{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T22:16:41Z","timestamp":1767997001434,"version":"3.49.0"},"reference-count":35,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,7,28]],"date-time":"2023-07-28T00:00:00Z","timestamp":1690502400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2022YFA1005000"],"award-info":[{"award-number":["2022YFA1005000"]}]},{"name":"National Key Research and Development Program of China","award":["12141108"],"award-info":[{"award-number":["12141108"]}]},{"name":"National Key Research and Development Program of China","award":["20JCYBJC01390"],"award-info":[{"award-number":["20JCYBJC01390"]}]},{"name":"National Key Research and Development Program of China","award":["050-63233070"],"award-info":[{"award-number":["050-63233070"]}]},{"name":"National Key Research and Development Program of China","award":["CUHK SRFS2223-4S03"],"award-info":[{"award-number":["CUHK SRFS2223-4S03"]}]},{"name":"Natural Science Foundation of China","award":["2022YFA1005000"],"award-info":[{"award-number":["2022YFA1005000"]}]},{"name":"Natural Science Foundation of China","award":["12141108"],"award-info":[{"award-number":["12141108"]}]},{"name":"Natural Science Foundation of China","award":["20JCYBJC01390"],"award-info":[{"award-number":["20JCYBJC01390"]}]},{"name":"Natural Science Foundation of China","award":["050-63233070"],"award-info":[{"award-number":["050-63233070"]}]},{"name":"Natural Science Foundation of China","award":["CUHK SRFS2223-4S03"],"award-info":[{"award-number":["CUHK SRFS2223-4S03"]}]},{"name":"Natural Science Foundation of Tianjin","award":["2022YFA1005000"],"award-info":[{"award-number":["2022YFA1005000"]}]},{"name":"Natural Science Foundation of Tianjin","award":["12141108"],"award-info":[{"award-number":["12141108"]}]},{"name":"Natural Science Foundation of Tianjin","award":["20JCYBJC01390"],"award-info":[{"award-number":["20JCYBJC01390"]}]},{"name":"Natural Science Foundation of Tianjin","award":["050-63233070"],"award-info":[{"award-number":["050-63233070"]}]},{"name":"Natural Science Foundation of Tianjin","award":["CUHK SRFS2223-4S03"],"award-info":[{"award-number":["CUHK SRFS2223-4S03"]}]},{"name":"Fundamental Research Funds for the Central Universities of China","award":["2022YFA1005000"],"award-info":[{"award-number":["2022YFA1005000"]}]},{"name":"Fundamental Research Funds for the Central Universities of China","award":["12141108"],"award-info":[{"award-number":["12141108"]}]},{"name":"Fundamental Research Funds for the Central Universities of China","award":["20JCYBJC01390"],"award-info":[{"award-number":["20JCYBJC01390"]}]},{"name":"Fundamental Research Funds for the Central Universities of China","award":["050-63233070"],"award-info":[{"award-number":["050-63233070"]}]},{"name":"Fundamental Research Funds for the Central Universities of China","award":["CUHK SRFS2223-4S03"],"award-info":[{"award-number":["CUHK SRFS2223-4S03"]}]},{"name":"Research Grants Council of the Hong Kong Special Administrative Region, China","award":["2022YFA1005000"],"award-info":[{"award-number":["2022YFA1005000"]}]},{"name":"Research Grants Council of the Hong Kong Special Administrative Region, China","award":["12141108"],"award-info":[{"award-number":["12141108"]}]},{"name":"Research Grants Council of the Hong Kong Special Administrative Region, China","award":["20JCYBJC01390"],"award-info":[{"award-number":["20JCYBJC01390"]}]},{"name":"Research Grants Council of the Hong Kong Special Administrative Region, China","award":["050-63233070"],"award-info":[{"award-number":["050-63233070"]}]},{"name":"Research Grants Council of the Hong Kong Special Administrative Region, China","award":["CUHK SRFS2223-4S03"],"award-info":[{"award-number":["CUHK SRFS2223-4S03"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we put forward the model of multiple linear-combination security multicast network coding, where the wiretapper desires to obtain some information about a predefined set of multiple linear combinations of the source symbols by eavesdropping any one (but not more than one) channel subset up to a certain size r, referred to as the security level. For this model, the security capacity is defined as the maximum average number of source symbols that can be securely multicast to all sink nodes for one use of the network under the linear-combination security constraint. For any security level and any linear-combination security constraint, we fully characterize the security capacity in terms of the ratio of the rank of the linear-combination security constraint to the number of source symbols. Also, we develop a general construction of linear security network codes. Finally, we investigate the asymptotic behavior of the security capacity for a sequence of linear-combination security models and discuss the asymptotic optimality of our code construction.<\/jats:p>","DOI":"10.3390\/e25081135","type":"journal-article","created":{"date-parts":[[2023,7,28]],"date-time":"2023-07-28T07:58:52Z","timestamp":1690531132000},"page":"1135","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Multiple Linear-Combination Security Network Coding"],"prefix":"10.3390","volume":"25","author":[{"given":"Yang","family":"Bai","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xuan","family":"Guang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Raymond W.","family":"Yeung","sequence":"additional","affiliation":[{"name":"Institute of Network Coding and the Department of Information Engineering, The Chinese University of Hong Kong, Hong Kong SAR, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1204","DOI":"10.1109\/18.850663","article-title":"Network Information Flow","volume":"46","author":"Ahlswede","year":"2000","journal-title":"IEEE Trans. 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