{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:31:22Z","timestamp":1760059882319,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,18]],"date-time":"2025-07-18T00:00:00Z","timestamp":1752796800000},"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":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"],"award-info":[{"award-number":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"2025 Major Project of the Natural Science Foundation of Inner Mongolia","award":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"],"award-info":[{"award-number":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"]}]},{"name":"Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region","award":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"],"award-info":[{"award-number":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"]}]},{"name":"Central Government Guides Local Science and Technology Development Fund Project of China","award":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"],"award-info":[{"award-number":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"]}]},{"name":"2024 Inner Mongolia Archives Technology Project","award":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"],"award-info":[{"award-number":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"]}]},{"name":"Tianjin Renai College &amp; Tianjin University Teacher Joint Development Fund Cooperation Project","award":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"],"award-info":[{"award-number":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"]}]},{"name":"2022 Chinese Academy of Sciences \u201cWestern Light\u201d Talent Training Program \u201cWestern Young Scholars\u201d Project","award":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"],"award-info":[{"award-number":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"]}]},{"name":"Inner Mongolia Autonomous Region Natural Science Foundation","award":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"],"award-info":[{"award-number":["62441212","2025ZD008","NJYT23106","NJYT23057","2022ZY0024","2025ZY0007","2024-47","FZ231001","22040601","2024MS06008"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Federated learning promotes the development of cross-domain intelligent applications under the premise of protecting data privacy, but there are still problems of sensitive parameter information leakage of multi-party data temporal alignment and resource scheduling process, and traditional symmetric encryption schemes suffer from low efficiency and poor security. To this end, in this paper, based on the modified NTRU-type multi-key fully homomorphic encryption scheme, an asymmetric algorithm, a secure computation scheme of multi-party least common multiple and greatest common divisor without full set under the semi-honest model is proposed. Participants strictly follow the established process. Nevertheless, considering that malicious participants may engage in poisoning attacks such as tampering with or uploading incorrect data to disrupt the protocol process and cause incorrect results, a scheme against malicious spoofing is further proposed, which resists malicious spoofing behaviors and not all malicious attacks, to verify the correctness of input parameters or data through hash functions and zero-knowledge proof, ensuring it can run safely and stably. Experimental results show that our semi-honest model scheme improves the efficiency by 39.5% and 45.6% compared to similar schemes under different parameter conditions, and it is able to efficiently process small and medium-sized data in real time under high bandwidth; although there is an average time increase of 1.39 s, the anti-malicious spoofing scheme takes into account both security and efficiency, achieving the design expectations.<\/jats:p>","DOI":"10.3390\/sym17071151","type":"journal-article","created":{"date-parts":[[2025,7,18]],"date-time":"2025-07-18T11:12:34Z","timestamp":1752837154000},"page":"1151","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["HE\/MPC-Based Scheme for Secure Computing LCM\/GCD and Its Application to Federated Learning"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3450-3808","authenticated-orcid":false,"given":"Xin","family":"Liu","sequence":"first","affiliation":[{"name":"School of Digital and Intelligent Industry (School of Cyber Science and Technology), Inner Mongolia University of Science and Technology, Baotou 014010, China"},{"name":"School of Intelligent Computing Engineering, Tianjin Ren\u2019ai College, Tianjin 301636, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinyuan","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Digital and Intelligent Industry (School of Cyber Science and Technology), Inner Mongolia University of Science and Technology, Baotou 014010, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dan","family":"Luo","sequence":"additional","affiliation":[{"name":"School of Intelligent Computing Engineering, Tianjin Ren\u2019ai College, Tianjin 301636, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lanying","family":"Liang","sequence":"additional","affiliation":[{"name":"School of Digital and Intelligent Industry (School of Cyber Science and Technology), Inner Mongolia University of Science and Technology, Baotou 014010, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Ye","sequence":"additional","affiliation":[{"name":"Beijing Institute of Computing Technology and Applications, Beijing 100854, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuchen","family":"Zhang","sequence":"additional","affiliation":[{"name":"Beijing Institute of Computing Technology and Applications, Beijing 100854, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Baohua","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Digital and Intelligent Industry (School of Cyber Science and Technology), Inner Mongolia University of Science and Technology, Baotou 014010, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4345-6932","authenticated-orcid":false,"given":"Yu","family":"Gu","sequence":"additional","affiliation":[{"name":"School of Digital and Intelligent Industry (School of Cyber Science and Technology), Inner Mongolia University of Science and Technology, Baotou 014010, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Digital and Intelligent Industry (School of Cyber Science and Technology), Inner Mongolia University of Science and Technology, Baotou 014010, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bell, J. (2022). What Is Machine Learning?. Machine Learning and the City, John Wiley & Sons Ltd.","DOI":"10.1002\/9781119815075.ch18"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Khan, M.M., and Alkhathami, M. (2024). Anomaly Detection in IoT-Based Healthcare: Machine Learning for Enhanced Security. Sci. Rep., 14.","DOI":"10.1038\/s41598-024-56126-x"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1061","DOI":"10.1007\/s10479-021-04410-8","article-title":"Past, Present, and Future of Sustainable Finance: Insights from Big Data Analytics through Machine Learning of Scholarly Research","volume":"345","author":"Kumar","year":"2025","journal-title":"Ann. Oper. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.iotcps.2023.12.003","article-title":"Machine Learning Techniques for IoT Security: Current Research and Future Vision with Generative AI and Large Language Models","volume":"4","author":"Alwahedi","year":"2024","journal-title":"Internet Things Cyber-Phys. Syst."},{"key":"ref_5","first-page":"46","article-title":"Advancements in Federated Learning: Models, Methods, and Privacy","volume":"57","author":"Chen","year":"2024","journal-title":"ACM Comput. Surv."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1007\/s10462-024-10846-8","article-title":"An Overview of Implementing Security and Privacy in Federated Learning","volume":"57","author":"Hu","year":"2024","journal-title":"Artif. Intell. Rev."},{"key":"ref_7","unstructured":"Borana, A., Mohanty, A., Mukade, S., Roy, D.A., and Vaidya, U.W. (2023). Indigenous Secure NTP Server for Time Synchronization. BARC Newsl., 7\u201310. Available online: https:\/\/barc.gov.in\/barc_nl\/2023\/2023010202.pdf."},{"key":"ref_8","first-page":"3419","article-title":"A Privacy-preserving Data Alignment Framework for Vertical Federated Learning","volume":"46","author":"Gao","year":"2024","journal-title":"J. Electron. Inf. Technol."},{"key":"ref_9","first-page":"69","article-title":"Temporal Rule Distribution Mining of Privacy-preserving","volume":"34","author":"Gen","year":"2008","journal-title":"Comput. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Yao, A.C. (1982, January 3\u20135). Protocols for Secure Computations. Proceedings of the 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982), Chicago, IL, USA.","DOI":"10.1109\/SFCS.1982.38"},{"key":"ref_11","first-page":"186","article-title":"Optimization of NTRU-type multi-key fully homomorphic encryption scheme","volume":"52","author":"Che","year":"2020","journal-title":"Adv. Eng. Sci."},{"key":"ref_12","unstructured":"Goldreich, O. (2009). Foundations of Cryptography: Volume 2, Basic Applications, Cambridge University Press. [1st ed.]."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"121192","DOI":"10.1016\/j.eswa.2023.121192","article-title":"Beyond Data Poisoning in Federated Learning","volume":"235","author":"Kasyap","year":"2024","journal-title":"Expert. Syst. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"120377","DOI":"10.1016\/j.ins.2024.120377","article-title":"BPFL: Blockchain-Based Privacy-Preserving Federated Learning against Poisoning Attack","volume":"665","author":"Ren","year":"2024","journal-title":"Inf. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Fiege, U., Fiat, A., and Shamir, A. (1987, January 1). Zero Knowledge Proofs of Identity. Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, New York, NY, USA.","DOI":"10.1145\/28395.28419"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ames, S., Hazay, C., Ishai, Y., and Venkitasubramaniam, M. (2017\u20133, January 30). Ligero: Lightweight Sublinear Arguments Without a Trusted Setup. Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, Dallas, TX, USA.","DOI":"10.1145\/3133956.3134104"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1360\/SSI-2019-0226","article-title":"Protocol for millionaires\u2019 problem in malicious models","volume":"51","author":"Li","year":"2021","journal-title":"Sci. Sin. Informationis"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"127794","DOI":"10.1016\/j.neucom.2024.127794","article-title":"Private-Preserving Language Model Inference Based on Secure Multi-Party Computation","volume":"592","author":"Song","year":"2024","journal-title":"Neurocomputing"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3505","DOI":"10.1109\/TIFS.2025.3546843","article-title":"Efficient and Privacy-Preserving Feature Selection Based on Multiparty Computation","volume":"20","author":"Wang","year":"2025","journal-title":"IEEE Trans. Inf. Forensics Secur."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Rahaman, M., Arya, V., Orozco, S.M., and Pappachan, P. (2024). Secure Multi-Party Computation (SMPC) Protocols and Privacy. Innovations in Modern Cryptography, IGI Global Scientific Publishing.","DOI":"10.4018\/979-8-3693-5330-1.ch008"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1016\/j.procs.2024.10.152","article-title":"Secure Multi-Party Key-Value Data Statistics against Malicious Models","volume":"247","author":"Zhao","year":"2024","journal-title":"Procedia Comput. Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2111","DOI":"10.1109\/TKDE.2025.3535944","article-title":"SMPCache: Towards More Efficient SQL Queries in Multi-Party Collaborative Data Analysis","volume":"37","author":"Shi","year":"2025","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"025023","DOI":"10.1088\/2058-9565\/ad34f5","article-title":"Quantum Secure Multi-Party Computational Geometry Based on Multi-Party Summation and Multiplication","volume":"9","author":"Dou","year":"2024","journal-title":"Quantum Sci. Technol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2129","DOI":"10.1109\/TC.2025.3557968","article-title":"Efficient Quantum Secure Vector Dominance and Its Applications in Computational Geometry","volume":"74","author":"Liu","year":"2025","journal-title":"IEEE Trans. Comput."},{"key":"ref_25","first-page":"9","article-title":"Efficient and Privacy-Preserving Min and k Th Min Computations in Mobile Sensing Systems","volume":"14","author":"Zhang","year":"2017","journal-title":"IEEE Trans. Depend. Sec. Comput."},{"key":"ref_26","first-page":"49","article-title":"The Essential Role of Trusted Third Parties in Electronic Commerce","volume":"75","author":"Froomkin","year":"1996","journal-title":"Or. Law. Rev."},{"key":"ref_27","first-page":"1715","article-title":"Secure Multi-Party Computation for Minimum and its applications","volume":"45","author":"Dou","year":"2017","journal-title":"Acts Electron. Sin."},{"key":"ref_28","first-page":"219","article-title":"Secure multiparty computation of the maximum and the minimum in cloud environment and its statistics application","volume":"6","author":"Li","year":"2019","journal-title":"J. Cryptologic Res."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"L\u00f3pez-Alt, A., Tromer, E., and Vaikuntanathan, V. (2012, January 19\u201322). On-the-Fly Multiparty Computation on the Cloud via Multikey Fully Homomorphic Encryption. Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, New York, NY, USA.","DOI":"10.1145\/2213977.2214086"},{"key":"ref_30","first-page":"483","article-title":"Private maximum and minimum computation","volume":"7","author":"Yang","year":"2020","journal-title":"J. Cryptologic Res."},{"key":"ref_31","first-page":"2076","article-title":"Secure Maximum (Minimum) Computation in Malicious Model","volume":"44","author":"Li","year":"2021","journal-title":"Chin. J. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1109\/TIT.1985.1057074","article-title":"A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms","volume":"31","author":"Elgamal","year":"1985","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Li, S., Luo, D., Liu, X., and Luo, R. (2024, January 26\u201327). Secure Computation of Maximum and Minimum Values in Data Aggregation Based on Cloud Computing. Proceedings of the 2024 Third International Conference on Distributed Computing and Electrical Circuits and Electronics (ICDCECE), Ballari, India.","DOI":"10.1109\/ICDCECE60827.2024.10549651"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"095111","DOI":"10.1088\/1402-4896\/ad6aee","article-title":"A Novel Quantum Security Multi-Party Extremum Protocol in a d-Dimensional Quantum System","volume":"99","author":"Lu","year":"2024","journal-title":"Phys. Scr."},{"key":"ref_35","unstructured":"Najarkolaei, S.R.H., Mojahedian, M.M., and Aref, M.R. (2024). Beyond Yao\u2019s Millionaires: Secure Multi-Party Computation of Non-Polynomial Functions. arXiv."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3042","DOI":"10.1109\/TCE.2024.3360455","article-title":"Extremely Efficient and Privacy-Preserving MAX\/MIN Protocol Based on Multiparty Computation in Big Data","volume":"70","author":"Park","year":"2024","journal-title":"IEEE Trans. Consum. Electron."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"16263","DOI":"10.1364\/OE.555634","article-title":"Secure Multiparty Computation for Maximum and Minimum Values Based on Quantum Homomorphic Encryption","volume":"33","author":"Li","year":"2025","journal-title":"Opt. Express"},{"key":"ref_38","unstructured":"Liu, Y. (2019). Research on Two Basic Problems in Secure Multi-Party Computation. [Master\u2019s Thesis, Xidian University]."},{"key":"ref_39","first-page":"1132","article-title":"Private Substitution and Its Application in Private Scientific Computation","volume":"41","author":"Yang","year":"2018","journal-title":"Chin. J. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2340006","DOI":"10.1142\/S0219749923400063","article-title":"Quantum Multi-Party Private Set Union Protocol Based on Least Common Multiple and Shor\u2019s Algorithm","volume":"21","author":"Liu","year":"2023","journal-title":"Int. J. Quantum Inf."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1140\/epjqt\/s40507-024-00268-4","article-title":"Efficient Quantum Secure Multi-Party Greatest Common Divisor Protocol and Its Applications in Private Set Operations","volume":"11","author":"Li","year":"2024","journal-title":"EPJ Quantum Technol."},{"key":"ref_42","unstructured":"Fan, Y., Zhu, R., Wang, Z., Wang, C., Tang, H., Dong, Y., Cho, H., and Ohno-Machado, L. (2025). ByzSFL: Achieving Byzantine-Robust Secure Federated Learning with Zero-Knowledge Proofs. arXiv."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Kokaj, A., and Mollakuqe, E. (2025). Mathematical Proposal for Securing Split Learning Using Homomorphic Encryption and Zero-Knowledge Proofs. Appl. Sci., 15.","DOI":"10.3390\/app15062913"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Zhang, R., Li, H., Qian, X., Jiang, W., and Zhang, X. (2024, January 9\u201313). An Efficient and Secure Privacy-Preserving Federated Learning via Lattice-Based Functional Encryption. Proceedings of the ICC 2024\u2014IEEE International Conference on Communications, Denver, CO, USA.","DOI":"10.1109\/ICC51166.2024.10622766"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"103765","DOI":"10.1016\/j.csi.2023.103765","article-title":"Lattice Based Distributed Threshold Additive Homomorphic Encryption with Application in Federated Learning","volume":"87","author":"Tian","year":"2024","journal-title":"Comput. Stand. Interfaces"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/7\/1151\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:12:04Z","timestamp":1760033524000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/7\/1151"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,18]]},"references-count":45,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["sym17071151"],"URL":"https:\/\/doi.org\/10.3390\/sym17071151","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,7,18]]}}}