Quantum-secure Key Exchange Using Heisenberg Lie Algebras: Design, Analysis and Implementation

Quantum-secure Key Exchange Using Heisenberg Lie Algebras: Design, Analysis and Implementation | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Quantum-secure Key Exchange Using Heisenberg Lie Algebras: Design, Analysis and Implementation Aybeyan Selim, Muzafer Saracevic, Suad Becirovic, Dzenis Pepic This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7600264/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This work introduces a new post-quantum key exchange protocol based on the computational hardness of the Nilpotent Commutator Inversion Problem (NCIP) in Heisenberg Lie algebras over GF(p), where p ≈ 2256. Compared to traditional methods based on integer factorization or discrete logarithms, our scheme relies on the non-linear algebraic properties of nilpotent matrices for quantum security. The protocol performs extremely well in sparse environments with 64-byte public keys requiring 0.024ms for 3 × 3 matrices and 128-bit post-quantum security. Through comprehensive comparison with NIST-standardized schemes (Kyber, SPHINCS+), we show superior performance in memory footprint (0.01KB) and computational overhead, particularly for IoT applications. Experimental evaluation confirms 95.1% success rates for 8 × 8 matrices (256-bit security) with sub-millisecond latency, while security analysis establishes resistance to both classical linear algebra attacks and quantum algorithms (Shor’s, Grover’s) through the NCIP hardness assumption. The work provides concrete parameter recommendations aligned with NIST security levels and demonstrates practical implementation in Python using optimized matrix operations over GF(pn). This Lie algebra-based approach offers a mathematically distinct alternative to existing post-quantum primitives, with particular advantages for scalable, lightweight deployments in edge computing and large-scale networks. Future work directions include hardware acceleration, formal security reductions, and integration with low-power network protocols. Theoretical Computer Science Lie algebras key exchange post-quantum cryptography nilpotent matrices and commutators Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7600264","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":514191925,"identity":"8625b679-e180-4a6c-8950-065091c71bd1","order_by":0,"name":"Aybeyan Selim","email":"","orcid":"","institution":"International Vision University, North Macedonia","correspondingAuthor":false,"prefix":"","firstName":"Aybeyan","middleName":"","lastName":"Selim","suffix":""},{"id":514191926,"identity":"765263b5-5a66-420e-9a98-7186ccfdde64","order_by":1,"name":"Muzafer Saracevic","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4ElEQVRIiWNgGAWjYLCCCiDmB+IDYN4BYrScAWLJBpK1GMBVEtKiOyP94YcDFffsjW/kHjxcuINBju9GAtvHH3i0mN1ISJY4cKY4cduNvITDM88wGEveSGCezYNfywHpj20JCWY3cgwO87YxJG4AamHG5zCzG4nNPw7+S7A3ngHRUg/SwojfYclsEgcbEhg3SEC0JBgAtTDgddiZZ2wWB44lJM448wao5YyE4cwzD5uZ8Wo5nv74xoGaBHv+9hzjz7w7bOT5jicfxuswVMDYIAEmidZAouJRMApGwSgYMQAAipVUojMh7vAAAAAASUVORK5CYII=","orcid":"","institution":"University of Novi Pazar, Serbia","correspondingAuthor":true,"prefix":"","firstName":"Muzafer","middleName":"","lastName":"Saracevic","suffix":""},{"id":514191927,"identity":"5572f7c7-96d4-4604-88db-19788543c30a","order_by":2,"name":"Suad Becirovic","email":"","orcid":"","institution":"University of Novi Pazar, Serbia","correspondingAuthor":false,"prefix":"","firstName":"Suad","middleName":"","lastName":"Becirovic","suffix":""},{"id":514191928,"identity":"756c4126-0784-47d4-8f7a-932172eb69c8","order_by":3,"name":"Dzenis Pepic","email":"","orcid":"","institution":"University of Novi Pazar, Serbia","correspondingAuthor":false,"prefix":"","firstName":"Dzenis","middleName":"","lastName":"Pepic","suffix":""}],"badges":[],"createdAt":"2025-09-12 12:15:27","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7600264/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7600264/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":91295034,"identity":"4989b0e7-ac8e-4a97-8989-0f0eab0d06b3","added_by":"auto","created_at":"2025-09-15 02:50:04","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":831710,"visible":true,"origin":"","legend":"","description":"","filename":"QUANTUMSECUREKEYEXCHANGEUSINGHEISENBERGLIEALGEBRASDESIGNANALYSISANDIMPLE2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7600264/v1_covered_fe70d3c7-4391-4f93-80f0-6b8c5f5b361d.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eQuantum-secure Key Exchange Using Heisenberg Lie Algebras: Design, Analysis and Implementation\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Lie algebras, key exchange, post-quantum cryptography, nilpotent matrices and commutators","lastPublishedDoi":"10.21203/rs.3.rs-7600264/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7600264/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis work introduces a new post-quantum key exchange protocol based on the computational hardness of the Nilpotent Commutator Inversion Problem (NCIP) in Heisenberg Lie algebras over GF(p), where p ≈ 2256. Compared to traditional methods based on integer factorization or discrete logarithms, our scheme relies on the non-linear algebraic properties of nilpotent matrices for quantum security. The protocol performs extremely well in sparse environments with 64-byte public keys requiring 0.024ms for 3 × 3 matrices and 128-bit post-quantum security. Through comprehensive comparison with NIST-standardized schemes (Kyber, SPHINCS+), we show superior performance in memory footprint (0.01KB) and computational overhead, particularly for IoT applications. Experimental evaluation confirms 95.1% success rates for 8 × 8 matrices (256-bit security) with sub-millisecond latency, while security analysis establishes resistance to both classical linear algebra attacks and quantum algorithms (Shor’s, Grover’s) through the NCIP hardness assumption. 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