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The quantum digital signature (QDS) has attracted much attention as it ensures the nonrepudiation, unforgeability, and transferability of signature messages based on information-theoretic security. Amiri et al. ( Phys. Rev. A 93 032325 ) proposed the first practical QDS protocol based on orthogonal coding, which has realized information-theoretic security and become the mainstream paradigm in QDS research. The procedure of QDS involves two essential stages, the one is the distribution stage, in which Alice-Bob and Alice-Charlie individually utilize the three-intensity decoy-state quantum key distribution protocol but without error correction or privacy amplification, namely, key-generation protocol, to generate correlated bit strings, the other is the messaging stage, in which Alice transmits signature messages to the two recipients.However, previous theoretical and experimental studies both overlooked the modulation errors that may be introduced in the state preparation process due to the imperfections in modulator devices. Under the traditional framework of GLLP analysis method, these errors will significantly reduce the actual signature rates. Therefore, we propose a state-preparation-error tolerant QDS and use parameter analysis to characterize the state preparation error to make the simulation analysis more realistic. In addition, we analyze the signature rates of the present scheme by using the three-intensity decoy-state method. Compared with previous QDS protocols, our protocol almost shows no performance degradation under practical state preparation errors and exhibits a maximum transmission distance around 180 km. Furthermore, state preparation errors do not have a significant influence on the bit error rate induced by normal communication between the legitimate users or the one produced by an eavesdropper. These results prove that the method proposed in this paper has excellent robustness against state preparation errors and it can achieve much higher signature rates and signature distances than other standard methods. Besides, signature rates are basically unchanged under different total pulse numbers, which shows that our protocol also has good robustness against the finite-size effect. Additionally, in the key generation process, our method is only required to prepare three quantum states, which will reduce the difficulty of experiment realizations. Furthermore, the proposed method can also be combined with the measurement-device-independent QDS protocol and the twin-field QDS protocol to further increase the security level of QDS protocol. Therefore, our work will provide an important reference value for realizing the practical application of QDS in the future. -
Keywords:
- quantum digital signature /
- state preparation errors /
- decoy state
[1] Diffie W, Helman M E 1976 IEEE Trans. Inf. Theory 22 644
Google Scholar
[2] Gottesman D, Chuang I 2001 arXiv: quant-ph/0105032v2
[3] Clarke P J, Collins R J, Dunjko V, Andersson E, Jeffers J, Buller G S 2012 Nat. Commun. 3 1174
Google Scholar
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Google Scholar
[5] Amiri R, Wallden P, Kent A, Andersson E 2016 Phys. Rev. A 93 032325
Google Scholar
[6] Puthoor I V, Amiri R, Wallden P, Curty M, Andersson Erika 2016 Phys. Rev. A 94 022328
Google Scholar
[7] An X B, Zhang H, Zhang C M, Chen W, Wang S, Yin Z Q, Wang Q, He D Y, Hao P L, Liu S F, Zhou X Y, Guo G C, Han Z F 2019 Opt. Lett. 44 139
Google Scholar
[8] Zhang C H, Zhou X Y, Ding H J, Zhang C M, Guo G C, Wang Q 2018 Phys. Rev. Appl. 10 034033
Google Scholar
[9] Ding H J, Chen J J, Li J, Zhou X Y, Zhang C H, Zhang C M, Wang Q 2020 Opt. Lett. 45 1711
Google Scholar
[10] 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 Natl. Sci. Rev. 10 1093
[11] Lo H K, Ma X F, Chen K 2005 Phys. Rev. Lett. 94 230504
Google Scholar
[12] Zeng G, Keitel C H 2002 Phys. Rev. A 65 042312
Google Scholar
[13] Tamaki K, Lo H K, Fung C H F, Qi B 2011 Phys. Rev. A 85 042307
[14] Koashi M 2009 New J. Phys. 11 045018
Google Scholar
[15] Lo H K, Preskill J 2007 Quant. Inf. Comput. 8 431
[16] Gottesman D, Lo H K, Lütkenhaus N, Preskill J 2004 Quant. Inf. Comput. 4 325
[17] Tamaki K, Curty M, Kato G, Lo H K, Azuma K 2014 Phys. Rev. A 90 052314
Google Scholar
[18] Serfling R J 1974 Ann. Statist. 2 39
[19] 马啸, 孙铭烁, 刘靖阳, 丁华建, 王琴 2022 71 030301
Google Scholar
Ma X, Sun M S, Liu J Y, Ding H J, Wang Q 2022 Acta Phys. Sin. 71 030301
Google Scholar
[20] Hoeffding W 1994 Probability Inequalities for Sums of Bounded Random Variables (New York: Springer) pp409–426
[21] Gobby C, Yuan Z L, Shields A J 2004 Appl. Phys. Lett. 84 3672
Google Scholar
[22] Zhang C H, Zhou X Y, Zhang C M, Li J, Wang Q 2021 Opt. Lett. 46 3757
Google Scholar
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图 1 态制备误差容忍方法和GLLP分析方法签名率大小对比结果
Figure 1. Comparison on the signature rate between the state-preparation-error tolerance scheme and GLLP method.
图 2 态制备误差容忍方法和GLLP分析方法的错误率对比结果
Figure 2. Comparison on the error rate between the state-preparation-error tolerance scheme and GLLP method.
图 3 传输距离为20 km时, 不同总脉冲数下, 态制备误差容忍方法和GLLP分析方法签名率随着态制备误差变化对比
Figure 3. The signature rate vs. state preparation error for the state-preparation-error tolerance scheme and GLLP method under different total number of pulses. Here the transmission distance is fixed at 20 km.
表 1 基于量子数字签名的态制备误差容忍协议仿真使用的参数列表[21]
Table 1. The parameter list used for simulation of state preparation error tolerance protocol based on quantum digital signature protocol[21].
接收方探测器
暗计数率 Pd接收方探测器
探测效率 ηd信道损耗系数
α/(dB·km–1)KGP过程中估参
长度比例 d发射的总
脉冲数Ntot失败概率
${\varepsilon _{{\text{PE}}}}$最弱诱骗态
强度 w1.5×10–6 0.145 0.2 1/21 1014 10–5 0.002 
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[1] Diffie W, Helman M E 1976 IEEE Trans. Inf. Theory 22 644
Google Scholar
[2] Gottesman D, Chuang I 2001 arXiv: quant-ph/0105032v2
[3] Clarke P J, Collins R J, Dunjko V, Andersson E, Jeffers J, Buller G S 2012 Nat. Commun. 3 1174
Google Scholar
[4] Collins R J, Donaldson R J, Dunjko V, Wallden P, Clarke P J, Andersson E, Jeffers J, Buller G S 2014 Phys. Rev. Lett. 113 040502
Google Scholar
[5] Amiri R, Wallden P, Kent A, Andersson E 2016 Phys. Rev. A 93 032325
Google Scholar
[6] Puthoor I V, Amiri R, Wallden P, Curty M, Andersson Erika 2016 Phys. Rev. A 94 022328
Google Scholar
[7] An X B, Zhang H, Zhang C M, Chen W, Wang S, Yin Z Q, Wang Q, He D Y, Hao P L, Liu S F, Zhou X Y, Guo G C, Han Z F 2019 Opt. Lett. 44 139
Google Scholar
[8] Zhang C H, Zhou X Y, Ding H J, Zhang C M, Guo G C, Wang Q 2018 Phys. Rev. Appl. 10 034033
Google Scholar
[9] Ding H J, Chen J J, Li J, Zhou X Y, Zhang C H, Zhang C M, Wang Q 2020 Opt. Lett. 45 1711
Google Scholar
[10] 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 Natl. Sci. Rev. 10 1093
[11] Lo H K, Ma X F, Chen K 2005 Phys. Rev. Lett. 94 230504
Google Scholar
[12] Zeng G, Keitel C H 2002 Phys. Rev. A 65 042312
Google Scholar
[13] Tamaki K, Lo H K, Fung C H F, Qi B 2011 Phys. Rev. A 85 042307
[14] Koashi M 2009 New J. Phys. 11 045018
Google Scholar
[15] Lo H K, Preskill J 2007 Quant. Inf. Comput. 8 431
[16] Gottesman D, Lo H K, Lütkenhaus N, Preskill J 2004 Quant. Inf. Comput. 4 325
[17] Tamaki K, Curty M, Kato G, Lo H K, Azuma K 2014 Phys. Rev. A 90 052314
Google Scholar
[18] Serfling R J 1974 Ann. Statist. 2 39
[19] 马啸, 孙铭烁, 刘靖阳, 丁华建, 王琴 2022 71 030301
Google Scholar
Ma X, Sun M S, Liu J Y, Ding H J, Wang Q 2022 Acta Phys. Sin. 71 030301
Google Scholar
[20] Hoeffding W 1994 Probability Inequalities for Sums of Bounded Random Variables (New York: Springer) pp409–426
[21] Gobby C, Yuan Z L, Shields A J 2004 Appl. Phys. Lett. 84 3672
Google Scholar
[22] Zhang C H, Zhou X Y, Zhang C M, Li J, Wang Q 2021 Opt. Lett. 46 3757
Google Scholar
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