{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,7]],"date-time":"2025-12-07T05:18:57Z","timestamp":1765084737865,"version":"3.46.0"},"reference-count":23,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T00:00:00Z","timestamp":1764892800000},"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":"crossref","award":["62201080"],"award-info":[{"award-number":["62201080"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study the problem of locally encoded secure distributed batch matrix multiplication (LESDBMM), where M pairs of sources each encode their respective batches of massive matrices and distribute the generated shares to a subset of N worker nodes. Each worker node computes a response from the received shares and sends the result to a sink node, which must be able to recover all M batches of pairwise matrix products in the presence of up to S stragglers. Additionally, any set of up to X colluding workers cannot learn any information about the matrices. Based on the idea of cross-subspace (CSA) codes and CSA null shaper, we propose the first LESDBMM scheme for batch processing. When the problem reduces to the coded distributed batch matrix multiplication (CDBMM) setting where M=1,X=0 and every source distributes its share to all worker nodes, the proposed scheme achieves performance matching that of the cross-subspace alignment (CSA) codes for CDBMM in terms of the maximum number of tolerable stragglers, communication cost, and computational complexity. Therefore, our scheme can be viewed as a generalization of CSA codes for CDBMM to the LESDBMM setting.<\/jats:p>","DOI":"10.3390\/e27121231","type":"journal-article","created":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T10:50:38Z","timestamp":1764931838000},"page":"1231","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Locally Encoded Secure Distributed Batch Matrix Multiplication"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-8018-9011","authenticated-orcid":false,"given":"Haobo","family":"Jia","sequence":"first","affiliation":[{"name":"School of Artificial Intelligence, Beijing University of Posts and Telecommunications, Beijing 100876, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8329-9911","authenticated-orcid":false,"given":"Zhuqing","family":"Jia","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Beijing University of Posts and Telecommunications, Beijing 100876, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1514","DOI":"10.1109\/TIT.2017.2736066","article-title":"Speeding Up Distributed Machine Learning Using Codes","volume":"64","author":"Lee","year":"2018","journal-title":"IEEE Trans. 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