{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T11:26:35Z","timestamp":1769081195496,"version":"3.49.0"},"reference-count":15,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,10]],"date-time":"2020-11-10T00:00:00Z","timestamp":1604966400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Information"],"abstract":"<jats:p>Ultra-large-scale matrix inversion has been applied as the fundamental operation of numerous domains, owing to the growth of big data and matrix applications. Using cryptography as an example, the solution of ultra-large-scale linear equations over finite fields is important in many cryptanalysis schemes. However, inverting matrices of extremely high order, such as in millions, is challenging; nonetheless, the need has become increasingly urgent. Hence, we propose a parallel distributed block recursive computing method that can process matrices at a significantly increased scale, based on Strassen\u2019s method; furthermore, we describe the related well-designed algorithm herein. Additionally, the experimental results based on comparison show the efficiency and the superiority of our method. Using our method, up to 140,000 dimensions can be processed in a supercomputing center.<\/jats:p>","DOI":"10.3390\/info11110523","type":"journal-article","created":{"date-parts":[[2020,11,10]],"date-time":"2020-11-10T10:47:28Z","timestamp":1605005248000},"page":"523","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["A Method of Ultra-Large-Scale Matrix Inversion Using Block Recursion"],"prefix":"10.3390","volume":"11","author":[{"given":"HouZhen","family":"Wang","sequence":"first","affiliation":[{"name":"Key Laboratory of Aerospace Information Security and Trusted Computing, Ministry of Education, School of Cyber Science and Engineering, Wuhan University, Wuhan 430072, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8672-1241","authenticated-orcid":false,"given":"Yan","family":"Guo","sequence":"additional","affiliation":[{"name":"Key Laboratory of Aerospace Information Security and Trusted Computing, Ministry of Education, School of Cyber Science and Engineering, Wuhan University, Wuhan 430072, China"}]},{"given":"HuanGuo","family":"Zhang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Aerospace Information Security and Trusted Computing, Ministry of Education, School of Cyber Science and Engineering, Wuhan University, Wuhan 430072, China"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Yan, M., Sang, J., and Xu, C. (2015, January 23\u201326). Unified youtube video recommendation via cross-network collaboration. Proceedings of the 5th ACM on International Conference on Multimedia Retrieval, Shanghai, China.","DOI":"10.1145\/2671188.2749344"},{"key":"ref_2","unstructured":"Matsue, T., Sekitsuka, T., and Shingyoji, R. (2019). Satellite Radiowave Receiving Device, Electronic Timepiece, Method for Controlling Positioning Operations, and Storage Device. (App. 16\/135,383), U.S. Patent."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Porras, J., Baena, J., and Ding, J. (2014, January 1\u20133). ZHFE, a new multivariate public key encryption scheme. Proceedings of the International Workshop on Post-Quantum Cryptography, Waterloo, ON, Canada.","DOI":"10.1007\/978-3-319-11659-4_14"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1080\/00029890.1987.12000605","article-title":"Gauss-Jordan reduction: A brief history","volume":"94","author":"Althoen","year":"1987","journal-title":"Am. Math. Mon."},{"key":"ref_5","unstructured":"Krishnamoorthy, A., and Menon, D. (2013, January 26\u201328). Matrix inversion using Cholesky decomposition. Proceedings of the 2013 Signal Processing: Algorithms, Architectures, Arrangements, and Applications (SPA), Poznan, Poland."},{"key":"ref_6","unstructured":"Press, W., Teukolsky, S., Vetterling, W., and Flannery, B. (2007). Numerical Recipes: The Art of Scientific Computing, Cambridge University Press. [3rd ed.]."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.amc.2007.02.082","article-title":"A way to obtain Monte Carlo matrix inversion with minimal error","volume":"191","author":"Vajargah","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"7919","DOI":"10.1088\/0305-4470\/30\/22\/026","article-title":"Analytical inversion of general tridiagonal matrices","volume":"30","author":"Huang","year":"1997","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1007\/s11227-010-0487-y","article-title":"Parallel algorithm on inversion for adjacent pentadiagonal matrices with mpi","volume":"59","author":"Kanal","year":"2012","journal-title":"J. Supercomput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1109\/TPDS.2011.103","article-title":"Triangular matrix inversion on heterogeneous multicore systems","volume":"23","author":"Ries","year":"2012","journal-title":"IEEE Trans. Parallel Distrib. Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1007\/BF02165411","article-title":"Gaussian elimination is not optimal","volume":"13","author":"Strassen","year":"1969","journal-title":"Numer. Math."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Pan, V.Y. (1978, January 16\u201318). Strassen\u2019s algorithm is not optimal trilinear technique of aggregating, uniting and canceling for constructing fast algorithms for matrix operations. Proceedings of the 19th Annual Symposium on Foundations of Computer Science (sfcs 1978), Ann Arbor, MI, USA.","DOI":"10.1109\/SFCS.1978.34"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1137\/0211038","article-title":"On the asymptotic complexity of matrix multiplication","volume":"11","author":"Coppersmith","year":"1982","journal-title":"SIAM J. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Xiang, J., Meng, H., and Aboulnaga, A. (2014, January 23\u201327). Scalable matrix inversion using mapreduce. Proceedings of the 23rd International Symposium on High-Performance Parallel and Distributed Computing, Vancouver, BC, Canada.","DOI":"10.1145\/2600212.2600220"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2166","DOI":"10.1109\/ACCESS.2016.2546544","article-title":"Spark-based large-scale matrix inversion for big data processing","volume":"4","author":"Liu","year":"2016","journal-title":"IEEE Access"}],"container-title":["Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2078-2489\/11\/11\/523\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:31:28Z","timestamp":1760178688000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2078-2489\/11\/11\/523"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,10]]},"references-count":15,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["info11110523"],"URL":"https:\/\/doi.org\/10.3390\/info11110523","relation":{},"ISSN":["2078-2489"],"issn-type":[{"value":"2078-2489","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,10]]}}}