Crossref]

2014 (3)

2013 (3)

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, and L. Li, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

K. Kravtsov, Z. Wang, W. Trappe, and P. R. Prucnal, “Physical layer secret key generation for fiber-optical networks,” Opt. Express 21(20), 23756–23771 (2013).
[Crossref]

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

2011 (2)

Y.-S. Shiu, S. Y. Chang, H.-C. Wu, S. C.-H. Huang, and H.-H. Chen, “Physical layer security in wireless networks: A tutorial,” IEEE Wireless Commun. 18(2), 66–74 (2011).
[Crossref]

M. P. Fok, Z. Wang, Y. Deng, and P. R. Prucnal, “Optical Layer Security in Fiber-Optic Networks,” IEEE Transactions on Inf. Forensics Secur. 6(3), 725–736 (2011).
[Crossref]

2010 (4)

C. Ye, S. Mathur, A. Reznik, Y. Shah, W. Trappe, and N. B. Mandayam, “Information-Theoretically Secret Key Generation for Fading Wireless Channels,” IEEE Trans. Inform. Forensic Secur. 5(2), 240–254 (2010).
[Crossref]

N. Patwari, J. Croft, S. Jana, and S. K. Kasera, “High-Rate Uncorrelated Bit Extraction for Shared Secret Key Generation from Channel Measurements,” IEEE Transactions on Mob. Comput. 9(1), 17–30 (2010).
[Crossref]

J. Du, Y. Dai, G. K. Lei, W. Tong, and C. Shu, “Photonic crystal fiber based mach-zehnder interferometer for dpsk signal demodulation,” Opt. Express 18(8), 7917–7922 (2010).
[Crossref]

S. Zhang, P.-Y. Kam, C. Yu, and J. Chen, “Decision-Aided Carrier Phase Estimation for Coherent Optical Communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010).
[Crossref]

2009 (2)

M. G. Taylor, “Phase Estimation Methods for Optical Coherent Detection Using Digital Signal Processing,” J. Lightwave Technol. 27(7), 901–914 (2009).
[Crossref]

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]

2008 (1)

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

2006 (1)

2003 (1)

W.-Y. Hwang, “Quantum Key Distribution with High Loss: Toward Global Secure Communication,” Phys. Rev. Lett. 91(5), 057901 (2003).
[Crossref]

2002 (1)

R. Raz, O. Reingold, and S. Vadhan, “Extracting all the randomness and reducing the error in trevisan’s extractors,” J. Comput. Syst. Sci. 65(1), 97–128 (2002).
[Crossref]

1988 (1)

C. H. Bennett, G. Brassard, and J.-M. Robert, “Privacy amplification by public discussion,” SIAM J. Comput. 17(2), 210–229 (1988).
[Crossref]

Al Faruque, M. A.

I. U. Zaman, A. B. Lopez, M. A. Al Faruque, and O. Boyraz, “Polarization mode dispersion-based physical layer key generation for optical fiber link security,” in Optical Sensors, (Optical Society of America, 2017), pp. JTu4A–20

Andresen, E. R.

Barros, J.

M. Bloch and J. Barros, Physical-Layer Security: From Information Theory to Security Engineering (Cambridge University, 2011).

Bechmann-Pasquinucci, H.

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]

Belhadj, N.

M. Z. Iqbal, H. Fathallah, and N. Belhadj, “Optical fiber tapping: Methods and precautions,” in High Capacity Optical Networks and Enabling Technologies (HONET), 2011 (IEEE, 2011), pp. 164–168.

Bennett, C. H.

C. H. Bennett, G. Brassard, and J.-M. Robert, “Privacy amplification by public discussion,” SIAM J. Comput. 17(2), 210–229 (1988).
[Crossref]

Bloch, M.

M. Bloch and J. Barros, Physical-Layer Security: From Information Theory to Security Engineering (Cambridge University, 2011).

Bossuet, L.

B. Colombier, L. Bossuet, V. Fischer, and D. Hély, “Key reconciliation protocols for error correction of silicon puf responses,” IEEE Trans. Inform. Forensic Secur. 12(8), 1988–2002 (2017).
[Crossref]

Boyraz, O.

I. U. Zaman, A. B. Lopez, M. A. Al Faruque, and O. Boyraz, “Polarization mode dispersion-based physical layer key generation for optical fiber link security,” in Optical Sensors, (Optical Society of America, 2017), pp. JTu4A–20

Brassard, G.

C. H. Bennett, G. Brassard, and J.-M. Robert, “Privacy amplification by public discussion,” SIAM J. Comput. 17(2), 210–229 (1988).
[Crossref]

G. Brassard and L. Salvail, “Secret-key reconciliation by public discussion,” in Workshop on the Theory and Application of of Cryptographic Techniques, (Springer, 1993), pp. 410–423.

Bromberg, Y.

Y. Bromberg, B. Redding, S. M. Popoff, and H. Cao, “Remote Key Establishment by Mode Mixing in Multimode Fibres and Optical Reciprocity,” arXiv:1506.07892 [physics, physics:quant-ph] (2015). ArXiv: 1506.07892.

Cao, G.

H. Song, L. Xie, S. Zhu, and G. Cao, “Sensor node compromise detection: the location perspective,” in Proceedings of the 2007 international conference on Wireless communications and mobile computing, (ACM, 2007), pp. 242–247.

Cao, H.

Y. Bromberg, B. Redding, S. M. Popoff, and H. Cao, “Remote Key Establishment by Mode Mixing in Multimode Fibres and Optical Reciprocity,” arXiv:1506.07892 [physics, physics:quant-ph] (2015). ArXiv: 1506.07892.

Cerf, N. J.

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]

Chang, M. P.

Chang, S. Y.

Y.-S. Shiu, S. Y. Chang, H.-C. Wu, S. C.-H. Huang, and H.-H. Chen, “Physical layer security in wireless networks: A tutorial,” IEEE Wireless Commun. 18(2), 66–74 (2011).
[Crossref]

Chen, H.-H.

Y.-S. Shiu, S. Y. Chang, H.-C. Wu, S. C.-H. Huang, and H.-H. Chen, “Physical layer security in wireless networks: A tutorial,” IEEE Wireless Commun. 18(2), 66–74 (2011).
[Crossref]

Chen, J.

Chen, T.-Y.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, and L. Li, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Colombier, B.

B. Colombier, L. Bossuet, V. Fischer, and D. Hély, “Key reconciliation protocols for error correction of silicon puf responses,” IEEE Trans. Inform. Forensic Secur. 12(8), 1988–2002 (2017).
[Crossref]

Croft, J.

N. Patwari, J. Croft, S. Jana, and S. K. Kasera, “High-Rate Uncorrelated Bit Extraction for Shared Secret Key Generation from Channel Measurements,” IEEE Transactions on Mob. Comput. 9(1), 17–30 (2010).
[Crossref]

Cui, K.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, and L. Li, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Curty, M.

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
[Crossref]

Dai, Y.

de Riedmatten, H.

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

Deng, Y.

M. P. Fok, Z. Wang, Y. Deng, and P. R. Prucnal, “Optical Layer Security in Fiber-Optic Networks,” IEEE Transactions on Inf. Forensics Secur. 6(3), 725–736 (2011).
[Crossref]

Diamanti, E.

E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” npj Quantum Inf. 2(1), 16025 (2016).
[Crossref]

Du, J.

Duong, T. Q.

J. Zhang, T. Q. Duong, A. Marshall, and R. Woods, “Key Generation From Wireless Channels: A Review,” IEEE Access 4, 614–626 (2016).
[Crossref]

Dušek, M.

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]

Dynes, J. F.

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate–distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

Fathallah, H.

M. Z. Iqbal, H. Fathallah, and N. Belhadj, “Optical fiber tapping: Methods and precautions,” in High Capacity Optical Networks and Enabling Technologies (HONET), 2011 (IEEE, 2011), pp. 164–168.

Fischer, V.

B. Colombier, L. Bossuet, V. Fischer, and D. Hély, “Key reconciliation protocols for error correction of silicon puf responses,” IEEE Trans. Inform. Forensic Secur. 12(8), 1988–2002 (2017).
[Crossref]

Fok, M. P.

M. P. Fok, Z. Wang, Y. Deng, and P. R. Prucnal, “Optical Layer Security in Fiber-Optic Networks,” IEEE Transactions on Inf. Forensics Secur. 6(3), 725–736 (2011).
[Crossref]

Frost, N. A.

Furdek, M.

N. Skorin-Kapov, M. Furdek, S. Zsigmond, and L. Wosinska, “Physical-layer security in evolving optical networks,” IEEE Commun. Mag. 54(8), 110–117 (2016).
[Crossref]

Gisin, N.

B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015).
[Crossref]

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

Hajomer, A. A. E.

A. A. E. Hajomer, X. Yang, A. Sultan, and W. Hu, “Key Distribution Based on Phase Fluctuation Between Polarization Modes in Optical Channel,” IEEE Photonics Technol. Lett. 30(8), 704–707 (2018).
[Crossref]

Hardjono, T.

J. Pieprzyk, T. Hardjono, and J. Seberry, Fundamentals of Computer Security (Springer Science & Business Media, 2013).

Hély, D.

B. Colombier, L. Bossuet, V. Fischer, and D. Hély, “Key reconciliation protocols for error correction of silicon puf responses,” IEEE Trans. Inform. Forensic Secur. 12(8), 1988–2002 (2017).
[Crossref]

Houlmann, R.

B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015).
[Crossref]

Hu, W.

A. A. E. Hajomer, X. Yang, A. Sultan, and W. Hu, “Key Distribution Based on Phase Fluctuation Between Polarization Modes in Optical Channel,” IEEE Photonics Technol. Lett. 30(8), 704–707 (2018).
[Crossref]

Huang, C.

C. Huang, P. Y. Ma, B. J. Shastri, P. Mittal, and P. R. Prucnal, “Robustness of optical steganographic communication under coherent detection attack,” IEEE Photonics Technol. Lett. 31(4), 327–330 (2019).
[Crossref]

Huang, S. C.-H.

Y.-S. Shiu, S. Y. Chang, H.-C. Wu, S. C.-H. Huang, and H.-H. Chen, “Physical layer security in wireless networks: A tutorial,” IEEE Wireless Commun. 18(2), 66–74 (2011).
[Crossref]

Huang, Y.-K.

B. Wu, Y.-K. Huang, S. Zhang, B. J. Shastri, and P. R. Prucnal, “Long range secure key distribution over multiple amplified fiber spans based on environmental instabilities,” in CLEO: Science and Innovations (Optical Society of America, 2016), pp. SF1F–4.

Hwang, W.-Y.

W.-Y. Hwang, “Quantum Key Distribution with High Loss: Toward Global Secure Communication,” Phys. Rev. Lett. 91(5), 057901 (2003).
[Crossref]

Iqbal, M. Z.

M. Z. Iqbal, H. Fathallah, and N. Belhadj, “Optical fiber tapping: Methods and precautions,” in High Capacity Optical Networks and Enabling Technologies (HONET), 2011 (IEEE, 2011), pp. 164–168.

Jana, S.

N. Patwari, J. Croft, S. Jana, and S. K. Kasera, “High-Rate Uncorrelated Bit Extraction for Shared Secret Key Generation from Channel Measurements,” IEEE Transactions on Mob. Comput. 9(1), 17–30 (2010).
[Crossref]

Kam, P.-Y.

Kasera, S. K.

N. Patwari, J. Croft, S. Jana, and S. K. Kasera, “High-Rate Uncorrelated Bit Extraction for Shared Secret Key Generation from Channel Measurements,” IEEE Transactions on Mob. Comput. 9(1), 17–30 (2010).
[Crossref]

Keiding, S. R.

Korzh, B.

B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015).
[Crossref]

Kravtsov, K.

Krishnamachari, V. V.

Laperle, C.

Lei, G. K.

Li, L.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, and L. Li, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Li, M. J.

B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015).
[Crossref]

Liang, H.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, and L. Li, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Lim, C. C. W.

B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015).
[Crossref]

Liu, N.-L.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, and L. Li, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Liu, Y.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, and L. Li, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Lo, H.-K.

E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” npj Quantum Inf. 2(1), 16025 (2016).
[Crossref]

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
[Crossref]

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

Lopez, A. B.

I. U. Zaman, A. B. Lopez, M. A. Al Faruque, and O. Boyraz, “Polarization mode dispersion-based physical layer key generation for optical fiber link security,” in Optical Sensors, (Optical Society of America, 2017), pp. JTu4A–20

Lucamarini, M.

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate–distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

Lütkenhaus, N.

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]

Ma, P. Y.

C. Huang, P. Y. Ma, B. J. Shastri, P. Mittal, and P. R. Prucnal, “Robustness of optical steganographic communication under coherent detection attack,” IEEE Photonics Technol. Lett. 31(4), 327–330 (2019).
[Crossref]

Ma, X.

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

Mandayam, N. B.

C. Ye, S. Mathur, A. Reznik, Y. Shah, W. Trappe, and N. B. Mandayam, “Information-Theoretically Secret Key Generation for Fading Wireless Channels,” IEEE Trans. Inform. Forensic Secur. 5(2), 240–254 (2010).
[Crossref]

Marshall, A.

J. Zhang, T. Q. Duong, A. Marshall, and R. Woods, “Key Generation From Wireless Channels: A Review,” IEEE Access 4, 614–626 (2016).
[Crossref]

Mathur, S.

C. Ye, S. Mathur, A. Reznik, Y. Shah, W. Trappe, and N. B. Mandayam, “Information-Theoretically Secret Key Generation for Fading Wireless Channels,” IEEE Trans. Inform. Forensic Secur. 5(2), 240–254 (2010).
[Crossref]

Minár, J.

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

Mittal, P.

C. Huang, P. Y. Ma, B. J. Shastri, P. Mittal, and P. R. Prucnal, “Robustness of optical steganographic communication under coherent detection attack,” IEEE Photonics Technol. Lett. 31(4), 327–330 (2019).
[Crossref]

Nolan, D.

B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015).
[Crossref]

O’Sullivan, M.

Patwari, N.

N. Patwari, J. Croft, S. Jana, and S. K. Kasera, “High-Rate Uncorrelated Bit Extraction for Shared Secret Key Generation from Channel Measurements,” IEEE Transactions on Mob. Comput. 9(1), 17–30 (2010).
[Crossref]

Peev, M.

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]

Pieprzyk, J.

J. Pieprzyk, T. Hardjono, and J. Seberry, Fundamentals of Computer Security (Springer Science & Business Media, 2013).

Popoff, S. M.

Y. Bromberg, B. Redding, S. M. Popoff, and H. Cao, “Remote Key Establishment by Mode Mixing in Multimode Fibres and Optical Reciprocity,” arXiv:1506.07892 [physics, physics:quant-ph] (2015). ArXiv: 1506.07892.

Potma, E. O.

Prucnal, P. R.

C. Huang, P. Y. Ma, B. J. Shastri, P. Mittal, and P. R. Prucnal, “Robustness of optical steganographic communication under coherent detection attack,” IEEE Photonics Technol. Lett. 31(4), 327–330 (2019).
[Crossref]

B. Wu, Z. Wang, B. J. Shastri, M. P. Chang, N. A. Frost, and P. R. Prucnal, “Temporal phase mask encrypted optical steganography carried by amplified spontaneous emission noise,” Opt. Express 22(1), 954–961 (2014).
[Crossref]

K. Kravtsov, Z. Wang, W. Trappe, and P. R. Prucnal, “Physical layer secret key generation for fiber-optical networks,” Opt. Express 21(20), 23756–23771 (2013).
[Crossref]

M. P. Fok, Z. Wang, Y. Deng, and P. R. Prucnal, “Optical Layer Security in Fiber-Optic Networks,” IEEE Transactions on Inf. Forensics Secur. 6(3), 725–736 (2011).
[Crossref]

B. Wu, Y.-K. Huang, S. Zhang, B. J. Shastri, and P. R. Prucnal, “Long range secure key distribution over multiple amplified fiber spans based on environmental instabilities,” in CLEO: Science and Innovations (Optical Society of America, 2016), pp. SF1F–4.

Qi, B.

E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” npj Quantum Inf. 2(1), 16025 (2016).
[Crossref]

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

Raz, R.

R. Raz, O. Reingold, and S. Vadhan, “Extracting all the randomness and reducing the error in trevisan’s extractors,” J. Comput. Syst. Sci. 65(1), 97–128 (2002).
[Crossref]

Redding, B.

Y. Bromberg, B. Redding, S. M. Popoff, and H. Cao, “Remote Key Establishment by Mode Mixing in Multimode Fibres and Optical Reciprocity,” arXiv:1506.07892 [physics, physics:quant-ph] (2015). ArXiv: 1506.07892.

Reingold, O.

R. Raz, O. Reingold, and S. Vadhan, “Extracting all the randomness and reducing the error in trevisan’s extractors,” J. Comput. Syst. Sci. 65(1), 97–128 (2002).
[Crossref]

Reznik, A.

C. Ye, S. Mathur, A. Reznik, Y. Shah, W. Trappe, and N. B. Mandayam, “Information-Theoretically Secret Key Generation for Fading Wireless Channels,” IEEE Trans. Inform. Forensic Secur. 5(2), 240–254 (2010).
[Crossref]

Robert, J.-M.

C. H. Bennett, G. Brassard, and J.-M. Robert, “Privacy amplification by public discussion,” SIAM J. Comput. 17(2), 210–229 (1988).
[Crossref]

Salomaa, A.

A. Salomaa, Public-Key Cryptography (Springer Science & Business Media, 2013).

Salvail, L.

G. Brassard and L. Salvail, “Secret-key reconciliation by public discussion,” in Workshop on the Theory and Application of of Cryptographic Techniques, (Springer, 1993), pp. 410–423.

Sanguinetti, B.

B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015).
[Crossref]

Scarani, V.

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]

Seberry, J.

J. Pieprzyk, T. Hardjono, and J. Seberry, Fundamentals of Computer Security (Springer Science & Business Media, 2013).

Shah, Y.

C. Ye, S. Mathur, A. Reznik, Y. Shah, W. Trappe, and N. B. Mandayam, “Information-Theoretically Secret Key Generation for Fading Wireless Channels,” IEEE Trans. Inform. Forensic Secur. 5(2), 240–254 (2010).
[Crossref]

Shastri, B. J.

C. Huang, P. Y. Ma, B. J. Shastri, P. Mittal, and P. R. Prucnal, “Robustness of optical steganographic communication under coherent detection attack,” IEEE Photonics Technol. Lett. 31(4), 327–330 (2019).
[Crossref]

B. Wu, Z. Wang, B. J. Shastri, M. P. Chang, N. A. Frost, and P. R. Prucnal, “Temporal phase mask encrypted optical steganography carried by amplified spontaneous emission noise,” Opt. Express 22(1), 954–961 (2014).
[Crossref]

B. Wu, Y.-K. Huang, S. Zhang, B. J. Shastri, and P. R. Prucnal, “Long range secure key distribution over multiple amplified fiber spans based on environmental instabilities,” in CLEO: Science and Innovations (Optical Society of America, 2016), pp. SF1F–4.

Shentu, G.-L.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, and L. Li, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Shields, A. J.

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate–distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

Shiu, Y.-S.

Y.-S. Shiu, S. Y. Chang, H.-C. Wu, S. C.-H. Huang, and H.-H. Chen, “Physical layer security in wireless networks: A tutorial,” IEEE Wireless Commun. 18(2), 66–74 (2011).
[Crossref]

Shu, C.

Simon, C.

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

Skorin-Kapov, N.

N. Skorin-Kapov, M. Furdek, S. Zsigmond, and L. Wosinska, “Physical-layer security in evolving optical networks,” IEEE Commun. Mag. 54(8), 110–117 (2016).
[Crossref]

Song, H.

H. Song, L. Xie, S. Zhu, and G. Cao, “Sensor node compromise detection: the location perspective,” in Proceedings of the 2007 international conference on Wireless communications and mobile computing, (ACM, 2007), pp. 242–247.

Sultan, A.

A. A. E. Hajomer, X. Yang, A. Sultan, and W. Hu, “Key Distribution Based on Phase Fluctuation Between Polarization Modes in Optical Channel,” IEEE Photonics Technol. Lett. 30(8), 704–707 (2018).
[Crossref]

Tamaki, K.

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
[Crossref]

Tan, X.

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

Taylor, M. G.

Thew, R.

B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015).
[Crossref]

Tong, W.

Trappe, W.

W. Trappe, “The challenges facing physical layer security,” IEEE Commun. Mag. 53(6), 16–20 (2015).
[Crossref]

K. Kravtsov, Z. Wang, W. Trappe, and P. R. Prucnal, “Physical layer secret key generation for fiber-optical networks,” Opt. Express 21(20), 23756–23771 (2013).
[Crossref]

C. Ye, S. Mathur, A. Reznik, Y. Shah, W. Trappe, and N. B. Mandayam, “Information-Theoretically Secret Key Generation for Fading Wireless Channels,” IEEE Trans. Inform. Forensic Secur. 5(2), 240–254 (2010).
[Crossref]

Trevisan, L.

L. Trevisan and S. Vadhan, “Extracting randomness from samplable distributions,” in Proceedings 41st Annual Symposium on Foundations of Computer Science, (IEEE, 2000), pp. 32–42.

Vadhan, S.

R. Raz, O. Reingold, and S. Vadhan, “Extracting all the randomness and reducing the error in trevisan’s extractors,” J. Comput. Syst. Sci. 65(1), 97–128 (2002).
[Crossref]

L. Trevisan and S. Vadhan, “Extracting randomness from samplable distributions,” in Proceedings 41st Annual Symposium on Foundations of Computer Science, (IEEE, 2000), pp. 32–42.

Wang, J.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, and L. Li, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Wang, L.-J.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, and L. Li, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Wang, Z.

Woods, R.

J. Zhang, T. Q. Duong, A. Marshall, and R. Woods, “Key Generation From Wireless Channels: A Review,” IEEE Access 4, 614–626 (2016).
[Crossref]

Wosinska, L.

N. Skorin-Kapov, M. Furdek, S. Zsigmond, and L. Wosinska, “Physical-layer security in evolving optical networks,” IEEE Commun. Mag. 54(8), 110–117 (2016).
[Crossref]

Wu, B.

B. Wu, Z. Wang, B. J. Shastri, M. P. Chang, N. A. Frost, and P. R. Prucnal, “Temporal phase mask encrypted optical steganography carried by amplified spontaneous emission noise,” Opt. Express 22(1), 954–961 (2014).
[Crossref]

B. Wu, Y.-K. Huang, S. Zhang, B. J. Shastri, and P. R. Prucnal, “Long range secure key distribution over multiple amplified fiber spans based on environmental instabilities,” in CLEO: Science and Innovations (Optical Society of America, 2016), pp. SF1F–4.

Wu, H.-C.

Y.-S. Shiu, S. Y. Chang, H.-C. Wu, S. C.-H. Huang, and H.-H. Chen, “Physical layer security in wireless networks: A tutorial,” IEEE Wireless Commun. 18(2), 66–74 (2011).
[Crossref]

Xie, L.

H. Song, L. Xie, S. Zhu, and G. Cao, “Sensor node compromise detection: the location perspective,” in Proceedings of the 2007 international conference on Wireless communications and mobile computing, (ACM, 2007), pp. 242–247.

Xu, F.

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

Xu, H.

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

Yang, X.

A. A. E. Hajomer, X. Yang, A. Sultan, and W. Hu, “Key Distribution Based on Phase Fluctuation Between Polarization Modes in Optical Channel,” IEEE Photonics Technol. Lett. 30(8), 704–707 (2018).
[Crossref]

Ye, C.

C. Ye, S. Mathur, A. Reznik, Y. Shah, W. Trappe, and N. B. Mandayam, “Information-Theoretically Secret Key Generation for Fading Wireless Channels,” IEEE Trans. Inform. Forensic Secur. 5(2), 240–254 (2010).
[Crossref]

Yin, H.-L.

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, and L. Li, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

Yu, C.

Yuan, Z.

E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” npj Quantum Inf. 2(1), 16025 (2016).
[Crossref]

Yuan, Z. L.

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate–distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

Zaman, I. U.

I. U. Zaman, A. B. Lopez, M. A. Al Faruque, and O. Boyraz, “Polarization mode dispersion-based physical layer key generation for optical fiber link security,” in Optical Sensors, (Optical Society of America, 2017), pp. JTu4A–20

Zbinden, H.

B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015).
[Crossref]

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

Zhang, J.

J. Zhang, T. Q. Duong, A. Marshall, and R. Woods, “Key Generation From Wireless Channels: A Review,” IEEE Access 4, 614–626 (2016).
[Crossref]

Zhang, S.

S. Zhang, P.-Y. Kam, C. Yu, and J. Chen, “Decision-Aided Carrier Phase Estimation for Coherent Optical Communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010).
[Crossref]

B. Wu, Y.-K. Huang, S. Zhang, B. J. Shastri, and P. R. Prucnal, “Long range secure key distribution over multiple amplified fiber spans based on environmental instabilities,” in CLEO: Science and Innovations (Optical Society of America, 2016), pp. SF1F–4.

Zhu, S.

H. Song, L. Xie, S. Zhu, and G. Cao, “Sensor node compromise detection: the location perspective,” in Proceedings of the 2007 international conference on Wireless communications and mobile computing, (ACM, 2007), pp. 242–247.

Zsigmond, S.

N. Skorin-Kapov, M. Furdek, S. Zsigmond, and L. Wosinska, “Physical-layer security in evolving optical networks,” IEEE Commun. Mag. 54(8), 110–117 (2016).
[Crossref]

IEEE Access (1)

J. Zhang, T. Q. Duong, A. Marshall, and R. Woods, “Key Generation From Wireless Channels: A Review,” IEEE Access 4, 614–626 (2016).
[Crossref]

IEEE Commun. Mag. (2)

W. Trappe, “The challenges facing physical layer security,” IEEE Commun. Mag. 53(6), 16–20 (2015).
[Crossref]

N. Skorin-Kapov, M. Furdek, S. Zsigmond, and L. Wosinska, “Physical-layer security in evolving optical networks,” IEEE Commun. Mag. 54(8), 110–117 (2016).
[Crossref]

IEEE Photonics Technol. Lett. (2)

A. A. E. Hajomer, X. Yang, A. Sultan, and W. Hu, “Key Distribution Based on Phase Fluctuation Between Polarization Modes in Optical Channel,” IEEE Photonics Technol. Lett. 30(8), 704–707 (2018).
[Crossref]

C. Huang, P. Y. Ma, B. J. Shastri, P. Mittal, and P. R. Prucnal, “Robustness of optical steganographic communication under coherent detection attack,” IEEE Photonics Technol. Lett. 31(4), 327–330 (2019).
[Crossref]

IEEE Trans. Inform. Forensic Secur. (2)

B. Colombier, L. Bossuet, V. Fischer, and D. Hély, “Key reconciliation protocols for error correction of silicon puf responses,” IEEE Trans. Inform. Forensic Secur. 12(8), 1988–2002 (2017).
[Crossref]

C. Ye, S. Mathur, A. Reznik, Y. Shah, W. Trappe, and N. B. Mandayam, “Information-Theoretically Secret Key Generation for Fading Wireless Channels,” IEEE Trans. Inform. Forensic Secur. 5(2), 240–254 (2010).
[Crossref]

IEEE Transactions on Inf. Forensics Secur. (1)

M. P. Fok, Z. Wang, Y. Deng, and P. R. Prucnal, “Optical Layer Security in Fiber-Optic Networks,” IEEE Transactions on Inf. Forensics Secur. 6(3), 725–736 (2011).
[Crossref]

IEEE Transactions on Mob. Comput. (1)

N. Patwari, J. Croft, S. Jana, and S. K. Kasera, “High-Rate Uncorrelated Bit Extraction for Shared Secret Key Generation from Channel Measurements,” IEEE Transactions on Mob. Comput. 9(1), 17–30 (2010).
[Crossref]

IEEE Wireless Commun. (1)

Y.-S. Shiu, S. Y. Chang, H.-C. Wu, S. C.-H. Huang, and H.-H. Chen, “Physical layer security in wireless networks: A tutorial,” IEEE Wireless Commun. 18(2), 66–74 (2011).
[Crossref]

J. Comput. Syst. Sci. (1)

R. Raz, O. Reingold, and S. Vadhan, “Extracting all the randomness and reducing the error in trevisan’s extractors,” J. Comput. Syst. Sci. 65(1), 97–128 (2002).
[Crossref]

J. Lightwave Technol. (3)

Nat. Photonics (2)

H.-K. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014).
[Crossref]

B. Korzh, C. C. W. Lim, R. Houlmann, N. Gisin, M. J. Li, D. Nolan, B. Sanguinetti, R. Thew, and H. Zbinden, “Provably secure and practical quantum key distribution over 307 km of optical fibre,” Nat. Photonics 9(3), 163–168 (2015).
[Crossref]

Nature (1)

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate–distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

npj Quantum Inf. (1)

E. Diamanti, H.-K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” npj Quantum Inf. 2(1), 16025 (2016).
[Crossref]

Opt. Express (4)

Phys. Rev. A (2)

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

J. Minář, H. de Riedmatten, C. Simon, H. Zbinden, and N. Gisin, “Phase-noise measurements in long-fiber interferometers for quantum-repeater applications,” Phys. Rev. A 77(5), 052325 (2008).
[Crossref]

Phys. Rev. Lett. (2)

Y. Liu, T.-Y. Chen, L.-J. Wang, H. Liang, G.-L. Shentu, J. Wang, K. Cui, H.-L. Yin, N.-L. Liu, and L. Li, “Experimental measurement-device-independent quantum key distribution,” Phys. Rev. Lett. 111(13), 130502 (2013).
[Crossref]

W.-Y. Hwang, “Quantum Key Distribution with High Loss: Toward Global Secure Communication,” Phys. Rev. Lett. 91(5), 057901 (2003).
[Crossref]

Rev. Mod. Phys. (1)

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]

SIAM J. Comput. (1)

C. H. Bennett, G. Brassard, and J.-M. Robert, “Privacy amplification by public discussion,” SIAM J. Comput. 17(2), 210–229 (1988).
[Crossref]

Other (10)

L. Trevisan and S. Vadhan, “Extracting randomness from samplable distributions,” in Proceedings 41st Annual Symposium on Foundations of Computer Science, (IEEE, 2000), pp. 32–42.

G. Brassard and L. Salvail, “Secret-key reconciliation by public discussion,” in Workshop on the Theory and Application of of Cryptographic Techniques, (Springer, 1993), pp. 410–423.

M. Z. Iqbal, H. Fathallah, and N. Belhadj, “Optical fiber tapping: Methods and precautions,” in High Capacity Optical Networks and Enabling Technologies (HONET), 2011 (IEEE, 2011), pp. 164–168.

H. Song, L. Xie, S. Zhu, and G. Cao, “Sensor node compromise detection: the location perspective,” in Proceedings of the 2007 international conference on Wireless communications and mobile computing, (ACM, 2007), pp. 242–247.

A. Salomaa, Public-Key Cryptography (Springer Science & Business Media, 2013).

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Figures (8)

Fig. 1.
Fig. 1. System illustration: The key is generated from the environmental noise by using two localized MZIs collectively. The key is securely exchanged between Alice and Bob in public fiber links without encryption. ASE: amplified spontaneous emission, serves as the key carrier; PD: photo-detector, is used to detect the key; PC: polarization controller.
Fig. 2.
Fig. 2. (the key waveforms obtained by (a) Alice and (b) Bob, respectively; (c) correlation measurement of two waveforms as a function of time lag. The peak correlation coefficient is 0.86, indicating the key waveforms obtained by Alice and Bob have significant similarity.
Fig. 3.
Fig. 3. (a) Autocorrelation functions for the two signals measured by Alice and Bob. (b) 8-bit key histogram of Alice’s extracted key
Fig. 4.
Fig. 4. Temporal phase in (a) a distributed MZI and (b) a localized MZI; temporal waveforms in (c) a distributed MZI and (d) a localized MZI; frequency spectra of the signal generated in (e) a distributed MZI and (f) a localized MZI. The two MZIs are compared at the same fiber length and phase deviation. The bandwidth of the signal in the localized MZI is 5.7 times higher than that in the distributed MZI.
Fig. 5.
Fig. 5. (a) the signal tapped from point A at the transmission link by Eve; (b) the signal waveform received by Bob.
Fig. 6.
Fig. 6. The measurement of cross-correlation coefficient against the MZI mismatch.
Fig. 7.
Fig. 7. The relation between the MZI arm length and (a) cross-correlation coefficient of the keys received by Alice and Bob; and (b) the supported transmission distance (defined as the transmission distance when the cross-correlation coefficient is 0.86)
Fig. 8.
Fig. 8. Illustration of the attack model of tapping the fiber link and recovering the key signal.

Equations (9)

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E A ( t ) = 1 2 A 0 A exp ( i ω A t ) { exp ( i θ n A ( t ) ) exp [ i ( θ n A ( t + t M Z I ) + Δ θ e A ( t ) ] } .
E A B ( t ) = 1 4 ( α L T ) 1 2 A 0 A exp ( i ω A t ) { exp [ i ( θ n A ( t + t T + t M Z I ) + Δ θ e B ( t + t M Z I + t T ) ) ] exp [ i ( θ n A ( t + 2 t M Z I + t T ) + Δ θ e A ( t ) + Δ θ e B ( t + t M Z I + t T ) ) ] exp [ i θ n A ( t + t T ) ] exp [ i ( θ n A ( t + t M Z I + t T ) + Δ θ e A ( t ) ) ] } .
I A B ( t ) = | E A B ( t ) | 2 = 1 8 α L T | A 0 A | 2 [ 2 cos ( Δ θ e A ( t ) Δ θ e B ( t + t M Z I + t T ) ) ] .
I B A ( t ) = | E B A ( t ) | 2 = 1 8 α L T | A 0 B | 2 [ 2 cos ( Δ θ e A ( t + t M Z I + t T ) Δ θ e B ( t ) ) ] .
K A B ( t ) = cos ( Δ θ e A ( t ) Δ θ e B ( t + t M Z I + t T ) ) K B A ( t ) = cos ( Δ θ e B ( t ) Δ θ e A ( t + t M Z I + t T ) )
I A ( t ) = | E A ( t ) | 2 = 1 2 | A 0 A | 2 { cos [ θ n A ( t + t M Z I ) θ n A ( t ) + Δ θ e A ( t ) ] + 1 } .
S T A ( t ) = cos [ θ n A ( t + t M Z I ) θ n A ( t ) + Δ θ e A ( t ) ]
Δ θ ( J τ ) = j = 1 J i = 1 I δ φ τ i j
I A E ( t ) = 1 2 | A 0 A | 2 [ cos ( Δ θ e A ( t ) ) + 1 ] I B E ( t ) = 1 2 | A 0 B | 2 [ cos ( Δ θ e B ( t ) ) + 1 ]

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