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Process.</a> <a href="/?q=in%3A491312" title="Articles in this Issue">22, No. 1, Paper No. 40, 23 p. (2023)</a>. </div> <div class="abstract">Summary: In this paper, we present a novel measurement-device-independent quantum protocol for electronic payment (E-payment), which can remove side channel attacks for the detectors. Especially, we use Bell states instead of three or more-qubit entangled states to reduce the complexity of quantum resources. Furthermore, we employ both quantum public-key encryption and blockchain technologies to ensure more perfect security properties. Compared with the previously proposed protocols, our proposed protocol has not only higher security but also better feasibility with current technologies.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7657152">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A81P94" title="MSC2020">81P94</a> </td> <td class="space"> Quantum cryptography (quantum-theoretic aspects) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AE-payment">E-payment</a>; <a href="/?q=ut%3Aquantum+public+key">quantum public key</a>; <a href="/?q=ut%3Ameasurement-device-independent">measurement-device-independent</a>; <a href="/?q=ut%3Ablockchain">blockchain</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1508.81700" data-ciurl="/ci/07657152" data-biburl="/bibtex/07657152.bib" data-amsurl="/amsrefs/07657152.bib" data-xmlurl="/xml/07657152.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07657152.pdf" title="Zbl 1508.81700 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s11128-022-03770-9" aria-label="DOI for “Measurement-device-independent quantum protocol for E-payment based on blockchain”" title="10.1007/s11128-022-03770-9">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 212-219 (1996) · <a href="/0922.68044" class="nowrap">Zbl 0922.68044</a></td> </tr><tr> <td>[2]</td> <td class="space">Shor, P.W.: Algorithms for quantum computation discrete logarithms and factoring. In: Proceedings 35th Annual Symposium on Foundations of Computer Science, pp. 124-134 (1994)</td> </tr><tr> <td>[3]</td> <td class="space">Lo, HK, Insecurity of quantum secure computations, Phys. Rev. A., 56, 2, 021154 (1997) · <a href="https://doi.org/10.1103/PhysRevA.56.1154" class="nowrap">doi:10.1103/PhysRevA.56.1154</a></td> </tr><tr> <td>[4]</td> <td class="space">Zhang, Y.; Deng, RH; Liu, X., Outsourcing service fair payment based on blockchain and its applications in cloud computing, IEEE Trans. Serv. Comput., 14, 4, 1152-1166 (2018) · <a href="https://doi.org/10.1109/TSC.2018.2864191" class="nowrap">doi:10.1109/TSC.2018.2864191</a></td> </tr><tr> <td>[5]</td> <td class="space">Coffie, CPK; Hongjiang, Z.; Mensah, IA, Determinants of FinTech payment services diffusion by SMEs in Sub-Saharan Africa: evidence from Ghana, Inf. Technol. Dev., 27, 3, 539-560 (2021) · <a href="https://doi.org/10.1080/02681102.2020.1840324" class="nowrap">doi:10.1080/02681102.2020.1840324</a></td> </tr><tr> <td>[6]</td> <td class="space">Wen, XJ; Nie, Z., An E-payment system based on quantum blind and group signature, Phys. Scr., 82, 6, 5468-5478 (2010)</td> </tr><tr> <td>[7]</td> <td class="space">Wen, XJ; Chen, YZ; Fang, JB, An inter-bank E-payment protocol based on quantum proxy blind signature, Quantum Inf. Process., 12, 1, 549-558 (2013) · <a href="/1277.94044" class="nowrap">Zbl 1277.94044</a> · <a href="https://doi.org/10.1007/s11128-012-0398-3" class="nowrap">doi:10.1007/s11128-012-0398-3</a></td> </tr><tr> <td>[8]</td> <td class="space">Cai, XQ; Wei, CY, Cryptanalysis of an inter-bank E-payment protocol based on quantum proxy blind signature, Quantum Inf. Process., 12, 4, 1651-1657 (2013) · <a href="/1281.94077" class="nowrap">Zbl 1281.94077</a> · <a href="https://doi.org/10.1007/s11128-012-0477-5" class="nowrap">doi:10.1007/s11128-012-0477-5</a></td> </tr><tr> <td>[9]</td> <td class="space">Shao, AX; Zhang, JZ; Xie, SC, An E-payment protocol based on quantum multi-proxy blind signature, Int. J. Theor. Phys., 56, 4, 1241-1248 (2017) · <a href="/1390.81131" class="nowrap">Zbl 1390.81131</a> · <a href="https://doi.org/10.1007/s10773-016-3266-6" class="nowrap">doi:10.1007/s10773-016-3266-6</a></td> </tr><tr> <td>[10]</td> <td class="space">Niu, XF; Zhang, JZ; Xie, SC, A third-party E-payment protocol based on quantum multi-proxy blind signature, Int. J. Theor. Phys., 57, 8, 2563-2573 (2018) · <a href="/1401.81029" class="nowrap">Zbl 1401.81029</a> · <a href="https://doi.org/10.1007/s10773-018-3778-3" class="nowrap">doi:10.1007/s10773-018-3778-3</a></td> </tr><tr> <td>[11]</td> <td class="space">Tiliwalidi, K.; Zhang, JZ; Xie, SC, A multi-bank E-payment protocol based on quantum proxy blind signature, Int. J. Theor. Phys., 58, 10, 3510-3520 (2019) · <a href="/1428.81072" class="nowrap">Zbl 1428.81072</a> · <a href="https://doi.org/10.1007/s10773-019-04217-1" class="nowrap">doi:10.1007/s10773-019-04217-1</a></td> </tr><tr> <td>[12]</td> <td class="space">Xie, SC; Niu, XF; Zhang, JZ, An improved quantum E-payment system, Int. J. Theor. Phys., 59, 2, 445-453 (2020) · <a href="https://doi.org/10.1007/s10773-019-04338-7" class="nowrap">doi:10.1007/s10773-019-04338-7</a></td> </tr><tr> <td>[13]</td> <td class="space">Gou, X.; Shi, R.; Gao, W., A novel quantum E-payment protocol based on blockchain, Quantum Inf. Process., 20, 5, 1-17 (2021) · <a href="/1509.81369" class="nowrap">Zbl 1509.81369</a> · <a href="https://doi.org/10.1007/s11128-021-03126-9" class="nowrap">doi:10.1007/s11128-021-03126-9</a></td> </tr><tr> <td>[14]</td> <td class="space">Wang, TY; Wei, ZL, One-time proxy signature based on quantum cryptography, Quantum Inf. Process., 11, 2, 455-463 (2012) · <a href="https://doi.org/10.1007/s11128-011-0258-6" class="nowrap">doi:10.1007/s11128-011-0258-6</a></td> </tr><tr> <td>[15]</td> <td class="space">Lo, HK; Curty, M.; Qi, B., Measurement device independent quantum key distribution, Phys. Rev. Lett., 108, 13, 130503 (2012) · <a href="https://doi.org/10.1103/PhysRevLett.108.130503" class="nowrap">doi:10.1103/PhysRevLett.108.130503</a></td> </tr><tr> <td>[16]</td> <td class="space">Zhou, ZR; Sheng, YB; Niu, PH, Measurement-device-independent quantum secure direct communication, Sci. China Phys. Mech. Astron., 63, 3, 1-6 (2020) · <a href="https://doi.org/10.1007/s11433-019-1450-8" class="nowrap">doi:10.1007/s11433-019-1450-8</a></td> </tr><tr> <td>[17]</td> <td class="space">Curty, M.; Xu, F.; Cui, W., Finite-key analysis for measurement-device-independent quantum key distribution, Nat. Commun., 5, 1, 1-7 (2014) · <a href="https://doi.org/10.1038/ncomms4732" class="nowrap">doi:10.1038/ncomms4732</a></td> </tr><tr> <td>[18]</td> <td class="space">Nikolopoulos, GM, Applications of single-qubit rotations in quantum public-key cryptography, Phys. Rev. A., 77, 3, 032348 (2008) · <a href="/1255.81126" class="nowrap">Zbl 1255.81126</a> · <a href="https://doi.org/10.1103/PhysRevA.77.032348" class="nowrap">doi:10.1103/PhysRevA.77.032348</a></td> </tr><tr> <td>[19]</td> <td class="space">Bennett, CH; Brassard, G., Quantum public key distribution reinvented, ACM SIGACT News, 18, 4, 51-53 (1987) · <a href="https://doi.org/10.1145/36068.36070" class="nowrap">doi:10.1145/36068.36070</a></td> </tr><tr> <td>[20]</td> <td class="space">Paul, T., Quantum computation and quantum information, Math. Struct. Comput. Sci., 17, 6, 1115-1115 (2007) · <a href="/1118.83015" class="nowrap">Zbl 1118.83015</a> · <a href="https://doi.org/10.1017/S0960129507006317" class="nowrap">doi:10.1017/S0960129507006317</a></td> </tr><tr> <td>[21]</td> <td class="space">Wang, S.; Ouyang, L.; Yuan, Y., Blockchain-enabled smart contracts: architecture, applications, and future trends, IEEE Trans. Syst. Man Cybern. Syst., 49, 11, 2266-2277 (2019) · <a href="https://doi.org/10.1109/TSMC.2019.2895123" class="nowrap">doi:10.1109/TSMC.2019.2895123</a></td> </tr><tr> <td>[22]</td> <td class="space">Shi, RH, Anonymous quantum sealed-bid auction, IEEE Trans. Circuits Syst. II Express Briefs, 69, 2, 414-418 (2021)</td> </tr><tr> <td>[23]</td> <td class="space">Nofer, M.; Gomber, P.; Hinz, O., Blockchain, Bus. Inf. Syst. Eng., 59, 3, 183-187 (2017) · <a href="https://doi.org/10.1007/s12599-017-0467-3" class="nowrap">doi:10.1007/s12599-017-0467-3</a></td> </tr><tr> <td>[24]</td> <td class="space">Zheng, Z.; Xie, S.; Dai, HN, Blockchain challenges and opportunities: a survey, Int. J. Web Grid Serv., 14, 4, 352-375 (2018) · <a href="https://doi.org/10.1504/IJWGS.2018.095647" class="nowrap">doi:10.1504/IJWGS.2018.095647</a></td> </tr><tr> <td>[25]</td> <td class="space">Liang, W.; Zhang, D.; Lei, X., Circuit copyright blockchain: blockchain-based homomorphic encryption for IP circuit protection, IEEE Trans. Emerg. Top. Comput., 9, 3, 1410-1420 (2020) · <a href="https://doi.org/10.1109/TETC.2020.2993032" class="nowrap">doi:10.1109/TETC.2020.2993032</a></td> </tr><tr> <td>[26]</td> <td class="space">Boaron, A.; Boso, G.; Rusca, D., Secure quantum key distribution over 421 km of optical fiber, Phys. Rev. Lett., 121, 19, 190520 (2018) · <a href="https://doi.org/10.1103/PhysRevLett.121.190502" class="nowrap">doi:10.1103/PhysRevLett.121.190502</a></td> </tr><tr> <td>[27]</td> <td class="space">Lucamarini, M.; Yuan, ZL; Dynes, JF; Shields, AJ, Overcoming the rate-distance limit of quantum key distribution without quantum repeaters, Nature, 557, 7705, 400-403 (2018) · <a href="https://doi.org/10.1038/s41586-018-0066-6" class="nowrap">doi:10.1038/s41586-018-0066-6</a></td> </tr><tr> <td>[28]</td> <td class="space">Shi, RH, Useful equations about Bell states and their applications to quantum secret sharing, IEEE Commun. Lett., 24, 2, 386-390 (2019) · <a href="https://doi.org/10.1109/LCOMM.2019.2954134" class="nowrap">doi:10.1109/LCOMM.2019.2954134</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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