{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T15:08:06Z","timestamp":1767625686905,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,11]],"date-time":"2024-01-11T00:00:00Z","timestamp":1704931200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["61931004","62072250","2021QY0700","BK20230415","23KJB120007","2021r039"],"award-info":[{"award-number":["61931004","62072250","2021QY0700","BK20230415","23KJB120007","2021r039"]}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["61931004","62072250","2021QY0700","BK20230415","23KJB120007","2021r039"],"award-info":[{"award-number":["61931004","62072250","2021QY0700","BK20230415","23KJB120007","2021r039"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Jiangsu Province Natural Science Foundation","doi-asserted-by":"publisher","award":["61931004","62072250","2021QY0700","BK20230415","23KJB120007","2021r039"],"award-info":[{"award-number":["61931004","62072250","2021QY0700","BK20230415","23KJB120007","2021r039"]}],"id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of the Jiangsu Higher Education Institutions of China","award":["61931004","62072250","2021QY0700","BK20230415","23KJB120007","2021r039"],"award-info":[{"award-number":["61931004","62072250","2021QY0700","BK20230415","23KJB120007","2021r039"]}]},{"name":"The Startup Foundation for Introducing Talent of NUIST","award":["61931004","62072250","2021QY0700","BK20230415","23KJB120007","2021r039"],"award-info":[{"award-number":["61931004","62072250","2021QY0700","BK20230415","23KJB120007","2021r039"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We propose a secure user pairing (UP) and power allocation (PA) strategy for a downlink Non-Orthogonal Multiple Access (NOMA) system when there exists an external eavesdropper. The secure transmission of data through the downlink is constructed to optimize both UP and PA. This optimization aims to maximize the achievable sum secrecy rate (ASSR) while adhering to a limit on the rate for each user. However, this poses a challenge as it involves a mixed integer nonlinear programming (MINLP) problem, which cannot be efficiently solved through direct search methods due to its complexity. To handle this gracefully, we first divide the original problem into two smaller issues, i.e., an optimal PA problem for two paired users and an optimal UP problem. Next, we obtain the closed-form optimal solution for PA between two users and UP in a simplified NOMA system involving four users. Finally, the result is extended to a general 2K-user NOMA system. The proposed UP and PA method satisfies the minimum rate constraints with an optimal ASSR as shown theoretically and as validated by numerical simulations. According to the results, the proposed method outperforms random UP and that in a standard OMA system in terms of the ASSR and the average ASSR. It is also interesting to find that increasing the number of user pairs will bring more performance gain in terms of the average ASSR.<\/jats:p>","DOI":"10.3390\/e26010064","type":"journal-article","created":{"date-parts":[[2024,1,11]],"date-time":"2024-01-11T08:27:07Z","timestamp":1704961627000},"page":"64","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Secure User Pairing and Power Allocation for Downlink Non-Orthogonal Multiple Access against External Eavesdropping"],"prefix":"10.3390","volume":"26","author":[{"given":"Yuxuan","family":"Li","sequence":"first","affiliation":[{"name":"School of Electronics and Information Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China"}]},{"given":"Yanqiu","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Electronics and Information Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6094-4626","authenticated-orcid":false,"given":"Xiaopeng","family":"Ji","sequence":"additional","affiliation":[{"name":"School of Electronics and Information Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2678","DOI":"10.1109\/LCOMM.2021.3077219","article-title":"On the Application of BAC-NOMA to 6G umMTC","volume":"25","author":"Ding","year":"2021","journal-title":"IEEE Commun. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1109\/MWC.2019.1800598","article-title":"Non-Orthogonal Multiple Access: Common Myths and Critical Questions","volume":"26","author":"Vaezi","year":"2019","journal-title":"IEEE Wirel. Commun."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1109\/LCOMM.2013.123113.132450","article-title":"Non-Orthogonal Multiple Access in Downlink Coordinated Two-Point Systems","volume":"18","author":"Choi","year":"2014","journal-title":"IEEE Commun. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1501","DOI":"10.1109\/LSP.2014.2343971","article-title":"On the Performance of Non-Orthogonal Multiple Access in 5G Systems with Randomly Deployed Users","volume":"21","author":"Ding","year":"2014","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/MVT.2006.307294","article-title":"Comparison of orthogonal and non-orthogonal approaches to future wireless cellular systems","volume":"1","author":"Wang","year":"2006","journal-title":"IEEE Veh. Technol. Mag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1109\/MCOM.2015.7263349","article-title":"Non-orthogonal multiple access for 5G: Solutions, challenges, opportunities, and future research trends","volume":"53","author":"Dai","year":"2015","journal-title":"IEEE Commun. Mag."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1109\/LCOMM.2017.2786252","article-title":"On the Performance of Non-Orthogonal Multiple Access in Short-Packet Communications","volume":"22","author":"Yu","year":"2018","journal-title":"IEEE Commun. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1037","DOI":"10.1109\/JSAC.2022.3145234","article-title":"Evolution of NOMA Toward Next Generation Multiple Access (NGMA) for 6G","volume":"40","author":"Liu","year":"2022","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1109\/MWC.001.1900159","article-title":"Secure Transmission via Beamforming Optimization for NOMA Networks","volume":"27","author":"Cao","year":"2020","journal-title":"IEEE Wirel. Commun."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1073\/pnas.1618130114","article-title":"Wireless physical layer security","volume":"114","author":"Poor","year":"2017","journal-title":"Proc. Nat. Acad. Sci. USA"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1708","DOI":"10.1109\/LCOMM.2018.2841378","article-title":"Secure Beamforming for MIMO-NOMA-Based Cognitive Radio Network","volume":"22","author":"Nandan","year":"2018","journal-title":"IEEE Commun. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2639","DOI":"10.1109\/TWC.2019.2906202","article-title":"Beamforming Design and Power Allocation for Secure Transmission With NOMA","volume":"18","author":"Feng","year":"2019","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"732","DOI":"10.1109\/JSYST.2022.3155786","article-title":"Joint Trajectory and Power Optimization for Jamming-Aided NOMA-UAV Secure Networks","volume":"17","author":"Li","year":"2023","journal-title":"IEEE Syst. J."},{"key":"ref_14","first-page":"531","article-title":"Secure Downlink Transmission Strategies against Active Eavesdropping in NOMA Systems: A Zero-Sum Game Approach","volume":"136","author":"Chen","year":"2023","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1557","DOI":"10.1109\/TWC.2021.3104856","article-title":"Beamforming and Jamming Optimization for IRS-Aided Secure NOMA Networks","volume":"21","author":"Wang","year":"2022","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1109\/LWC.2018.2853741","article-title":"Optimal User Pairing for Downlink Non-Orthogonal Multiple Access (NOMA)","volume":"8","author":"Zhu","year":"2019","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3944","DOI":"10.1109\/TCOMM.2017.2709301","article-title":"Optimal Resource Allocation for Power-Efficient MC-NOMA With Imperfect Channel State Information","volume":"65","author":"Wei","year":"2017","journal-title":"IEEE Trans. Commun."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"6010","DOI":"10.1109\/TVT.2015.2480766","article-title":"Impact of User Pairing on 5G Nonorthogonal Multiple-Access Downlink Transmissions","volume":"65","author":"Ding","year":"2016","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"5319","DOI":"10.1109\/TCOMM.2017.2744640","article-title":"User Pairing for Downlink Non-Orthogonal Multiple Access Networks Using Matching Algorithm","volume":"65","author":"Liang","year":"2017","journal-title":"IEEE Trans. Commun."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3234","DOI":"10.1109\/LCOMM.2021.3102116","article-title":"Graph-Theoretical Dynamic User Pairing for Downlink NOMA Systems","volume":"25","author":"Koca","year":"2021","journal-title":"IEEE Commun. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1109\/JAS.2021.1003886","article-title":"Resource Allocation Based on User Pairing and Subcarrier Matching for Downlink Non-Orthogonal Multiple Access Networks","volume":"8","author":"Liu","year":"2021","journal-title":"IEEE\/CAA J. Autom. Sin."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"930","DOI":"10.1109\/LCOMM.2016.2539162","article-title":"Secrecy Sum Rate Maximization in Non-orthogonal Multiple Access","volume":"20","author":"Zhang","year":"2016","journal-title":"IEEE Commun. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2000","DOI":"10.1109\/LWC.2020.3011191","article-title":"Covert Communication in Downlink NOMA Systems With Random Transmit Power","volume":"9","author":"Tao","year":"2020","journal-title":"IEEE Wirel. Commun. Lett."},{"unstructured":"Cormen, T.H., Leiserson, C.E., Rivest, R.L., and Stein, C. (2022). Introduction to Algorithms, The MIT Press. [4th ed.].","key":"ref_24"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/1\/64\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:44:47Z","timestamp":1760103887000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/1\/64"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,11]]},"references-count":24,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["e26010064"],"URL":"https:\/\/doi.org\/10.3390\/e26010064","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,1,11]]}}}