# Attacks against a Simplified Experimentally Feasible Semiquantum Key Distribution Protocol

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Mirror Protocol

- $\mathbf{I}$ (
**CTRL**) Reflect all the photons towards Bob, without measuring any photon. The mathematical description is:$${I\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}|{m}_{\mathfrak{1}},{m}_{\mathfrak{0}}{\rangle}_{\mathrm{B}}={\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}{|{m}_{\mathfrak{1}},{m}_{\mathfrak{0}}\rangle}_{\mathrm{B}}.$$ - ${\mathbf{S}}_{\mathbf{1}}$ (
**SWAP-10**) Reflect all photons in the $|\mathfrak{0}\rangle $ mode towards Bob, and measure all photons in the $|\mathfrak{1}\rangle $ mode. The mathematical description is:$${S}_{1}{\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}|{m}_{\mathfrak{1}},{m}_{\mathfrak{0}}{\rangle}_{\mathrm{B}}=\phantom{\rule{0.166667em}{0ex}}|{m}_{\mathfrak{1}},0{\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}{|0,{m}_{\mathfrak{0}}\rangle}_{\mathrm{B}}.$$ - ${\mathbf{S}}_{\mathbf{0}}$ (
**SWAP-01**) Reflect all photons in the $|\mathfrak{1}\rangle $ mode towards Bob, and measure all photons in the $|\mathfrak{0}\rangle $ mode. The mathematical description is:$${S}_{0}{\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}|{m}_{\mathfrak{1}},{m}_{\mathfrak{0}}{\rangle}_{\mathrm{B}}=\phantom{\rule{0.166667em}{0ex}}|0,{m}_{\mathfrak{0}}{\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}{|{m}_{\mathfrak{1}},0\rangle}_{\mathrm{B}}.$$ - $\mathbf{S}$ (
**SWAP-ALL**) Measure all the photons, without reflecting any photon towards Bob. The mathematical description is:$${S\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}|{m}_{\mathfrak{1}},{m}_{\mathfrak{0}}{\rangle}_{\mathrm{B}}=\phantom{\rule{0.166667em}{0ex}}|{m}_{\mathfrak{1}},{m}_{\mathfrak{0}}{\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}{|0,0\rangle}_{\mathrm{B}}.$$

- It can be implemented as a mechanically moved mirror. Such mirror is trivial to implement, but it is very slow.
- It can be implemented by using optical elements: an electronically-triggered Pockels cell, which changes the polarization of the photon(s) in one of the pulses, and a polarizing beam splitter, which can split the two different pulses (that now have different polarizations) into two paths. This implementation is feasible and gives much higher bit rates than the mechanical implementation.

## 3. The “Simplified Mirror Protocol”: A Simpler and Non-Robust Variant of the Mirror Protocol

- $\mathbf{I}$ (
**CTRL**) Reflect all the photons towards Bob, without measuring any photon. The mathematical description is:$${I\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}|{m}_{\mathfrak{1}},{m}_{\mathfrak{0}}{\rangle}_{\mathrm{B}}={\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}{|{m}_{\mathfrak{1}},{m}_{\mathfrak{0}}\rangle}_{\mathrm{B}}.$$ - ${\mathbf{S}}_{\mathbf{1}}$ (
**SWAP-10**) Reflect all photons in the $|\mathfrak{0}\rangle $ mode towards Bob, and measure all photons in the $|\mathfrak{1}\rangle $ mode. The mathematical description is:$${S}_{1}{\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}|{m}_{\mathfrak{1}},{m}_{\mathfrak{0}}{\rangle}_{\mathrm{B}}=\phantom{\rule{0.166667em}{0ex}}|{m}_{\mathfrak{1}},0{\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}{|0,{m}_{\mathfrak{0}}\rangle}_{\mathrm{B}}.$$ - ${\mathbf{S}}_{\mathbf{0}}$ (
**SWAP-01**) Reflect all photons in the $|\mathfrak{1}\rangle $ mode towards Bob, and measure all photons in the $|\mathfrak{0}\rangle $ mode. The mathematical description is:$${S}_{0}{\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}|{m}_{\mathfrak{1}},{m}_{\mathfrak{0}}{\rangle}_{\mathrm{B}}=\phantom{\rule{0.166667em}{0ex}}|0,{m}_{\mathfrak{0}}{\rangle}_{\mathrm{A}}\phantom{\rule{0.166667em}{0ex}}{|{m}_{\mathfrak{1}},0\rangle}_{\mathrm{B}}.$$

## 4. Attacks against the Simplified Mirror Protocol

#### 4.1. A Full Attack on the Simplified Protocol that Gives Eve Full Information

#### 4.2. A Weaker Attack on the Simplified Protocol Causing Fewer Losses of the CTRL Bits

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

QKD | Quantum Key Distribution |

SQKD | Semiquantum Key Distribution |

## References

- Boyer, M.; Kenigsberg, D.; Mor, T. Quantum Key Distribution with Classical Bob. Phys. Rev. Lett.
**2007**, 99, 140501. [Google Scholar] [CrossRef] [PubMed] - Boyer, M.; Gelles, R.; Kenigsberg, D.; Mor, T. Semiquantum key distribution. Phys. Rev. A
**2009**, 79, 032341. [Google Scholar] [CrossRef] - Zou, X.; Qiu, D.; Li, L.; Wu, L.; Li, L. Semiquantum-key distribution using less than four quantum states. Phys. Rev. A
**2009**, 79, 052312. [Google Scholar] [CrossRef] - Boyer, M.; Mor, T. Comment on “Semiquantum-key distribution using less than four quantum states”. Phys. Rev. A
**2011**, 83, 046301. [Google Scholar] [CrossRef] - Lu, H.; Cai, Q.Y. Quantum key distribution with classical Alice. Int. J. Quantum Inf.
**2008**, 06, 1195–1202. [Google Scholar] [CrossRef] - Sun, Z.W.; Du, R.G.; Long, D.Y. Quantum key distribution with limited classical Bob. Int. J. Quantum Inf.
**2013**, 11, 1350005. [Google Scholar] [CrossRef] - Yu, K.F.; Yang, C.W.; Liao, C.H.; Hwang, T. Authenticated semi-quantum key distribution protocol using Bell states. Quantum Inf. Process.
**2014**, 13, 1457–1465. [Google Scholar] [CrossRef] - Krawec, W.O. Mediated semiquantum key distribution. Phys. Rev. A
**2015**, 91, 032323. [Google Scholar] [CrossRef] - Zou, X.; Qiu, D.; Zhang, S.; Mateus, P. Semiquantum key distribution without invoking the classical party’s measurement capability. Quantum Inf. Process.
**2015**, 14, 2981–2996. [Google Scholar] [CrossRef] - Krawec, W.O. Security proof of a semi-quantum key distribution protocol. In Proceedings of the 2015 IEEE International Symposium on Information Theory (ISIT), Hong Kong, China, 14–19 June 2015; pp. 686–690. [Google Scholar] [CrossRef]
- Krawec, W.O. Security of a semi-quantum protocol where reflections contribute to the secret key. Quantum Inf. Process.
**2016**, 15, 2067–2090. [Google Scholar] [CrossRef] [Green Version] - Zhang, W.; Qiu, D.; Mateus, P. Security of a single-state semi-quantum key distribution protocol. Quantum Inf. Process.
**2018**, 17, 135. [Google Scholar] [CrossRef] [Green Version] - Krawec, W.O. Practical security of semi-quantum key distribution. In Proceedings of SPIE, Quantum Information Science, Sensing, and Computation X; Donkor, E., Ed.; SPIE: Washington, DC, USA, 2018; Volumme 10660, p. 1066009. [Google Scholar] [CrossRef]
- Tan, Y.G.; Lu, H.; Cai, Q.Y. Comment on “Quantum Key Distribution with Classical Bob”. Phys. Rev. Lett.
**2009**, 102, 098901. [Google Scholar] [CrossRef] [PubMed] - Boyer, M.; Kenigsberg, D.; Mor, T. Boyer, Kenigsberg, and Mor Reply. Phys. Rev. Lett.
**2009**, 102, 098902. [Google Scholar] [CrossRef] - Boyer, M.; Katz, M.; Liss, R.; Mor, T. Experimentally feasible protocol for semiquantum key distribution. Phys. Rev. A
**2017**, 96, 062335. [Google Scholar] [CrossRef] [Green Version] - Brassard, G.; Lütkenhaus, N.; Mor, T.; Sanders, B.C. Limitations on Practical Quantum Cryptography. Phys. Rev. Lett.
**2000**, 85, 1330–1333. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Table 1.**The state sent from Alice to Bob in the Mirror protocol without errors or losses, depending on Alice’s classical operation and on whether Alice detected a photon or not.

Alice’s Classical Operation | Did Alice Detect a Photon? | State Sent from Alice to Bob |
---|---|---|

CTRL | no (happens with certainty) | ${\phantom{\rule{0.166667em}{0ex}}|0,1\rangle}_{\mathrm{x}},\mathrm{B}=\frac{1}{\sqrt{2}}\left[{\phantom{\rule{0.166667em}{0ex}}|0,1\rangle}_{\mathrm{B}}+\phantom{\rule{0.166667em}{0ex}}{|1,0\rangle}_{\mathrm{B}}\right]$ |

SWAP-10 | no (happens with probability $\frac{1}{2}$) | ${\phantom{\rule{0.166667em}{0ex}}|0,1\rangle}_{\mathrm{B}}$ |

SWAP-10 | yes (happens with probability $\frac{1}{2}$) | ${\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}$ |

SWAP-01 | no (happens with probability $\frac{1}{2}$) | ${\phantom{\rule{0.166667em}{0ex}}|1,0\rangle}_{\mathrm{B}}$ |

SWAP-01 | yes (happens with probability $\frac{1}{2}$) | ${\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}$ |

SWAP-ALL | yes (happens with certainty) | ${\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}$ |

**Table 2.**The state sent from Alice to Bob in the simplified Mirror protocol without errors or losses, depending on Alice’s classical operation and on whether Alice detected a photon or not.

Alice’s Classical Operation | Did Alice Detect a Photon? | State Sent from Alice to Bob |
---|---|---|

CTRL | no (happens with certainty) | ${\phantom{\rule{0.166667em}{0ex}}|0,1\rangle}_{\mathrm{x}},\mathrm{B}=\frac{1}{\sqrt{2}}\left[{\phantom{\rule{0.166667em}{0ex}}|0,1\rangle}_{\mathrm{B}}+\phantom{\rule{0.166667em}{0ex}}{|1,0\rangle}_{\mathrm{B}}\right]$ |

SWAP-10 | no (happens with probability $\frac{1}{2}$) | ${\phantom{\rule{0.166667em}{0ex}}|0,1\rangle}_{\mathrm{B}}$ |

SWAP-10 | yes (happens with probability $\frac{1}{2}$) | ${\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}$ |

SWAP-01 | no (happens with probability $\frac{1}{2}$) | ${\phantom{\rule{0.166667em}{0ex}}|1,0\rangle}_{\mathrm{B}}$ |

SWAP-01 | yes (happens with probability $\frac{1}{2}$) | ${\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}$ |

**Table 3.**The state of Bob+Eve after Alice’s classical operation for the attacks described in Section 4.1 and Section 4.2, depending on Alice’s classical operation and on whether Alice detected a photon or not.

Alice’s Classical Operation | Did Alice Detect a Photon? | Bob+Eve State |
---|---|---|

CTRL | no (happens with certainty) | $\frac{1}{\sqrt{3}}\left[{\phantom{\rule{0.166667em}{0ex}}|0,1\rangle}_{\mathrm{B}}{\phantom{\rule{0.166667em}{0ex}}|1\rangle}_{\mathrm{E}}+{\phantom{\rule{0.166667em}{0ex}}|1,0\rangle}_{\mathrm{B}}{\phantom{\rule{0.166667em}{0ex}}|1\rangle}_{\mathrm{E}}+{\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}\phantom{\rule{0.166667em}{0ex}}{|0\rangle}_{\mathrm{E}}\right]$ |

SWAP-10 | no (happens with probability $\frac{2}{3}$) | $\frac{1}{\sqrt{2}}\left[{\phantom{\rule{0.166667em}{0ex}}|0,1\rangle}_{\mathrm{B}}{\phantom{\rule{0.166667em}{0ex}}|1\rangle}_{\mathrm{E}}+{\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}\phantom{\rule{0.166667em}{0ex}}{|0\rangle}_{\mathrm{E}}\right]$ |

SWAP-10 | yes (happens with probability $\frac{1}{3}$) | ${\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}\phantom{\rule{0.166667em}{0ex}}{|1\rangle}_{\mathrm{E}}$ |

SWAP-01 | no (happens with probability $\frac{2}{3}$) | $\frac{1}{\sqrt{2}}\left[{\phantom{\rule{0.166667em}{0ex}}|1,0\rangle}_{\mathrm{B}}{\phantom{\rule{0.166667em}{0ex}}|1\rangle}_{\mathrm{E}}+{\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}\phantom{\rule{0.166667em}{0ex}}{|0\rangle}_{\mathrm{E}}\right]$ |

SWAP-01 | yes (happens with probability $\frac{1}{3}$) | ${\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}\phantom{\rule{0.166667em}{0ex}}{|1\rangle}_{\mathrm{E}}$ |

**Table 4.**The state of Bob+Eve after completing Eve’s attack described in Section 4.1, depending on Alice’s classical operation and on whether Alice detected a photon or not.

Alice’s Classical Operation | Did Alice Detect a Photon? | Bob+Eve State |
---|---|---|

CTRL | no (happens with certainty) | ${\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}\phantom{\rule{0.166667em}{0ex}}{|+\rangle}_{\mathrm{E}}$ |

SWAP-10 | no (happens with probability $\frac{2}{3}$) | $\frac{1}{\sqrt{6}}{\phantom{\rule{0.166667em}{0ex}}|0,1\rangle}_{\mathrm{B}}{\phantom{\rule{0.166667em}{0ex}}|0\rangle}_{\mathrm{E}}+\phantom{\rule{0.166667em}{0ex}}{|0,0\rangle}_{\mathrm{B}}\frac{{3\phantom{\rule{0.166667em}{0ex}}|0\rangle}_{\mathrm{E}}+\phantom{\rule{0.166667em}{0ex}}{|1\rangle}_{\mathrm{E}}}{\sqrt{12}}$ |

SWAP-10 | yes (happens with probability $\frac{1}{3}$) | ${\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}\phantom{\rule{0.166667em}{0ex}}{|2\rangle}_{\mathrm{E}}$ |

SWAP-01 | no (happens with probability $\frac{2}{3}$) | $\frac{1}{\sqrt{6}}{\phantom{\rule{0.166667em}{0ex}}|1,0\rangle}_{\mathrm{B}}{\phantom{\rule{0.166667em}{0ex}}|1\rangle}_{\mathrm{E}}+\phantom{\rule{0.166667em}{0ex}}{|0,0\rangle}_{\mathrm{B}}\frac{{\phantom{\rule{0.166667em}{0ex}}|0\rangle}_{\mathrm{E}}+3\phantom{\rule{0.166667em}{0ex}}{|1\rangle}_{\mathrm{E}}}{\sqrt{12}}$ |

SWAP-01 | yes (happens with probability $\frac{1}{3}$) | ${\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}\phantom{\rule{0.166667em}{0ex}}{|2\rangle}_{\mathrm{E}}$ |

**Table 5.**The state of Bob+Eve after completing Eve’s attack described in Section 4.2, depending on Alice’s classical operation and on whether Alice detected a photon or not. The parameters a and b are defined in Equations (20) and (21).

Alice’s Classical Operation | Did Alice Detect a Photon? | Bob+Eve State |
---|---|---|

CTRL | no (happens with certainty) | $\sqrt{\frac{2{\u03f5}^{2}}{3}}{\phantom{\rule{0.166667em}{0ex}}|0,1\rangle}_{\mathrm{x}}{,\mathrm{B}\phantom{\rule{0.166667em}{0ex}}|2\rangle}_{\mathrm{E}}+\sqrt{1-\frac{2{\u03f5}^{2}}{3}}{\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}\phantom{\rule{0.166667em}{0ex}}{|+\rangle}_{\mathrm{E}}$ |

SWAP-10 | no (happens with probability $\frac{2}{3}$) | $\frac{1}{\sqrt{2}}\left[{\phantom{\rule{0.166667em}{0ex}}|0,1\rangle}_{\mathrm{B}}\left({\u03f5\phantom{\rule{0.166667em}{0ex}}|2\rangle}_{\mathrm{E}}+\kappa \phantom{\rule{0.166667em}{0ex}}{|0\rangle}_{\mathrm{E}}\right)+\phantom{\rule{0.166667em}{0ex}}{|0,0\rangle}_{\mathrm{B}}\left({a\phantom{\rule{0.166667em}{0ex}}|0\rangle}_{\mathrm{E}}+b\phantom{\rule{0.166667em}{0ex}}{|1\rangle}_{\mathrm{E}}\right)\right]$ |

SWAP-10 | yes (happens with probability $\frac{1}{3}$) | $\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}\phantom{\rule{0.166667em}{0ex}}{|2\rangle}_{\mathrm{E}$ |

SWAP-01 | no (happens with probability $\frac{2}{3}$) | $\frac{1}{\sqrt{2}}\left[{\phantom{\rule{0.166667em}{0ex}}|1,0\rangle}_{\mathrm{B}}\left({\u03f5\phantom{\rule{0.166667em}{0ex}}|2\rangle}_{\mathrm{E}}+\kappa \phantom{\rule{0.166667em}{0ex}}{|1\rangle}_{\mathrm{E}}\right)+\phantom{\rule{0.166667em}{0ex}}{|0,0\rangle}_{\mathrm{B}}\left({b\phantom{\rule{0.166667em}{0ex}}|0\rangle}_{\mathrm{E}}+a\phantom{\rule{0.166667em}{0ex}}{|1\rangle}_{\mathrm{E}}\right)\right]$ |

SWAP-01 | yes (happens with probability $\frac{1}{3}$) | $\phantom{\rule{0.166667em}{0ex}}|0,0\rangle}_{\mathrm{B}}\phantom{\rule{0.166667em}{0ex}}{|2\rangle}_{\mathrm{E}$ |

**Table 6.**The probability p of Eve obtaining an information bit, and the loss rates ${R}_{\mathrm{CTRL}}$ and ${R}_{\mathrm{SWAP}-\mathrm{x}}$ of CTRL and SWAP-x bits (where $x\in \{01,10\}$), respectively, for several values of the attack’s parameter $\u03f5$.

$\u03f5$ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 |

$\mathbf{p}$ | 1 | 0.97 | 0.89 | 0.78 | 0.66 | 0.55 | 0.44 | 0.34 | 0.25 | 0.15 | 0 |

${\mathbf{R}}_{\mathrm{CTRL}}$ | 1 | 0.99 | 0.97 | 0.94 | 0.89 | 0.83 | 0.76 | 0.67 | 0.57 | 0.46 | 0.33 |

${\mathbf{R}}_{\mathrm{SWAP}-\mathrm{x}}$ | 0.83 | 0.83 | 0.82 | 0.79 | 0.76 | 0.73 | 0.68 | 0.63 | 0.58 | 0.53 | 0.5 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Boyer, M.; Liss, R.; Mor, T.
Attacks against a Simplified Experimentally Feasible Semiquantum Key Distribution Protocol. *Entropy* **2018**, *20*, 536.
https://doi.org/10.3390/e20070536

**AMA Style**

Boyer M, Liss R, Mor T.
Attacks against a Simplified Experimentally Feasible Semiquantum Key Distribution Protocol. *Entropy*. 2018; 20(7):536.
https://doi.org/10.3390/e20070536

**Chicago/Turabian Style**

Boyer, Michel, Rotem Liss, and Tal Mor.
2018. "Attacks against a Simplified Experimentally Feasible Semiquantum Key Distribution Protocol" *Entropy* 20, no. 7: 536.
https://doi.org/10.3390/e20070536