{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T22:03:06Z","timestamp":1770156186714,"version":"3.49.0"},"reference-count":48,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,6]],"date-time":"2025-10-06T00:00:00Z","timestamp":1759708800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, the use of nilpotent Lie algebras as the basis for homomorphic encryption based on additive operations is explored. The g-setting is set up over gln(Zq)) and the group G=exp(g), and it is noted that the exponential and logarithm series are truncated by nilpotency in a natural way. From this, an additive symmetric conjugation scheme is constructed: given a message element M and a central randomizer U\u2208zg, we encrypt =KexpM+UK\u22121 and decrypt to M=log(K\u22121CK)\u2212U. The scheme is additive in nature, with the security defined in the IND-CPA model. Integrity is ensured using an encrypt-then-MAC construction. These properties together provide both confidentiality and robustness while preserving the homomorphic functionality. The scheme realizes additive homomorphism through a truncated BCH-sum, so it is suitable for ciphertext summations. We implemented a prototype and took reproducible measurements (Python 3.11\/NumPy) of the series {10,102,103,104,105} over 10 iterations, reporting the medians and 95% confidence intervals. The graphs exhibit that the latency per operation remains constant at fixed values, and the total time scales approximately linearly with the batch size; we also report the throughput, peak memory usage, \u2223C\u2223\/\u2223M\u2223 expansion rate, and achievable aggregation depth. The applications are federated reporting, IoT telemetry, and privacy-preserving aggregations in DBMS; the limitations include its additive nature (lacking general multiplicative homomorphism), IND-CPA (but not CCA), and side-channel resistance requirements. We place our approach in contrast to the standard FHE building blocks BFV\/BGV\/CKKS nd the emerging NIST PQC standards (FIPS 203\/204\/205), as a well-established security model with future engineering optimizations.<\/jats:p>","DOI":"10.3390\/sym17101666","type":"journal-article","created":{"date-parts":[[2025,10,6]],"date-time":"2025-10-06T08:10:51Z","timestamp":1759738251000},"page":"1666","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Homomorphic Cryptographic Scheme Based on Nilpotent Lie Algebras for Post-Quantum Security"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8285-2175","authenticated-orcid":false,"given":"Aybeyan","family":"Selim","sequence":"first","affiliation":[{"name":"Faculty of Engineering and Architecture, International Vision University, Major Cede Filipovski, 1230 Gostivar, North Macedonia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2577-7927","authenticated-orcid":false,"given":"Muzafer","family":"Sara\u010devi\u0107","sequence":"additional","affiliation":[{"name":"Department of Economics and Computer Sciences, University of Novi Pazar, Dimitrija Tucovi\u0107a 65, 36300 Novi Pazar, Serbia"}]},{"given":"Azra","family":"\u0106atovi\u0107","sequence":"additional","affiliation":[{"name":"Department of Economics and Computer Sciences, University of Novi Pazar, Dimitrija Tucovi\u0107a 65, 36300 Novi Pazar, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,6]]},"reference":[{"key":"ref_1","unstructured":"Stallings, W. (2022). Cryptography and Network Security: Principles and Practice, Pearson Education. [8th ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Schneier, B. (2015). Applied Cryptography: Protocols, Algorithms, and Source Code in C, John Wiley & Sons. [2nd ed.].","DOI":"10.1002\/9781119183471.ch10"},{"key":"ref_3","unstructured":"Goldwasser, S., and Bellare, M. (2025, June 19). Lecture notes on cryptography. In Summer Course on Cryptography; Santa Barbara, CA, USA, 2008. Available online: https:\/\/cseweb.ucsd.edu\/~mihir\/papers\/gb.pdf."},{"key":"ref_4","unstructured":"Wang, Y. (2012). Public Key Cryptography Standards: PKCS. arXiv."},{"key":"ref_5","unstructured":"Shor, P.W. (1994, January 20\u201322). Algorithms for Quantum Computation: Discrete Logarithms and Factoring. Proceedings of the 35th Annual Symposium on Foundation of Computer Science, Washington, DC, USA."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Bernstein, D.J., Buchmann, J., and Dahmen, E. (2009). Introduction to post-quantum cryptography. Post-Quantum Cryptography, Springer.","DOI":"10.1007\/978-3-540-88702-7"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kulynych, B., Lueks, W., Isaakidis, M., Danezis, G., and Troncoso, C. (2018, January 15\u201319). ClaimChain: Improving the security and privacy of in-band key distribution for messaging. Proceedings of the 2018 Workshop on Privacy in the Electronic Society, Alexandria, VA, USA.","DOI":"10.1145\/3267323.3268947"},{"key":"ref_8","first-page":"472","article-title":"Group-based cryptography","volume":"175","author":"Myasnikov","year":"2011","journal-title":"J. Math. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Lee, J.M. (2013). Introduction to Smooth Manifolds, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4419-9982-5_1"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"287","DOI":"10.4310\/MRL.1999.v6.n3.a3","article-title":"An algebraic method for public-key cryptography","volume":"6","author":"Anshel","year":"1999","journal-title":"Math. Res. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/3-540-48910-X_16","article-title":"Public-Key Cryptosystems Based on Composite Degree Residuosity Classes","volume":"Volume 1592","author":"Stern","year":"1999","journal-title":"Advances in Cryptology\u2014EU-ROCRYPT \u201999"},{"key":"ref_12","unstructured":"Humphreys, J. (2015). Introduction to Lie Algebras and Representation Theory, Springer. [3rd ed.]."},{"key":"ref_13","first-page":"482","article-title":"Applications of Lie Theory in Cryptographic Systems","volume":"18","author":"Michael","year":"2024","journal-title":"J. Gen. Lie Theory App"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1137\/S0097539701398521","article-title":"Identity-based encryption from the Weil pairing","volume":"32","author":"Boneh","year":"2003","journal-title":"SIAM J. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Wiener, M. (1999). Secure Integration of Asymmetric and Symmetric Encryption Schemes. Advances in Cryptology\u2014CRYPTO\u2019 99. CRYPTO 1999, Springer. Lecture Notes in Computer Science.","DOI":"10.1007\/3-540-48405-1"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"6","DOI":"10.1145\/1568318.1568324","article-title":"On lattices, learning with errors, random linear codes, and cryptography","volume":"56","author":"Regev","year":"2009","journal-title":"J. ACM"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1515\/jmc-2019-0032","article-title":"A framework for cryptographic problems from linear algebra","volume":"14","author":"Bootland","year":"2019","journal-title":"J. Math. Cryptol."},{"key":"ref_18","unstructured":"Serre, J.-P. (2006). Lie Algebras and Lie Groups: 1964 Lectures, Springer."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"868","DOI":"10.1007\/978-3-642-32009-5_50","article-title":"Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP","volume":"Volume 7417","author":"Canetti","year":"2012","journal-title":"Advances in Cryptology\u2014CRYPTO 2012"},{"key":"ref_20","first-page":"144","article-title":"Somewhat Practical Fully Homomorphic Encryption","volume":"2012","author":"Fan","year":"2012","journal-title":"IACR Cryptol. Eprint Arch."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2633600","article-title":"(Leveled) Fully Homomorphic Encryption without Bootstrapping","volume":"6","author":"Brakerski","year":"2014","journal-title":"ACM Trans. Comput. Theory"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1007\/978-3-319-70694-8_15","article-title":"Homomorphic Encryption for Arithmetic of Approximate Numbers","volume":"Volume 10624","author":"Takagi","year":"2017","journal-title":"Advances in Cryptology\u2014ASIACRYPT 2017"},{"key":"ref_23","unstructured":"National Institute of Standards and Technology (NIST) (2025, June 19). FIPS 203: Module-Lattice-Based Key-Encapsulation Mechanism (ML-KEM). Gaithersburg, MD, USA, 2024, Available online: https:\/\/csrc.nist.gov\/pubs\/fips\/203\/final."},{"key":"ref_24","unstructured":"National Institute of Standards and Technology (NIST) (2025, June 19). FIPS 204: Module-Lattice-Based Digital Signature Standard (ML-DSA). Gaithersburg, MD, USA, 2024, Available online: https:\/\/csrc.nist.gov\/pubs\/fips\/204\/final."},{"key":"ref_25","unstructured":"National Institute of Standards and Technology (NIST) (2025, June 19). FIPS 205: Stateless Hash-Based Digital Signature Standard (SLH-DSA). Gaithersburg, MD, USA, 2024, Available online: https:\/\/csrc.nist.gov\/pubs\/fips\/205\/final."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1007\/s00145-011-9115-0","article-title":"Practical Chosen Ciphertext Secure Encryption from Factoring","volume":"26","author":"Hofheinz","year":"2013","journal-title":"J. Cryptol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3763","DOI":"10.1016\/j.laa.2013.09.045","article-title":"The poset of the nilpotent commutator of a nilpotent matrix","volume":"439","author":"Khatami","year":"2013","journal-title":"Linear Algebra Its Appl."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kreuzer, M., and Robbiano, L. (2000). Computational Commutative Algebra, Springer.","DOI":"10.1007\/978-3-540-70628-1"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1016\/j.jalgebra.2006.02.015","article-title":"Non-commuting graph of a group","volume":"298","author":"Abdollahi","year":"2006","journal-title":"J. Algebra"},{"key":"ref_30","first-page":"71","article-title":"Bilinear cryptography using Lie algebras from p-groups","volume":"2","author":"Khamseh","year":"2021","journal-title":"Math. Comput. Sci."},{"key":"ref_31","first-page":"155","article-title":"Discrete logarithm for nilpotent groups and cryptanalysis of polylinear cryptographic system","volume":"12","year":"2019","journal-title":"Prikl. Diskretn. matematika. Prilozhenie"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Samiullah, F., Gan, M.-L., Akleylek, S., and Aun, Y. (2023, January 27\u201329). Quantum Resistance Group Key Management for IoTs. Proceedings of the 2023 2nd International Conference on Emerging Trends in Electrical, Control, and Telecommunication Engineering (ETECTE), Lahore, Pakistan.","DOI":"10.1109\/ETECTE59617.2023.10396800"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"103740","DOI":"10.1016\/j.csi.2023.103740","article-title":"Post-quantum signature algorithms on noncommutative algebras, using difficulty of solving systems of quadratic equations","volume":"86","author":"Duong","year":"2023","journal-title":"Comput. Stand. Interfaces"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"119102","DOI":"10.1007\/s11432-019-2704-7","article-title":"A CCA secure public key encryption scheme based on finite groups of Lie type","volume":"65","author":"Hong","year":"2021","journal-title":"Sci. China Inf. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Wang, N., and Liu, F. (2025). Application of the Adaptive Mixed-Order Cubature Particle Filter Algorithm Based on Matrix Lie Group Representation for the Initial Alignment of SINS. Information, 16.","DOI":"10.3390\/info16050416"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Zeng, H., Zou, S., Yao, C., and Xu, C. (2025). LGNet: A Symmetric Dual-Branch Lightweight Model for Remote Sensing Scene Classification Based on Lie Group Feature Extraction and Cross-Attention Mechanism. Symmetry, 17.","DOI":"10.3390\/sym17050780"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Sophocleous, C., and Tracin\u00e0, R. (2025). Lie Group Classification for a Reduced Burgers System. Symmetry, 17.","DOI":"10.3390\/sym17050763"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Xu, C., Shu, J., Wang, Z., and Wang, J. (2025). Lie Group Intrinsic Mean Feature Detectors for Real-Time Industrial Surface Defect Detection. Symmetry, 17.","DOI":"10.3390\/sym17040612"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Li, Y., Cherif, A.M., and Xie, Y. (2025). Characterization of Ricci Solitons and Harmonic Vector Fields on the Lie Group Nil4. Mathematics, 13.","DOI":"10.3390\/math13071155"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Fiori, S. (2025). Discrete-Time Dynamical Systems on Structured State Spaces: State-Transition Laws in Finite-Dimensional Lie Algebras. Symmetry, 17.","DOI":"10.3390\/sym17030463"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Bildirici, M., Ucan, Y., and Tekercioglu, R. (2025). Stochastic SO(2) Lie Group Method for Approximating Correlation Matrices. Mathematics, 13.","DOI":"10.3390\/math13091496"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Ayala, V., Torreblanca, M., and Valdivia, W. (2025). Conjectures on the Stability of Linear Control Systems on Matrix Lie Groups. Symmetry, 17.","DOI":"10.3390\/sym17040593"},{"key":"ref_43","unstructured":"Bauspie\u00df, P.V. (2024). Post-Quantum Secure Biometric Systems, NTNU."},{"key":"ref_44","unstructured":"Emerencia, C. (2024). A Mathematical Approach to Post-Quantum Cryptography. [Ph.D. Thesis, Vrije Universiteit Brussel]."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Hasi\u0107, A., Azizovi\u0107, M., Azizovi\u0107, E., and Sara\u010devi\u0107, M. (2025). Solvability and Nilpotency of Lie Algebras in Cryptography and Steganography. Mathematics, 13.","DOI":"10.3390\/math13111824"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"19","DOI":"10.69882\/adba.cs.2025014","article-title":"Biblio-metric Analysis of Studies on Cyber Crimes Between 2000\u20132023","volume":"2","author":"Akmese","year":"2025","journal-title":"ADBA Comput. Sci."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"188","DOI":"10.51537\/chaos.1348302","article-title":"Anomaly Detection in Cyber Security with Graph-Based LSTM in Log Analysis","volume":"5","author":"Alaca","year":"2023","journal-title":"Chaos Theory Appl."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1","DOI":"10.69882\/adba.cs.2025011","article-title":"The Role of Technological Approaches in Cyber Security of Autonomous Vehicles","volume":"2","author":"Seyyarer","year":"2025","journal-title":"ADBA Comput. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/10\/1666\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,6]],"date-time":"2025-10-06T08:22:59Z","timestamp":1759738979000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/10\/1666"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,6]]},"references-count":48,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["sym17101666"],"URL":"https:\/\/doi.org\/10.3390\/sym17101666","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,6]]}}}