## Differential-phase-shift quantum secret sharing

Optics Express, Vol. 16, Issue 20, pp. 15469-15476 (2008)

http://dx.doi.org/10.1364/OE.16.015469

Acrobat PDF (114 KB)

### Abstract

A quantum secret sharing (QSS) protocol based on a differential-phase-shift scheme is proposed, which quantum mechanically provides a full secret key to one party and partial keys to two other parties. A weak coherent pulse train is utilized instead of individual photons as in conventional schemes. Compared with previous QSS protocols, the present one features a simple setup, is suitable for fiber transmission, and offers the possibility for a high key creation rate. An experiment is also carried out to demonstrate the operation.

© 2008 Optical Society of America

## 1. Introduction

1. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. **74**, 145–195 (2002). [CrossRef]

2. K. Inoue, “Quantum key distribution technologies,” IEEE J. Sel. Top. Quantum Electron. **12**, 888–896 (2006). [CrossRef]

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

4. A. Karlsson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A **59**, 162 (1999). [CrossRef]

9. H. Takesue and K. Inoue, “Quantum secret sharing based on modulated high-dimension time-bin entanglement,” Phys. Rev. A **74**, 012315 (2006). [CrossRef]

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

*i*|V>)/√2, (|H>-

*i*|V>)/√2} with |H> and |V> being the horizontal and vertical polarization states. This scheme is a QSS variant of the traditional BB84 QKD protocol, and looks more practical than entanglement-based protocols. A primitive experiment was also demonstrated [8

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

10. J. Chen, G. Wu, Y. Li, E. Wu, and H. Zeng, “Active polarization in optical fibers suitable for quantum key distribution,” Opt. Express **15**, 17928–17936 (2007). [CrossRef] [PubMed]

11. G. B. Xavier, G. Vilela de Faria, G. P. Temporão, and J. P. von der Weid, “Full polarization control for fiber optical quantum communication systems using polarization encoding,” Opt. Express **16**, 1867–1873 (2008). [CrossRef] [PubMed]

12. N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution,” Phys. Rev. A **61**, 052304 (2000). [CrossRef]

13. K. Inoue, E. Waks, and Y. Yamamoto, “Differential-phase-shift quantum key distribution using coherent light,” Phys. Rev. A **68**, 022317 (2003). [CrossRef]

## 2. Differential-phase-shift quantum secret sharing

*π*} for each pulse. The optical power is set at less than 1 photon (e.g., 0.1–0.2) per pulse on average. Bob also phase-modulates the received signal by {0, π} for each pulse and then sends it to Charlie, while monitoring the received signal power by splitting and detecting a part of it. Charlie measures the phase difference of adjacent pulses with a one-pulse delayed Mach-Zehnder interferometer, such that detectors 1 and 2 count a photon for a phase difference of 0 and π, respectively. Here, a photon is detected occasionally and randomly because the received signal power is smaller than one photon per pulse. While measuring the signal, Charlie records the photon detection time and which detector counts a photon.

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

## 3. Eavesdropping against DPS QSS

14. E. Waks, H. Takesue, and Y. Yamamoto, “Security of differential-phase-shift quantum key distribution against individual attacks,” Phys. Rev. A **73**, 012344 (2006). [CrossRef]

12. N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution,” Phys. Rev. A **61**, 052304 (2000). [CrossRef]

14. E. Waks, H. Takesue, and Y. Yamamoto, “Security of differential-phase-shift quantum key distribution against individual attacks,” Phys. Rev. A **73**, 012344 (2006). [CrossRef]

*µT*[12

_{ab}12. N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution,” Phys. Rev. A **61**, 052304 (2000). [CrossRef]

*µ*is the average photon number per pulse sent from Alice and

*T*is the transmittance from Alice to Bob’s output. Then, the probability that Alice does not know Bob’s bit is (1 - 2

_{ab}*µT*), and the resultant bit error rate is (1 - 2

_{ab}*µT*)/2. Note that this bit error rate is dependent of Bob’s monitoring detector in practice. When the photon counting rate of Bob’s detector fluctuates, Alice can send photons of more than an initially designed number, utilizing this fluctuation, and reduce the bit error rate as a result. For a fluctuation of 10 %, for example, the upper bound of the error rate is given by (1 - 2

_{ab}*µ*′

*T*)/2 with

_{ab}*µ*’=1.1

*µ*. Then, the information leakage to Alice, which is described later, is increased by this amount.

*µ*, with a strategy that stores the pulse train from Alice and makes two corresponding pulses interfere with each other after Charlie discloses his photon detection time. Then, the probability that Alice does not know Bob’s bit is (1 - 2

*µ*), and the resultant bit error rate is (1 - 2

*µ*)/2.

## 4. Error correction and privacy amplification

*µT*)/2, and that by Bob’s betrayal is (1 - 2

_{ab}*µ*)/2, as discussed in the previous section. Malicious Bob introduces a smaller error rate than malicious Alice, meaning that Bob can obtain a larger amount of Charlie’s key. Thus, we just consider the amount of information leaked to malicious Bob in the following.

*e*, the upper bound for the allowable rate of Bob’s eavesdropping shown in Fig. 3,

*α*, is given by

*µ*is the probability for Bob to obtain Alice’s bit by the eavesdropping shown in Fig. 3. Thus, the information leakage to Bob through this partial eavesdropping is 2

*µα*. For the remaining part of Alice’s signal, (1 –

*α*), Bob phase-modulates and sends it to Charlie as in the normal condition. Here, however, he can also conduct a beam-splitting attack utilizing the transmission loss from Bob to Charlie, as shown in Fig. 4. Provided that the transmittance from Bob to Charlie is

*T*, Bob splits (1 –

*T*) Alice’s signal, stores it, and makes two pulses interfere with each other after Charlie discloses the photon detection time. This beam-splitting attack gives Bob partial information with a ratio of 2

*µ*(1 –

*α*)(1 –

*T*). Then, Bob obtains 2

*µα*+2

*µ*(1 –

*α*)(1 –

*T*) of Charlie’s key in total.

14. E. Waks, H. Takesue, and Y. Yamamoto, “Security of differential-phase-shift quantum key distribution against individual attacks,” Phys. Rev. A **73**, 012344 (2006). [CrossRef]

## 5. Experiment

17. T. Honjo, K. Inoue, and H. Takahashi, “Differential-phase-shift quantum key distribution experiment with a planar light-wave circuit Mach-Zehnder interferometer,” Opt. Lett. **29**, 2797–2799 (2004). [CrossRef] [PubMed]

_{3}phase modulator driven by a pseudo random binary stream, and attenuated to be 0.27 photon/pulse, which was an optimized number based on our security analysis. The signal from Alice was sent to Bob. At Bob’s site, part of the incoming signal was split to a monitoring photon detector with a splitting ratio of 1 %, and the main part was phase-modulated by {0, π} for each pulse. The phase-modulated signal was then sent to Charlie. At Charlie’s site, the incoming signal was input to a waveguide asymmetric Mach-Zehnder interferometer whose path length difference was 20 cm that corresponded to the pulse interval of 1 ns. The interferometer was a PLC (planar lightwave circuit) based device, which provided stable operation under the temperature control within 0.05 degree. The detail of the device is described in Ref. [17

17. T. Honjo, K. Inoue, and H. Takahashi, “Differential-phase-shift quantum key distribution experiment with a planar light-wave circuit Mach-Zehnder interferometer,” Opt. Lett. **29**, 2797–2799 (2004). [CrossRef] [PubMed]

## 6. Summary

## References and links

1. | N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. |

2. | K. Inoue, “Quantum key distribution technologies,” IEEE J. Sel. Top. Quantum Electron. |

3. | M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A |

4. | A. Karlsson, M. Koashi, and N. Imoto, “Quantum entanglement for secret sharing and secret splitting,” Phys. Rev. A |

5. | L. Xiao, G. Long, F. Long, J. Dejng, and Pan, “Efficient multiparty quantum-secret-sharing schemes,” Phys. Rev. A |

6. | S. K. Singh and R. Srikanth, “Generalized quantum secret sharing,” Phys. Rev. A |

7. | Z. Zhang, Y. Li, and Z. Man, “Multiparty quantum secret sharing,” Phys. Rev. A |

8. | C. Schmid, P. Trojek, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, “Experimental single qubit quantum secret sharing,” Phys. Rev. Lett. |

9. | H. Takesue and K. Inoue, “Quantum secret sharing based on modulated high-dimension time-bin entanglement,” Phys. Rev. A |

10. | J. Chen, G. Wu, Y. Li, E. Wu, and H. Zeng, “Active polarization in optical fibers suitable for quantum key distribution,” Opt. Express |

11. | G. B. Xavier, G. Vilela de Faria, G. P. Temporão, and J. P. von der Weid, “Full polarization control for fiber optical quantum communication systems using polarization encoding,” Opt. Express |

12. | N. Lütkenhaus, “Security against individual attacks for realistic quantum key distribution,” Phys. Rev. A |

13. | K. Inoue, E. Waks, and Y. Yamamoto, “Differential-phase-shift quantum key distribution using coherent light,” Phys. Rev. A |

14. | E. Waks, H. Takesue, and Y. Yamamoto, “Security of differential-phase-shift quantum key distribution against individual attacks,” Phys. Rev. A |

15. | G. Brassard, L. Salvail, and T. Helleseth, (Springer Verlag, Berlin, Germany, 1994), pp. 410–423. |

16. | C. H. Bennett, G. Brassard, C. Crepeau, and U. M. Maurer, “Generalized privacy amplification,” IEEE Trans. Info. Theory |

17. | T. Honjo, K. Inoue, and H. Takahashi, “Differential-phase-shift quantum key distribution experiment with a planar light-wave circuit Mach-Zehnder interferometer,” Opt. Lett. |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: August 1, 2008

Revised Manuscript: September 10, 2008

Manuscript Accepted: September 10, 2008

Published: September 16, 2008

**Citation**

K. Inoue, T. Ohashi, T. Kukita, K. Watanebe, S. Hayashi, T. Honjo, and H. Takesue, "Differential-phase-shift quantum secret sharing," Opt. Express **16**, 15469-15476 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15469

Sort: Year | Journal | Reset

### References

- N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002). [CrossRef]
- K. Inoue, "Quantum key distribution technologies," IEEE J. Sel. Top. Quantum Electron. 12, 888-896 (2006). [CrossRef]
- M. Hillery, V. Bužek, and A. Berthiaume, "Quantum secret sharing," Phys. Rev. A 59, 1829 (1999). [CrossRef]
- A. Karlsson, M. Koashi, and N. Imoto, "Quantum entanglement for secret sharing and secret splitting," Phys. Rev. A 59, 162 (1999). [CrossRef]
- L. Xiao, G. Long, F. Deng, and J. Pan, "Efficient multiparty quantum-secret-sharing schemes," Phys. Rev. A 69, 052307 (2004). [CrossRef]
- S. K. Singh and R. Srikanth, "Generalized quantum secret sharing," Phys. Rev. A 71, 012328 (2005). [CrossRef]
- Z. Zhang, Y. Li, and Z. Man, "Multiparty quantum secret sharing," Phys. Rev. A 71, 044301 (2005). [CrossRef]
- C. Schmid, P. Trojek, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, "Experimental single qubit quantum secret sharing," Phys. Rev. Lett. 95, 230505 (2005). [CrossRef] [PubMed]
- H. Takesue and K. Inoue, "Quantum secret sharing based on modulated high-dimension time-bin entanglement," Phys. Rev. A 74, 012315 (2006). [CrossRef]
- J. Chen, G. Wu, Y. Li, E. Wu, and H. Zeng, "Active polarization in optical fibers suitable for quantum key distribution," Opt. Express 15, 17928-17936 (2007). [CrossRef] [PubMed]
- G. B. Xavier, G. Vilela de Faria, G. P. Temporão, and J. P. von der Weid, "Full polarization control for fiber optical quantum communication systems using polarization encoding," Opt. Express 16, 1867-1873 (2008). [CrossRef] [PubMed]
- N. Lütkenhaus, "Security against individual attacks for realistic quantum key distribution," Phys. Rev. A 61, 052304 (2000). [CrossRef]
- K. Inoue, E. Waks, and Y. Yamamoto, "Differential-phase-shift quantum key distribution using coherent light," Phys. Rev. A 68, 022317 (2003). [CrossRef]
- E. Waks, H. Takesue, and Y. Yamamoto, "Security of differential-phase-shift quantum key distribution against individual attacks," Phys. Rev. A 73, 012344 (2006). [CrossRef]
- G. Brassard and L. Salvail, "Secret-key reconciliation by public discussion in advances," in Cryptography-EUROCRYPT???93, Lecture Notes in Computer Science, 765, T. Helleseth, (Springer Verlag, Berlin, Germany, 1994), pp. 410-423.
- C. H. Bennett, G. Brassard, C. Crepeau, and U. M. Maurer, "Generalized privacy amplification," IEEE Trans. Info. Theory 41, 1915-1923 (1995). [CrossRef]
- T. Honjo, K. Inoue and H. Takahashi, "Differential-phase-shift quantum key distribution experiment with a planar light-wave circuit Mach-Zehnder interferometer," Opt. Lett. 29, 2797-2799 (2004). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.