Finite-key analysis of decoy model semi-quantum key distribution based on four-state protocol

Zhan Shao-Kang; Wang Jin-Dong; Dong Shuang; Huang Si-Ying; Hou Qing-Cheng; Mo Nai-Da; Mi Shang; Xiang Li-Bing; Zhao Tian-Ming; Yu Ya-Fei; Wei Zheng-Jun; Zhang Zhi-Ming

doi:10.7498/aps.72.20230849

Abstract

Semi-quantum key distribution allows a full quantum user Alice and a classical user Bob to share a pair of security keys guaranteed by physical principles. Semi-quantum key distribution is proposed while verifying its robustness. Subsequently, its unconditional security of semi-quantum key distribution system is verified theoretically. In 2021, the feasibility of semi-quantum key distribution system based on mirror protocol was verified experimentally. However, the feasibility experimental system still uses the laser pulse with strong attenuation. It has been proved in the literature that the semi-quantum key distribution system still encounters the risk of secret key leakage under photon number splitting attack. Therefore, the actual security of key distribution can be further reasonably evaluated by introducing the temptation state and conducting the finite-key analysis in the key distribution process. In this work, for the model of adding one-decoy state only to Alice at the sending based on a four state semi-quantum key distribution system, the length of the security key in the case of finite-key is analyzed by using Hoeffding inequality, and then the formula of the security key rate is obtained. It is found in the numerical simulation that when the sample size is $ {10}^{5} $, the security key rate of $ {10}^{-4} $, which is close to the security key rate of the asymptotic limits, can be obtained in the case of close range. It is very important for the practical application of semi-quantum key distribution system.

Keywords:

Authors and contacts

1.
Guangdong Provincial Key Laboratory of Quantum Control Engineering and Materials, School of Information and Optoelectronic Science and Engineering, South China Normal University, Guangzhou 510006, China
2.
Guangdong Provincial Key Laboratory of Micro-nanophotonic Functional Materials and Devices, School of Information and Optoelectronic Science and Engineering, South China Normal University, Guangzhou 510006, China

Corresponding author: Wang Jin-Dong, wangjindong@m.scnu

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 62071186, 61771205) and the Key Laboratory Foundation of Guangdong Province, China (Grant No. 2020B1212060066).

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References

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[54]	Zhang X Z, Gong W G, Tan Y G, Ren Z Z, Guo X T 2009 Chin. Phys. B 18 2143 Google Scholar

Cited By

图 1 BKM-07协议模型

Figure 1. BKM-07 protocol model.

DownLoad: Full-Size Img PowerPoint

图 2 基于BKM-07协议的诱骗态SQKD模型

Figure 2. Decoy SQKD model based on BKM-07 protocol.

DownLoad: Full-Size Img PowerPoint

图 3 (a) 使用1 GHz的脉冲频率时不同码长$ {n}_{{\mathrm{Z}}} $之间的安全密钥率的比较, $ {n}_{{\mathrm{Z}}} $的取值为$ {10}^{s} $(s = [4, 5, 6, 7]), 当$ {n}_{{\mathrm{Z}}}={10}^{5} $时安全密钥速率与渐近情况的安全密钥率相近, 安全密钥率随光纤长度的增长而急剧衰减, 但在约30 km内能够保持10^–5的安全密钥率; (b) 考虑1 GHz的脉冲频率时三种不同$ {n}_{{\mathrm{Z}}} $之间近距离安全密钥率的比较, $ {n}_{{\mathrm{Z}}} $的取值为$ {10}^{s} $(s = [5, 6, 7])

Figure 3. (a) The comparison of secret key rate of different key sizes $ {n}_{{\mathrm{Z}}} $, when using the pulse frequency of 1 GHz. The value of $ {n}_{{\mathrm{Z}}} $ are $ {10}^{s} $(where s = [4, 5, 6, 7]). When $ {10}^{5} $ is chosen to be $ {n}_{{\mathrm{Z}}} $, the secret key rate is close to the asymptotic limit’s. The secret key rate decreases sharply with the increase of fiber length, but it can maintain a secret key rate of 10^–5 for about 30 km; (b) the comparison of the proximity security key rates between six different $ {n}_{{\mathrm{Z}}} $ when considering a pulse frequency of 1 GHz. The value of $ {n}_{{\mathrm{Z}}} $ are $ {10}^{s} $(where s = [5, 6, 7]).

DownLoad: Full-Size Img PowerPoint

表 1 50 km传播距离中每5 km处取得最大安全密钥率时脉冲强度$ {\mu }_{1} $, $ {v}_{1} $, $ {v}_{2} $的取值和得到的安全密钥率

Table 1. Pulse strength $ {\mu }_{1} $, $ {v}_{1} $, $ {v}_{2} $ and the number of Secret key ratio every 5 km in a 50 km transmission distance.

传输距离/km	$ {\mu }_{1} $	$ {v}_{1} $	$ {v}_{2} $	密钥率
0	0.68	0.48	0.07	0.001745822
5	0.68	0.48	0.07	0.000785235
10	0.68	0.48	0.07	0.000483058
15	0.68	0.48	0.07	0.000331894
20	0.68	0.48	0.07	0.000240990
25	0.68	0.48	0.07	0.000180301
30	0.68	0.48	0.08	0.000137404
35	0.68	0.48	0.08	0.000105438
40	0.68	0.49	0.09	0.000081846
45	0.68	0.49	0.10	0.000063533
50	0.68	0.49	0.10	0.000049545

DownLoad: CSV

[1]	Bennett C H, Brassard G 2014 Theor. Comput. Sci. 560 7 Google Scholar
[2]	Muller A, Herzog T, Huttner B, Tittel W, Zbinden H, Gisin N 1997 Appl. Phys. Lett. 70 793 Google Scholar
[3]	Wang J, Qin X, Jiang Y, Wang X, Chen L, Zhao F, Wei Z, Zhang Z 2016 Opt. Express 24 8302 Google Scholar
[4]	Mo X F, Zhu B, Han Z F, Gui Y Z, Guo G C 2005 Opt. Lett. 30 2632 Google Scholar
[5]	Kraus B, Gisin N, Renner R 2005 Phys. Rev. Lett. 95 080501 Google Scholar
[6]	Hwang W Y, Ahn D, Hwang S W 2001 Phys. Lett. A 279 133 Google Scholar
[7]	Duˇsek M, Haderka O, Hendrych M 1999 Opt. Commun. 169 103 Google Scholar
[8]	Lutkenhaus N, Jahma M 2002 New J. Phys. 4 44.1 Google Scholar
[9]	Bennett C H 1992 Phys. Rev. Lett. 68 3121 Google Scholar
[10]	Huttner B, Imoto N, Gisin N, Mor T 1995 Phys. Rev. A 51 1863 Google Scholar
[11]	Chaiwongkhot P, Zhong J Q, Huang A, Qin H, Shi S C, Makarov V 2022 EPJ Quantum Technol. 9 23 Google Scholar
[12]	Lydersen L, Wiechers C, Wittmann C, Elser D, Skaar J, Makarov A 2010 Nat. Photonics 4 686 Google Scholar
[13]	Lim C C W, Walenta N, Legré N, Gisin N, Zbinden H 2015 IEEE J. Sel. Top. Quantum Electron. 21 6601305 Google Scholar
[14]	Carlos N M, Juan Carlos G E 2021 Quantum Inf. Process. 20 196 Google Scholar
[15]	Kim C M, Kim Y W, Park Y J 2011 Curr. Appl. Phys. 11 1006 Google Scholar
[16]	Lu H, Fung C H F, Cai Q Y 2013 Phys. Rev. A 88 044302 Google Scholar
[17]	Chen Y P, Liu J Y, Sun M S, Zhou X X, Zhang C H, Li J, Wang Q 2021 Opt. Lett. 46 3729 Google Scholar
[18]	Zhou X Y, Zhang CH, Zhang C M, Wang Q 2019 Phys. Rev. A 99 062316 Google Scholar
[19]	Zeng P, Zhou H Y, Wu W J, Ma X F 2022 Nat. Commun. 13 3903 Google Scholar
[20]	Gu J, Cao X Y, Fu Y, He Z W, Yin Z J, Yin H L, Chen Z B 2022 Sci. Bull. 67 2167 Google Scholar
[21]	Cui C H, Yin Z Q, Wang R, Chen W, Wang S, Guo G C, Han Z F 2019 Phys. Rev. A 11 034053 Google Scholar
[22]	Xie Y M, Weng C X, Lu Y S, Fu Y, Wang Y, Yin H L, Chen Z B 2023 Phys. Rev. A 107 042603 Google Scholar
[23]	Curty M, Azuma K, Lo H K 2019 NPJ Quantum Inf. 5 64 Google Scholar
[24]	Xie Y M, Lu Y S, Weng C X, Cao X Y, Jia Z Y, Bao Y, Wang Y, Fu Y, Yin H L, Chen Z B 2022 PRX Quantum 3 020315 Google Scholar
[25]	Hwang W Y 2003 Phys. Rev. Lett. 91 057901 Google Scholar
[26]	Lo H K, Ma X, Chen K 2005 Phys. Rev. Lett. 94 230504 Google Scholar
[27]	Wang X B 2005 Phys. Rev. Lett. 94 230503 Google Scholar
[28]	Ma X, Qi B, Zhao Y, Lo H K 2005 Phys. Rev. A 72 012326 Google Scholar
[29]	Wang Q, Wang X B, Guo G C 2007 Phys. Rev. A 75 012312 Google Scholar
[30]	Ma X, Fung C H F, Dupuis F, Chen K, Tamaki K, Lo H K 2006 Phys. Rev. A 74 032330 Google Scholar
[31]	Scarani V, Ac´ın A, Ribordy G, Gisin N 2004 Phys. Rev. Lett. 92 057901 Google Scholar
[32]	Curty M, Xu F, Cui W, Lim C C W, Tamaki K, Lo H K 2014 Nat. Commun. 5 3732 Google Scholar
[33]	Mafu M, Garapo K, Petruccione F 2013 Phys. Rev. A 88 1 Google Scholar
[34]	Zhao L Y, Li H W, Yin Z Q, Chen W, You J, Han Z F 2014 Chin. Phys. B 23 100304 Google Scholar
[35]	Lim C C W, Curty M, Walenta N, Xu F H, Zbinden H 2014 Phys. Rev. A 89 022307 Google Scholar
[36]	Rusca D, Boaron A, Grünenfelder F, Martin A, Zbinden H 2018 Appl. Phys. Lett. 112 171104 Google Scholar
[37]	Boyer M, Kenigsberg D, Mor T 2007 Phys. Rev. Lett. 99 140501 Google Scholar
[38]	Zou X, Qiu D, Li L, Wu L, Li L 2009 Phys. Rev. A 79 052312 Google Scholar
[39]	Boyer M, Katz M, Liss R, Mor T 2017 Phys. Rev. A 96 062335 Google Scholar
[40]	Amer O, Krawec W O 2019 Phys. Rev. A 100 022319 Google Scholar
[41]	Krawec W O 2015 IEEE International Symposium Information Theory Hong Kong, China, June 14–19, 2015 p686
[42]	Boyer M, Liss R, Mor T 2018 Entropy 20 536 Google Scholar
[43]	Krawec W O, Liss R, Mor T 2023 IEEE Trans. Quantum Eng. 4 2100316 Google Scholar
[44]	Zhang W, Qiu D, Mateus P 2020 Int. J. Quantum Inf. 18 2050013 Google Scholar
[45]	Han S Y, Huang Y F, Mi S, Qin X, Wang J D, Yu Y F, Wei Z J, Zhang Z M 2021 EPJ Quantum Technol. 8 28 Google Scholar
[46]	Mi S, Dong S, Hou Q C, Wang J D, Yu Y F, Wei Z J, Zhang Z M 2022 Front. Phys. 10 1029552 Google Scholar
[47]	Hoeffding W 1963 J. Amer. Stat. Assoc. 58 13 Google Scholar
[48]	Renner R 2008 Int. J. Quantum Inf. 6 1 Google Scholar
[49]	Vitanov A, Dupuis F, Tomamichel M, Renner R 2013 IEEE Trans. Inf. Theory 59 2603 Google Scholar
[50]	Tomamichel M, Renner R 2011 Phys. Rev. Lett. 106 110506 Google Scholar
[51]	Fung C H F, Ma X F, Chau H F 2010 Phys. Rev. A 81 012318 Google Scholar
[52]	Dong S, Mi S, Hou Q C, Huang Y T, Wang J D, Yu Y F, Wei Z J, Zhang Z M, Fang J B 2023 EPJ Quantum Technol. 10 18 Google Scholar
[53]	Yin H L, Fu Y, Li C L, Weng C X, Li B H, Gu J, Lu Y S, Huang S, Chen Z B 2023 Nati. Sci. Rev. 10 nwac228 Google Scholar
[54]	Zhang X Z, Gong W G, Tan Y G, Ren Z Z, Guo X T 2009 Chin. Phys. B 18 2143 Google Scholar

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