{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,27]],"date-time":"2025-12-27T21:09:00Z","timestamp":1766869740247,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,11,20]],"date-time":"2019-11-20T00:00:00Z","timestamp":1574208000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Cryptography"],"abstract":"<jats:p>We introduce a private quantum money scheme with the note verification procedure based on sampling matching, a problem in a one-way communication complexity model. Our scheme involves a bank who produces and distributes quantum notes, noteholders who are untrusted, and trusted local verifiers of the bank to whom the holders send their notes in order to carry out transactions. The key aspects of our money scheme include: note verification procedure requiring a single round classical interaction between the local verifier and bank; fixed verification circuit that uses only passive linear optical components; re-usability of each note in our scheme which grows linearly with the size of note; and an unconditional security against any adversary trying to forge the banknote while tolerating the noise of up to 21.4%. We further describe a practical implementation technique of our money scheme using weak coherent states of light and the verification circuit involving a single 50\/50 beam splitter and two single-photon threshold detectors. Previous best-known matching based money scheme proposal involves a verification circuit where the number of optical components increase proportional to the increase in desired noise tolerance (robustness). In contrast, we achieve any desired noise tolerance (up to a maximal threshold value) with only a fixed number of optical components. This considerable reduction of components in our scheme enables us to reach the robustness values that is not feasible for any existing money scheme with the current technology.<\/jats:p>","DOI":"10.3390\/cryptography3040026","type":"journal-article","created":{"date-parts":[[2019,11,20]],"date-time":"2019-11-20T11:06:03Z","timestamp":1574247963000},"page":"26","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Practically Feasible Robust Quantum Money with Classical Verification"],"prefix":"10.3390","volume":"3","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3037-1083","authenticated-orcid":false,"given":"Niraj","family":"Kumar","sequence":"first","affiliation":[{"name":"School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, UK"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1145\/1008908.1008920","article-title":"Wiesner, Sigact News 15, 78 (1983)","volume":"15","author":"Wiesner","year":"1983","journal-title":"Sigact News"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1038\/299802a0","article-title":"A single quantum cannot be cloned","volume":"299","author":"Wootters","year":"1982","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.tcs.2014.05.025","article-title":"Quantum cryptography: Public key distribution and coin tossing","volume":"560","author":"Bennett","year":"2014","journal-title":"Theor. Comput. Sci."},{"key":"ref_4","unstructured":"Gottesman, D., and Chuang, I. (2001). Quantum digital signatures. arXiv."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1016\/j.jcss.2003.07.010","article-title":"A new protocol and lower bounds for quantum coin flipping","volume":"68","author":"Ambainis","year":"2004","journal-title":"J. Comput. Syst. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Broadbent, A., Fitzsimons, J., and Kashefi, E. (2009, January 24\u201327). Universal blind quantum computation. Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science, Atlanta, GA, USA.","DOI":"10.1109\/FOCS.2009.36"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Cr\u00e9peau, C., Gottesman, D., and Smith, A. (2002, January 19\u201321). Secure multi-party quantum computation. Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, Montreal, QC, Canada.","DOI":"10.1145\/509907.510000"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/s10623-015-0157-4","article-title":"Quantum cryptography beyond quantum key distribution","volume":"78","author":"Broadbent","year":"2016","journal-title":"Des. Codes Cryptogr."},{"key":"ref_9","unstructured":"Lutomirski, A. (2010). An online attack against Wiesner\u2019s quantum money. arXiv."},{"key":"ref_10","unstructured":"Brodutch, A., Nagaj, D., Sattath, O., and Unruh, D. (2014). An adaptive attack on Wiesner\u2019s quantum money. arXiv."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Gavinsky, D. (2012, January 26\u201329). Quantum money with classical verification. Proceedings of the 2012 IEEE 27th Annual Conference on Computational Complexity (CCC), Porto, Portugal.","DOI":"10.1109\/CCC.2012.10"},{"key":"ref_12","unstructured":"Georgiou, M., and Kerenidis, I. (2015, January 20\u201322). New constructions for quantum money. Proceedings of the 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015), Brussels, Belgium."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"062334","DOI":"10.1103\/PhysRevA.95.062334","article-title":"Quantum money with nearly optimal error tolerance","volume":"95","author":"Amiri","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Gavinsky, D., Kempe, J., Kerenidis, I., Raz, R., and De Wolf, R. (2007, January 11\u201313). Exponential separations for one-way quantum communication complexity, with applications to cryptography. Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, San Diego, CA, USA.","DOI":"10.1145\/1250790.1250866"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"062311","DOI":"10.1103\/PhysRevA.93.062311","article-title":"Practical quantum retrieval games","volume":"93","author":"Arrazola","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"16079","DOI":"10.1073\/pnas.1203552109","article-title":"Unforgeable noise-tolerant quantum tokens","volume":"109","author":"Pastawski","year":"2012","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Aaronson, S., and Christiano, P. (2012, January 19\u201322). Quantum money from hidden subspaces. Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, New York, NY, USA.","DOI":"10.1145\/2213977.2213983"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Farhi, E., Gosset, D., Hassidim, A., Lutomirski, A., and Shor, P. (2012, January 8\u201310). Quantum money from knots. Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, Cambridge, MA, USA.","DOI":"10.1145\/2090236.2090260"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2475","DOI":"10.1007\/s11128-016-1273-4","article-title":"Quantum cheques","volume":"15","author":"Moulick","year":"2016","journal-title":"Quantum Inf. Process."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Radian, R., and Sattath, O. (2019). Semi-Quantum Money. arXiv.","DOI":"10.1145\/3318041.3355462"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1038\/s41534-018-0058-2","article-title":"Experimental investigation of practical unforgeable quantum money","volume":"4","author":"Bozzio","year":"2018","journal-title":"npj Quantum Inf."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"032338","DOI":"10.1103\/PhysRevA.97.032338","article-title":"Experimental preparation and verification of quantum money","volume":"97","author":"Guan","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41467-019-12139-z","article-title":"Experimental demonstration of quantum advantage for one-way communication complexity surpassing best-known classical protocol","volume":"10","author":"Kumar","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_24","unstructured":"Ben-David, S., and Sattath, O. (2016). Quantum tokens for digital signatures. arXiv."},{"key":"ref_25","unstructured":"Goldreich, O. (2004). The Foundations of Cryptography, Volume 2, Chapter Encryption Schemes, Cambridge University Press."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Upfal, E., and Mitzenmacher, M. (2005). Probability and Computing: Randomized Algorithms and Probabilistic Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511813603"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Bar-Yossef, Z., Jayram, T.S., and Kerenidis, I. (2004, January 13\u201315). Exponential separation of quantum and classical one-way communication complexity. Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, Chicago, IL, USA.","DOI":"10.1145\/1007352.1007379"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Molina, A., Vidick, T., and Watrous, J. (2012, January 17\u201319). Optimal counterfeiting attacks and generalizations for Wiesner\u2019s quantum money. Proceedings of the Conference on Quantum Computation, Communication, and Cryptography, Tokyo, Japan.","DOI":"10.1007\/978-3-642-35656-8_4"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"052309","DOI":"10.1103\/PhysRevA.86.052309","article-title":"Security details for bit commitment by transmitting measurement outcomes","volume":"86","author":"Croke","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/0034-4877(72)90011-0","article-title":"Linear transformations which preserve trace and positive semidefiniteness of operators","volume":"3","year":"1972","journal-title":"Rep. Math. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Yao, A.C. (1983, January 7\u20139). Lower bounds by probabilistic arguments. Proceedings of the 24th Annual Symposium on Foundations of Computer Science (sfcs 1983), Tucson, AZ, USA.","DOI":"10.1109\/SFCS.1983.30"},{"key":"ref_32","first-page":"31","article-title":"Information-theoretical aspects of quantum measurement","volume":"9","author":"Holevo","year":"1973","journal-title":"Probl. Peredachi Informatsii"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"706","DOI":"10.1038\/nphoton.2009.231","article-title":"Optical quantum memory","volume":"3","author":"Lvovsky","year":"2009","journal-title":"Nat. Photonics"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1038\/nature03064","article-title":"Experimental demonstration of quantum memory for light","volume":"432","author":"Julsgaard","year":"2004","journal-title":"Nature"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"022314","DOI":"10.1103\/PhysRevA.65.022314","article-title":"Quantum memory for photons: Dark-state polaritons","volume":"65","author":"Fleischhauer","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"033809","DOI":"10.1103\/PhysRevA.62.033809","article-title":"Quantum memory for light","volume":"62","author":"Kozhekin","year":"2000","journal-title":"Phys. Rev. A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1038\/s41534-018-0103-1","article-title":"Quantum superiority for verifying NP-complete problems with linear optics","volume":"4","author":"Arrazola","year":"2018","journal-title":"npj Quantum Inf."}],"container-title":["Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2410-387X\/3\/4\/26\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:36:03Z","timestamp":1760189763000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2410-387X\/3\/4\/26"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,20]]},"references-count":37,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,12]]}},"alternative-id":["cryptography3040026"],"URL":"https:\/\/doi.org\/10.3390\/cryptography3040026","relation":{},"ISSN":["2410-387X"],"issn-type":[{"type":"electronic","value":"2410-387X"}],"subject":[],"published":{"date-parts":[[2019,11,20]]}}}