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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 14 — Jul. 15, 2013
  • pp: 16663–16669
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Experimental circular quantum secret sharing over telecom fiber network

Ke-Jin Wei, Hai-Qiang Ma, and Jian-Hui Yang  »View Author Affiliations


Optics Express, Vol. 21, Issue 14, pp. 16663-16669 (2013)
http://dx.doi.org/10.1364/OE.21.016663


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Abstract

We present a robust single photon circular quantum secret sharing (QSS) scheme with phase encoding over 50 km single mode fiber network using a circular QSS protocol. Our scheme can automatically provide a perfect compensation of birefringence and remain stable for a long time. A high visibility of 99.3% is obtained. Furthermore, our scheme realizes a polarization insensitive phase modulators. The visibility of this system can be maintained perpetually without any adjustment to the system every time we test the system.

© 2013 OSA

1. Introduction

Secret sharing [1

1. A. Shamir, “How to share a secret,” Commun. ACM 22(11), 612–613 (1979). [CrossRef]

] is an important branch of cryptography, its objective being to distribute a secret message among certain clients, such that all of them can share the secret if they all agree to collaborate, while if a subset tries to sidestep the others, no-one can obtain the message. Classical secret sharing is vulnerable to eavesdropping attacks, but the emergence of quantum cryptography promises a new path since it allows the unconditionally secure distribution of information between all the clients, based on the laws of quantum mechanics. Since Hillery et al. first presented a quantum secret sharing (QSS) scheme in 1999 using three-particle and four-particle Greenberger-Horne-Zeilinger (GHZ) entangled states [2

2. M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999). [CrossRef]

], many other QSS schemes [3

3. R. Cleve, D. Gottesman, and H. K. Lo, “How to share a quantum secret,” Phys. Rev. Lett. 83(3), 648–651 (1999). [CrossRef]

8

8. X. B. Chen, S. Yang, Y. Su, and Y. X. Yang, “Cryptanalysis on the improved multiparty quantum secret sharing protocol based on the GHZ state,” Phys. Scr. 86(5), 055002 (2012). [CrossRef]

] have been proposed in both theoretical and experimental aspects. In these schemes, multiparticle polarization entangled states are widely used and so are rather impractical since entangled states are difficult to generate and transmit. Some of the protocols that propose using a single photon qubit [9

9. F. G. Deng, H. Y. Zhou, and G. L. Long, “Bidirectional quantum secret sharing and secret splitting with polarized single photons,” Phys. Lett. A 337(4-6), 329–334 (2005). [CrossRef]

11

11. L. F. Han, Y. M. Liu, J. Liu, and Z. J. Zhang, “Multiparty quantum secret sharing of secure direct communication using single photons,” Opt. Commun. 281(9), 2690–2694 (2008). [CrossRef]

] have not been demonstrated experimentally. Moreover, since commercial single-mode fiber has birefringence, polarization encoding is impractical in applications over commercial networks. Recently, Bogdanski et al [12

12. J. Bogdanski, J. Ahrens, and M. Bourennane, “Sagnac secret sharing over telecom fiber networks,” Opt. Express 17(2), 1055–1063 (2009). [CrossRef] [PubMed]

, 13

13. J. Bogdanski, N. Rafiei, and M. Bourennane, “Experimental quantum secret sharing using telecommunication fiber,” Phys. Rev. A 78(6), 062307 (2008). [CrossRef]

] reported using a single qubit protocol [14

14. C. Schmid, P. Trojek, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, “Experimental single qubit quantum secret sharing,” Phys. Rev. Lett. 95(23), 230505 (2005). [CrossRef] [PubMed]

] with phase encoding in a 1550 nm fiber network, thus solving the above mentioned problems. However, the stability of their systems was based on a complicated polarization control system [15

15. J. Bogdanski, J. Ahrens, and M. Bourennane, “Single mode fiber birefringence compensation in Sagnac and “plug & play” interferometric setups,” Opt. Express 17(6), 4485–4494 (2009). [CrossRef] [PubMed]

] with a polarization insensitive phase modulator, and the bit generation efficiency was just 50%.

In this letter we present a robust single-photon QSS experiment for three parties over a single-mode fiber network in a Mach-Zehnder interferometric setup using the single qubit protocol proposed by Deng et al [16

16. F. G. Deng, H. Y. Zhou, and G. L. Long, “Circular quantum secret sharing,” J. Phys. Math. Gen. 39(45), 14089–14099 (2006). [CrossRef]

] with phase encoding. This protocol just requires a single qubit for information transmission, which has allowed its practical experimental realization and scalability, and its efficiency can reach nearly 100%. Moreover, our scheme provides autocompensation of birefringence with just a commercial polarization-sensitive phase modulator connected to a Faraday mirror, which has been shown to provide full compensation of the birefringence [17

17. D. S. Bethune and W. P. Risk, “Autocompensating quantum cryptography,” New J. Phys. 4, 421–4215 (2002).

]. Thanks to this design, our setup can also automatically provide completely polarization-insensitive phase modulation; to the best of our knowledge, there is no commercially available polarization-insensitive phase modulator, and our method is simpler than other previously reported ways to realize polarization-insensitive phase modulation [12

12. J. Bogdanski, J. Ahrens, and M. Bourennane, “Sagnac secret sharing over telecom fiber networks,” Opt. Express 17(2), 1055–1063 (2009). [CrossRef] [PubMed]

, 18

18. D. S. Bethune, M. Navarro, and W. P. Risk, “Enhanced autocompensating quantum cryptography system,” Appl. Opt. 41(9), 1640–1648 (2002). [CrossRef] [PubMed]

].

The scheme of this protocol for the three parties is shown in Fig. 1
Fig. 1 Scheme of the circular secret sharing protocol.
.

Using two sets of conjugate measuring bases, Alice randomly prepares her qubits which are polarized single photons into one of the following four states:
{|+z,|z,|+x,|x},
(1)
where

|+z=|0,           |z=|1,
(2)
|+x=12(|0+|1),        |x=12(|0|1).
(3)

Bob and Charlie can perform randomly one of the two unitary operations, U0 and U1 which represent the bits 0 and 1 respectively, on the single photons that they receive:

U0=|00|+|11|,
(4)
U1=|01||10|.
(5)

The identity operator U0 does nothing on the single photon. The nice feature of the U1 operation is that it flips or negates the states in both measuring bases. i. e.

U1|+z=|z,               U1|z=|+z,
(6)
U1|+x=|x,               U1|x=|+x.
(7)

To create the private key UA, Alice sends a quantum signal to Bob, who can chose the encoding mode or the control mode randomly. In the encoding mode, bob then encodes the photon with the two unitary operations U0 and U1 chosen randomly, marked by UB, and sends it to Charlie. Charlie performs a similar operation in the same way as Bob, and returns the photon to Alice after his operation marked as UC. For each photon that she receives, Alice performs a single-photon measurement with the same basis as the one she originally used to prepare it. As the two unitary operations U0 and U1 do not change the measuring bases, Alice obtains a deterministic outcome for almost all the photons returned, e. g., UA = UB ⊗ UC. In order to prevent any eavesdropper from getting information about UA, Bob and Charlie can chose the control mode, in the control mode, they measure a few photons he received with one of the two measuring bases randomly. Alice also can measure a few photons in this way. In the eavesdropping check Bob, Charlie or Alice will publish the result of measurements and negotiate with the parties who sent these photons to him. In essence, the security of this QSS protocol is ensured by the analysis of the error rates in a similar way to the BB84 and LM05 protocol [19

19. C. H. Bennett and G. Brassard, “Quantum Cryptography: Public key distribution and coin tossing,” in IEEE Int.Conf. on Computers, Systems, and Signal Processing, (Bangalore, 1984), 175–179.

, 20

20. M. Lucamarini and S. Mancini, “Secure deterministic communication without entanglement,” Phys. Rev. Lett. 94(14), 140501 (2005). [CrossRef] [PubMed]

].

In the current work only the encoding mode of the Deng’s protocol is experimentally realized. So, as a consequence of what said above, we cannot claim the security of the scheme against a number of specific attacks, like, for instance, the Trojan-horse. However, the fact is that the Trojan-horse attack is a serious threat for all the known QSS protocols and quantum key distribution [21

21. N. Gisin, S. Fasel, B. Kraus, H. Zbinden, and G. Ribordy, “Trojan-horse attacks on quantum-key-distribution systems,” Phys. Rev. A 73(2), 022320 (2006). [CrossRef]

]. The current partial implementation of the Deng’s protocol represents a meaningful result, as it is directly connected to the final secure rate achievable with a complete implementation.

2. Experimental setup

Figure 2
Fig. 2 Experimental setup. LD: laser diode; A: attenuator; Cir1, Cir2: circulators; PMA, PMB, PMC: phase-sensitive modulators; FM1, FM2, FM3: Faraday mirror; D1, D2: single photon detector; C: a 50 / 50 coupler; PBS1, PBS2, polarization beam splitter; PMA, PMB, PMC: phase modulator; SW1, SW2: fiber switch; CM1, CM2: the configuration of control mode; QC1; QC2: quantum channel.
shows the configuration of our QSS setup, which can be divided into two main parts:the sender Alice and the two other parties Bob and Charlie who can only reconstruct the secret code together. Alice’s station contains a pulsed 1310 nm laser diode LD of pulsewidth 20ps, an attenuator A, a phase-sensitive modulator PMA connected to a Faraday mirror FM1, two single-photon detectors D1 and D2, a 50 / 50 coupler, and two polarization beamsplitters PBS1 and PBS2 which are connected to the other parties. All Alice station’s components are polarization maintaining and aligned to the “horizontal” axis.

Bob and Charlie have identical components, including a phase sensitive modulator terminated by a Faraday mirror (PMB, FM2 and PMC, FM3, respectively), and a quantum channel. They also have a control mode box and a fiber switch use to choose the control mode.

The pulsed laser emits vertically polarized pulses at a repetition rate of 1 MHZ which are attenuated at A to the level of single photon, pass through the circulator Cir1, and are randomly modulated by the polarization-sensitive phase modulator PMA to have a phase ϕA of 0, π/2, π or 3π/2. After reflection by the Faraday mirror FM1 the polarization is rotated by 90° to the horizontal direction, and so the phase is not modulated in the return pass through PMA. The pulses then proceed through Cir2 and an ordinary 50/50 coupler C where they are divided along two paths and are transmitted through the two polarizing beamsplitters PBS1 and PBS2 to Bob and Charlie, respectively. The upper pulses, denoted by L1 in Fig. 2, enter SW1 in Bob’s port and passes through a quantum channel QC1 to the phase modulator PMB and Faraday mirror FM2. On the first pass through PMB there is no phase modulation since the photons are horizontally polarized, but after reflection at FM2 they have undergone a polarization rotation of 90° so on the return pass they can acquire a phase ϕB of 0 or π On returning to Alice’s station this vertically polarized pulse is reflected by PBS2 and PBS1 and reaches Charlie’s port. Charlie has the same setup as Bob, and also modulates the phase by ϕC of 0 or π; the pulse is reflected by FM3 back to PBS1 and arrive at the lower arm of coupler C. The lower pulse, denoted by L2 in Fig. 2, transmit over the opposite path so it arrives at the upper arm of coupler C at the same time of the upper pulse, but none of parties modulate phase in the lower pulse when it through the PMB and PMC. Thus interference occurs at C; a photon is detected at D1 when ϕB + ϕC = 0 or 2π, and at D2 if ϕB + ϕC = π. It should be particularly noted that our setup is based on phase encoding, so all three parties must modulate the phase of their photons, as mentioned above. After interference in the coupler C the photons arrive at detectors D1 or D2, depending on their phases shift. The detection probabilities of D1 and D2 for different phase values chosen by Alice, Bob and Charile are shown in Table 1

Table 1. Detection probabilities for different phase values of ϕA, ϕB and ϕC with the Deng’s protocol

table-icon
View This Table
. As Alice’s phase didn’t influence the detection probabilities of D1 and D2, the table just shown the result ϕA = 0 and ϕA = π. Thus Alice have an identical bit string according to their assumption such as: D1 click represents “0”, and D2 click represents “1”. Bob and Charlie must share its phase with each other when they want to know Alice’s sharing. The fiber switch is to switch between encoding mode and control mode, i.e. When Bob (Charlie) want to measure the photon he received, he can choose the SW1 (SW2) to CM1 (CM2) and completed the control mode presented in Deng’s paper. As said, the control mode is not currently implemented in our apparatus; however, its effect is simulated through the fiber switch since the counts are directed into CM1 or CM2 depicted in the figure. Compare to some paper using a beam splitter to realize a control mode [22

22. R. Kumar, M. Lucamarini, G. Giuseppe, R. Natali, G. Mancini, and P. Tombesi, “Two-way quantum key distribution at telecommunication wavelength,” Phys. Rev. A 77(2), 022304 (2008). [CrossRef]

, 23

23. M. F. Abdul Khir, M. Zain, I. Bahari, and S. Shaari, “Implementation of two way Quantum Key Distribution protocol with decoy state,” Opt. Commun. 285(5), 842–845 (2012). [CrossRef]

], the fiber switch provide a convenient way and the user can choose a mode alternatively according to the need of protocol.

3. Experimental data

The stability of this scheme was tested over a total fiber length of 50 km. Figure 3
Fig. 3 Counts of D1、D2 vs modulation voltage of PMB.
shows the count rates in D1 and D2 versus the voltage VPMB of Bob’s phase modulator, with the voltage of the other parties’ modulators fixed at 0. We can see that the there is good interference of the photons, and at the half-wave voltage (~4.2V), the counts in D1 and D2 reach their minimum and maximum, respectively, the photon pulses can contain the information all parties load in it and the “polarization insensitive phase modulator” modulator works well.

We measured the interference visibility when none of parties change their voltage of PM. The pluses sent from Alice’s port are attenuated at A to the level of single photon. At this, the visibility is equal to C1C2C1+C2, C1 and C2 mean the counts of D1 and D2 respectively.

Figure 4
Fig. 4 Visibility for the total fiber length of 50km during 3 hours.
shows the visibility measurement results tested over a total of 3 hours. The average visibility is 99.3% with a fluctuation of about 0.4%. The setup was found to be extremely stable and this visibility could be repeated without any adjustment to any part every time we started up the system.

4. Conclusion

Acknowledgments

This work was supported by the National Program for Basic Research in China Grant No. 2010CB923202; the National Natural Science Foundation of China Grant No. 61178010.

References and links

1.

A. Shamir, “How to share a secret,” Commun. ACM 22(11), 612–613 (1979). [CrossRef]

2.

M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59(3), 1829–1834 (1999). [CrossRef]

3.

R. Cleve, D. Gottesman, and H. K. Lo, “How to share a quantum secret,” Phys. Rev. Lett. 83(3), 648–651 (1999). [CrossRef]

4.

L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A 69(5), 052307 (2004). [CrossRef]

5.

S. Gaertner, C. Kurtsiefer, M. Bourennane, and H. Weinfurter, “Experimental demonstration of four-party quantum secret sharing,” Phys. Rev. Lett. 98(2), 020503, 4 (2007). [CrossRef] [PubMed]

6.

Y. Sun, Q. Y. Wen, F. Gao, X. B. Chen, and F. C. Zhu, “Multiparty quantum secret sharing based on bell measurement,” Opt. Commun. 282(17), 3647–3651 (2009). [CrossRef]

7.

L. Hao, C. Wang, and G. L. Long, “Quantum secret sharing protocol with four state Grover algorithm and its proof-of-principle experimental demonstration,” Opt. Commun. 284(14), 3639–3642 (2011). [CrossRef]

8.

X. B. Chen, S. Yang, Y. Su, and Y. X. Yang, “Cryptanalysis on the improved multiparty quantum secret sharing protocol based on the GHZ state,” Phys. Scr. 86(5), 055002 (2012). [CrossRef]

9.

F. G. Deng, H. Y. Zhou, and G. L. Long, “Bidirectional quantum secret sharing and secret splitting with polarized single photons,” Phys. Lett. A 337(4-6), 329–334 (2005). [CrossRef]

10.

F. L. Yan, T. Gao, and Y. C. Li, “Quantum secret sharing protocol between multiparty and multiparty with single photons and unitary transformations,” Chin. Phys. Lett. 25, 4 (2008).

11.

L. F. Han, Y. M. Liu, J. Liu, and Z. J. Zhang, “Multiparty quantum secret sharing of secure direct communication using single photons,” Opt. Commun. 281(9), 2690–2694 (2008). [CrossRef]

12.

J. Bogdanski, J. Ahrens, and M. Bourennane, “Sagnac secret sharing over telecom fiber networks,” Opt. Express 17(2), 1055–1063 (2009). [CrossRef] [PubMed]

13.

J. Bogdanski, N. Rafiei, and M. Bourennane, “Experimental quantum secret sharing using telecommunication fiber,” Phys. Rev. A 78(6), 062307 (2008). [CrossRef]

14.

C. Schmid, P. Trojek, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, “Experimental single qubit quantum secret sharing,” Phys. Rev. Lett. 95(23), 230505 (2005). [CrossRef] [PubMed]

15.

J. Bogdanski, J. Ahrens, and M. Bourennane, “Single mode fiber birefringence compensation in Sagnac and “plug & play” interferometric setups,” Opt. Express 17(6), 4485–4494 (2009). [CrossRef] [PubMed]

16.

F. G. Deng, H. Y. Zhou, and G. L. Long, “Circular quantum secret sharing,” J. Phys. Math. Gen. 39(45), 14089–14099 (2006). [CrossRef]

17.

D. S. Bethune and W. P. Risk, “Autocompensating quantum cryptography,” New J. Phys. 4, 421–4215 (2002).

18.

D. S. Bethune, M. Navarro, and W. P. Risk, “Enhanced autocompensating quantum cryptography system,” Appl. Opt. 41(9), 1640–1648 (2002). [CrossRef] [PubMed]

19.

C. H. Bennett and G. Brassard, “Quantum Cryptography: Public key distribution and coin tossing,” in IEEE Int.Conf. on Computers, Systems, and Signal Processing, (Bangalore, 1984), 175–179.

20.

M. Lucamarini and S. Mancini, “Secure deterministic communication without entanglement,” Phys. Rev. Lett. 94(14), 140501 (2005). [CrossRef] [PubMed]

21.

N. Gisin, S. Fasel, B. Kraus, H. Zbinden, and G. Ribordy, “Trojan-horse attacks on quantum-key-distribution systems,” Phys. Rev. A 73(2), 022320 (2006). [CrossRef]

22.

R. Kumar, M. Lucamarini, G. Giuseppe, R. Natali, G. Mancini, and P. Tombesi, “Two-way quantum key distribution at telecommunication wavelength,” Phys. Rev. A 77(2), 022304 (2008). [CrossRef]

23.

M. F. Abdul Khir, M. Zain, I. Bahari, and S. Shaari, “Implementation of two way Quantum Key Distribution protocol with decoy state,” Opt. Commun. 285(5), 842–845 (2012). [CrossRef]

OCIS Codes
(270.0270) Quantum optics : Quantum optics
(270.5565) Quantum optics : Quantum communications
(270.5568) Quantum optics : Quantum cryptography

ToC Category:
Quantum Optics

History
Original Manuscript: May 22, 2013
Manuscript Accepted: June 11, 2013
Published: July 3, 2013

Virtual Issues
August 6, 2013 Spotlight on Optics

Citation
Ke-Jin Wei, Hai-Qiang Ma, and Jian-Hui Yang, "Experimental circular quantum secret sharing over telecom fiber network," Opt. Express 21, 16663-16669 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-16663


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References

  1. A. Shamir, “How to share a secret,” Commun. ACM22(11), 612–613 (1979). [CrossRef]
  2. M. Hillery, V. Buzek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A59(3), 1829–1834 (1999). [CrossRef]
  3. R. Cleve, D. Gottesman, and H. K. Lo, “How to share a quantum secret,” Phys. Rev. Lett.83(3), 648–651 (1999). [CrossRef]
  4. L. Xiao, G. L. Long, F. G. Deng, and J. W. Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A69(5), 052307 (2004). [CrossRef]
  5. S. Gaertner, C. Kurtsiefer, M. Bourennane, and H. Weinfurter, “Experimental demonstration of four-party quantum secret sharing,” Phys. Rev. Lett.98(2), 020503, 4 (2007). [CrossRef] [PubMed]
  6. Y. Sun, Q. Y. Wen, F. Gao, X. B. Chen, and F. C. Zhu, “Multiparty quantum secret sharing based on bell measurement,” Opt. Commun.282(17), 3647–3651 (2009). [CrossRef]
  7. L. Hao, C. Wang, and G. L. Long, “Quantum secret sharing protocol with four state Grover algorithm and its proof-of-principle experimental demonstration,” Opt. Commun.284(14), 3639–3642 (2011). [CrossRef]
  8. X. B. Chen, S. Yang, Y. Su, and Y. X. Yang, “Cryptanalysis on the improved multiparty quantum secret sharing protocol based on the GHZ state,” Phys. Scr.86(5), 055002 (2012). [CrossRef]
  9. F. G. Deng, H. Y. Zhou, and G. L. Long, “Bidirectional quantum secret sharing and secret splitting with polarized single photons,” Phys. Lett. A337(4-6), 329–334 (2005). [CrossRef]
  10. F. L. Yan, T. Gao, and Y. C. Li, “Quantum secret sharing protocol between multiparty and multiparty with single photons and unitary transformations,” Chin. Phys. Lett.25, 4 (2008).
  11. L. F. Han, Y. M. Liu, J. Liu, and Z. J. Zhang, “Multiparty quantum secret sharing of secure direct communication using single photons,” Opt. Commun.281(9), 2690–2694 (2008). [CrossRef]
  12. J. Bogdanski, J. Ahrens, and M. Bourennane, “Sagnac secret sharing over telecom fiber networks,” Opt. Express17(2), 1055–1063 (2009). [CrossRef] [PubMed]
  13. J. Bogdanski, N. Rafiei, and M. Bourennane, “Experimental quantum secret sharing using telecommunication fiber,” Phys. Rev. A78(6), 062307 (2008). [CrossRef]
  14. C. Schmid, P. Trojek, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, “Experimental single qubit quantum secret sharing,” Phys. Rev. Lett.95(23), 230505 (2005). [CrossRef] [PubMed]
  15. J. Bogdanski, J. Ahrens, and M. Bourennane, “Single mode fiber birefringence compensation in Sagnac and “plug & play” interferometric setups,” Opt. Express17(6), 4485–4494 (2009). [CrossRef] [PubMed]
  16. F. G. Deng, H. Y. Zhou, and G. L. Long, “Circular quantum secret sharing,” J. Phys. Math. Gen.39(45), 14089–14099 (2006). [CrossRef]
  17. D. S. Bethune and W. P. Risk, “Autocompensating quantum cryptography,” New J. Phys.4, 421–4215 (2002).
  18. D. S. Bethune, M. Navarro, and W. P. Risk, “Enhanced autocompensating quantum cryptography system,” Appl. Opt.41(9), 1640–1648 (2002). [CrossRef] [PubMed]
  19. C. H. Bennett and G. Brassard, “Quantum Cryptography: Public key distribution and coin tossing,” in IEEE Int.Conf. on Computers, Systems, and Signal Processing, (Bangalore, 1984), 175–179.
  20. M. Lucamarini and S. Mancini, “Secure deterministic communication without entanglement,” Phys. Rev. Lett.94(14), 140501 (2005). [CrossRef] [PubMed]
  21. N. Gisin, S. Fasel, B. Kraus, H. Zbinden, and G. Ribordy, “Trojan-horse attacks on quantum-key-distribution systems,” Phys. Rev. A73(2), 022320 (2006). [CrossRef]
  22. R. Kumar, M. Lucamarini, G. Giuseppe, R. Natali, G. Mancini, and P. Tombesi, “Two-way quantum key distribution at telecommunication wavelength,” Phys. Rev. A77(2), 022304 (2008). [CrossRef]
  23. M. F. Abdul Khir, M. Zain, I. Bahari, and S. Shaari, “Implementation of two way Quantum Key Distribution protocol with decoy state,” Opt. Commun.285(5), 842–845 (2012). [CrossRef]

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