{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:31:09Z","timestamp":1760236269590,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,10]],"date-time":"2021-11-10T00:00:00Z","timestamp":1636502400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2021M691453"],"award-info":[{"award-number":["2021M691453"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62001203"],"award-info":[{"award-number":["62001203"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this article, the sum secure degrees-of-freedom (SDoF) of the multiple-input multiple-output (MIMO) X channel with confidential messages (XCCM) and arbitrary antenna configurations is studied, where there is no channel state information (CSI) at two transmitters and only delayed CSI at a multiple-antenna, full-duplex, and decode-and-forward relay. We aim at establishing the sum-SDoF lower and upper bounds. For the sum-SDoF lower bound, we design three relay-aided transmission schemes, namely, the relay-aided jamming scheme, the relay-aided jamming and one-receiver interference alignment scheme, and the relay-aided jamming and two-receiver interference alignment scheme, each corresponding to one case of antenna configurations. Moreover, the security and decoding of each scheme are analyzed. The sum-SDoF upper bound is proposed by means of the existing SDoF region of two-user MIMO broadcast channel with confidential messages (BCCM) and delayed channel state information at the transmitter (CSIT). As a result, the sum-SDoF lower and upper bounds are derived, and the sum-SDoF is characterized when the relay has sufficiently large antennas. Furthermore, even assuming no CSI at two transmitters, our results show that a multiple-antenna full-duplex relay with delayed CSI can elevate the sum-SDoF of the MIMO XCCM. This is corroborated by the fact that the derived sum-SDoF lower bound can be greater than the sum-SDoF of the MIMO XCCM with output feedback and delayed CSIT.<\/jats:p>","DOI":"10.3390\/e23111484","type":"journal-article","created":{"date-parts":[[2021,11,10]],"date-time":"2021-11-10T09:19:21Z","timestamp":1636535961000},"page":"1484","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Full-Duplex Relay with Delayed CSI Elevates the SDoF of the MIMO X Channel"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6770-0664","authenticated-orcid":false,"given":"Tong","family":"Zhang","sequence":"first","affiliation":[{"name":"The Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2978-0365","authenticated-orcid":false,"given":"Gaojie","family":"Chen","sequence":"additional","affiliation":[{"name":"5GIC & 6GIC, Institute for Communication Systems, University of Surrey, Guildford GU2 7XH, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0288-4212","authenticated-orcid":false,"given":"Shuai","family":"Wang","sequence":"additional","affiliation":[{"name":"The Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0044-8932","authenticated-orcid":false,"given":"Rui","family":"Wang","sequence":"additional","affiliation":[{"name":"The Department of Electrical and Electronic Engineering, Southern University of Science and Technology, Shenzhen 518055, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1109\/JSAIT.2021.3062755","article-title":"An Overview of Information-Theoretic Security and Privacy: Metrics, Limits and Applications","volume":"2","author":"Bloch","year":"2021","journal-title":"IEEE J. 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