{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,11]],"date-time":"2026-01-11T22:57:23Z","timestamp":1768172243297,"version":"3.49.0"},"reference-count":15,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,30]],"date-time":"2020-12-30T00:00:00Z","timestamp":1609286400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The revolutionary idea of asymmetric cryptography brings a fundamental change to our modern communication system. However, advances in quantum computers endanger the security of many asymmetric cryptosystems based on the hardness of factoring and discrete logarithm, while the complexity of the quantum algorithm makes it hard to implement in many applications. In this respect, novel asymmetric cryptosystems based on matrices over residue rings are in practice. In this article, a novel approach is introduced. Despite the matrix algebra M(k,\u2124n), the matrix algebra M(k,Rn\u2032),\u00a0Rn\u2032\u00a0=\u00a0\u21242[w]\u27e8wn\u22121\u27e9 as the chain ring is considered. In this technique, instead of exponentiation, the inner product automorphisms the use for key generation. The chain ring provides computational complexity to its algorithm, which improves the strength of the cryptosystem. However, the residue ring endangers the security of the original cryptosystem, while it is hard to break using Rn\u2032. The structure of the chain ring deals with the binary field \u21242, which simplifies its calculation and makes it capable of efficient execution in various applications.<\/jats:p>","DOI":"10.3390\/sym13010045","type":"journal-article","created":{"date-parts":[[2020,12,30]],"date-time":"2020-12-30T20:13:41Z","timestamp":1609359221000},"page":"45","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Asymmetric Cryptosystem on Matrix Algebra over a Chain Ring"],"prefix":"10.3390","volume":"13","author":[{"given":"Muzna","family":"Yumman","sequence":"first","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tariq","family":"Shah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2500-4214","authenticated-orcid":false,"given":"Iqtadar","family":"Hussain","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"644","DOI":"10.1109\/TIT.1976.1055638","article-title":"New directions in cryptography","volume":"22","author":"Diffie","year":"1976","journal-title":"IEEE Trans. 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Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/45\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:47:57Z","timestamp":1760179677000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/1\/45"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,30]]},"references-count":15,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2021,1]]}},"alternative-id":["sym13010045"],"URL":"https:\/\/doi.org\/10.3390\/sym13010045","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,30]]}}}