{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T11:10:59Z","timestamp":1762081859688,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,31]],"date-time":"2022-12-31T00:00:00Z","timestamp":1672444800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62175138","2020B0303040002","20200020115001","No","2021L258"],"award-info":[{"award-number":["62175138","2020B0303040002","20200020115001","No","2021L258"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100015956","name":"Key Research and Development Program of Guangdong Province","doi-asserted-by":"publisher","award":["62175138","2020B0303040002","20200020115001","No","2021L258"],"award-info":[{"award-number":["62175138","2020B0303040002","20200020115001","No","2021L258"]}],"id":[{"id":"10.13039\/501100015956","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012130","name":"Aeronautical Science Foundation of China","doi-asserted-by":"publisher","award":["62175138","2020B0303040002","20200020115001","No","2021L258"],"award-info":[{"award-number":["62175138","2020B0303040002","20200020115001","No","2021L258"]}],"id":[{"id":"10.13039\/501100012130","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Shanxi 1331KSC","award":["62175138","2020B0303040002","20200020115001","No","2021L258"],"award-info":[{"award-number":["62175138","2020B0303040002","20200020115001","No","2021L258"]}]},{"name":"Scientfic and Technological Innovation Programs of Higher Education Institutions in 269 Shanxi Province","award":["62175138","2020B0303040002","20200020115001","No","2021L258"],"award-info":[{"award-number":["62175138","2020B0303040002","20200020115001","No","2021L258"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We propose a multidimensional reconciliation encoding algorithm based on a field-programmable gate array (FPGA) with variable data throughput that enables quantum key distribution (QKD) systems to be adapted to different throughput requirements. Using the circulatory structure, data flow in the most complex pipeline operation in the same time interval, which enables the structural multiplexing of the algorithm. We handle the calculation and storage of eight-dimensional matrices cleverly to conserve resources and increase data processing speed. In order to obtain the syndrome more efficiently, we designed a simplified algorithm according to the characteristics of the FPGA and parity-check matrix, which omits the unnecessary operation of matrix multiplication. The simplified algorithm could adapt to different rates. We validated the feasibility and high speed of the algorithm by implementing the multidimensional reconciliation encoding algorithm on a Xilinx Virtex-7 FPGA. Our simulation results show that the maximum throughput could reach 4.88 M symbols\/s.<\/jats:p>","DOI":"10.3390\/e25010080","type":"journal-article","created":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T04:51:22Z","timestamp":1672635082000},"page":"80","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["FPGA-Based Implementation of Multidimensional Reconciliation Encoding in Quantum Key Distribution"],"prefix":"10.3390","volume":"25","author":[{"given":"Qing","family":"Lu","sequence":"first","affiliation":[{"name":"State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China"},{"name":"Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China"}]},{"given":"Zhenguo","family":"Lu","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China"},{"name":"Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China"}]},{"given":"Hongzhao","family":"Yang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China"},{"name":"Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China"}]},{"given":"Shenshen","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Physics and Information Engineering, Shanxi Normal University, Taiyuan 030031, China"},{"name":"Key Laboratory of Spectral Measurement and Analysis of Shanxi Province, Shanxi Normal University, Taiyuan 030031, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0228-7693","authenticated-orcid":false,"given":"Yongmin","family":"Li","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China"},{"name":"Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"025002","DOI":"10.1103\/RevModPhys.92.025002","article-title":"Secure quantum key distribution with realistic devices","volume":"92","author":"Xu","year":"2020","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1012","DOI":"10.1364\/AOP.361502","article-title":"Advances in quantum cryptography","volume":"12","author":"Pirandola","year":"2020","journal-title":"Adv. Opt. Photonics"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"010303","DOI":"10.1103\/PhysRevA.61.010303","article-title":"Continuous variable quantum cryptography","volume":"61","author":"Ralph","year":"1999","journal-title":"Phys. Rev. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"6072","DOI":"10.3390\/e17096072","article-title":"Distributing secret keys with quantum continuous variables: Principle, security and implementations","volume":"17","author":"Diamanti","year":"2015","journal-title":"Entropy"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"040303","DOI":"10.1088\/1674-1056\/26\/4\/040303","article-title":"Continuous variable quantum key distribution","volume":"26","author":"Li","year":"2017","journal-title":"Chin. Phys. B"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1800011","DOI":"10.1002\/qute.201800011","article-title":"Continuous-variable quantum key distribution with Gaussian modulation\u2014The theory of practical implementations","volume":"1","author":"Laudenbach","year":"2018","journal-title":"Adv. Quantum Technol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1364\/OPTICA.3.000634","article-title":"Experimental demonstration of Gaussian protocols for one-sided device-independent quantum key distribution","volume":"3","author":"Walk","year":"2016","journal-title":"Optica"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2794","DOI":"10.1364\/OE.26.002794","article-title":"High key rate continuous-variable quantum key distribution with a real local oscillator","volume":"26","author":"Wang","year":"2018","journal-title":"Opt. Express"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"010502","DOI":"10.1103\/PhysRevLett.125.010502","article-title":"Long-distance continuous-variable quantum key distribution over 202.81 km of fiber","volume":"125","author":"Zhang","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"054065","DOI":"10.1103\/PhysRevApplied.13.054065","article-title":"Experimental passive state preparation for continuous variable quantum communications","volume":"13","author":"Qi","year":"2020","journal-title":"Phys. Rev. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1038\/s41534-020-00336-4","article-title":"Feasibility of satellite-to-ground continuous-variable quantum key distribution","volume":"7","author":"Dequal","year":"2021","journal-title":"NPJ Quantum Inform."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1364\/OPTICA.450573","article-title":"Experimental demonstration of continuous-variable measurement-device-independent quantum key distribution over optical fiber","volume":"9","author":"Tian","year":"2022","journal-title":"Optica"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4740","DOI":"10.1038\/s41467-022-32161-y","article-title":"Practical continuous-variable quantum key distribution with composable security","volume":"13","author":"Jain","year":"2022","journal-title":"Nat. Commun."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3307","DOI":"10.1364\/OL.456978","article-title":"Experimental demonstration of high-rate discrete-modulated continuous-variable quantum key distribution system","volume":"47","author":"Pan","year":"2022","journal-title":"Opt. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3935","DOI":"10.1109\/JLT.2020.2985408","article-title":"High-speed post-processing in continuous-variable quantum key distribution based on FPGA implementation","volume":"38","author":"Yang","year":"2020","journal-title":"J. Lightw. Technol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1109\/TIT.2003.822618","article-title":"Reconciliation of a quantum-distributed Gaussian key","volume":"50","author":"Cardinal","year":"2004","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"042325","DOI":"10.1103\/PhysRevA.77.042325","article-title":"Multidimensional reconciliation for a continuous-variable quantum key distribution","volume":"77","author":"Leverrier","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"044401","DOI":"10.7567\/JJAP.56.044401","article-title":"High-efficiency reconciliation for continuous variable quantum key distribution","volume":"56","author":"Bai","year":"2017","journal-title":"Jpn. J. Appl. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"30455","DOI":"10.1364\/OE.461665","article-title":"Continuous-variable quantum key distribution with low-complexity information reconciliation","volume":"30","author":"Wang","year":"2022","journal-title":"Opt. Express"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1038\/s41534-021-00509-9","article-title":"Rate-compatible multi-edge type low-density parity-check code ensembles for continuous-variable quantum key distribution systems","volume":"8","author":"Jeong","year":"2022","journal-title":"NPJ Quantum Inform."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1038\/s41534-022-00650-z","article-title":"Multidimensional Bose quantum error correction based on neural network decoder","volume":"8","author":"Wang","year":"2022","journal-title":"NPJ Quantum Inform."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Richardson, T., and Urbanke, R. (2008). Modern Coding Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511791338"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1038\/s41534-018-0070-6","article-title":"Quasi-cyclic multi-edge LDPC codes for long-distance quantum cryptography","volume":"4","author":"Milicevic","year":"2018","journal-title":"NPJ Quantum Inform."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"10543","DOI":"10.1038\/s41598-018-28703-4","article-title":"High speed error correction for continuous-variable quantum key distribution with multi-edge type LDPC code","volume":"8","author":"Wang","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"14561","DOI":"10.1038\/s41598-020-71534-5","article-title":"High-throughput GPU layered decoder of quasi-cyclic multi-edge type low density parity check codes in continuous-variable quantum key distribution systems","volume":"10","author":"Li","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1038\/s42005-022-00941-z","article-title":"Sub-Gbps key rate four-state continuous-variable quantum key distribution within metropolitan area","volume":"5","author":"Wang","year":"2022","journal-title":"Commun. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"6122","DOI":"10.1109\/TCOMM.2022.3188018","article-title":"Optimised multithreaded CV-QKD reconciliation for global quantum networks","volume":"70","author":"Ai","year":"2022","journal-title":"IEEE Trans. Commun."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1109\/MM.2021.3085431","article-title":"FPGA-accelerated quantum computing emulation and quantum key distillation","volume":"41","author":"Li","year":"2021","journal-title":"IEEE Micro"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"062317","DOI":"10.1103\/PhysRevA.84.062317","article-title":"Long-distance continuous-variable quantum key distribution with a Gaussian modulation","volume":"84","author":"Jouguet","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_30","unstructured":"Jose, S. (2022, December 28). 7 Series FPGAs Overview. Available online: https:\/\/docs.xilinx.com\/v\/u\/en-US\/ds180_7Series_Overview."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"V\u00e1zquez-Castillo, J., Castillo-Atoche, A., Carrasco-Alvarez, R., Longoria-Gandara, O., and Orteg\u00f3n-Aguilar, J. (2020). FPGA-based hardware matrix inversion architecture using hybrid piecewise polynomial approximation systolic cells. Electronics, 9.","DOI":"10.3390\/electronics9010182"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Golub, G.H., and Van Loan, C.F. (2013). Matrix Computations, JHU Press.","DOI":"10.56021\/9781421407944"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1788","DOI":"10.1109\/TIT.2004.831841","article-title":"Quasicyclic low-density parity-check codes from circulant permutation matrices","volume":"50","author":"Fossorier","year":"2004","journal-title":"IEEE Trans. Inf. Theory"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/80\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:49:00Z","timestamp":1760147340000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/80"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,31]]},"references-count":33,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010080"],"URL":"https:\/\/doi.org\/10.3390\/e25010080","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,12,31]]}}}