{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T00:04:25Z","timestamp":1769040265764,"version":"3.49.0"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T00:00:00Z","timestamp":1768694400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Cryptography"],"abstract":"<jats:p>We present an algebraic framework for constructing challenge\u2013response authentication protocols based on powers of non-diagonalizable matrices over finite fields. The construction relies on upper triangular Toeplitz matrices with a single Jordan block and on their structured power expansions, which induce nonlinear relations between matrix parameters and exponents through an autopotency phenomenon. The protocol is built from a cyclic family of matrix products derived from secret matrices (Ai)i=1n\u2282GLk(Fp): for each index i, a product Pi=AiAi+1\u2026Ai+n\u22121 is formed (indices modulo n), and its power Pi(x) is published for a secret exponent x. The resulting family of powered products is linked by conjugation via the unknown factors Ai, enabling an interactive authentication mechanism in which the prover demonstrates the knowledge of selected factors by satisfying explicit conjugacy relations. We formalize the underlying algebraic problems in terms of factor recovery and conjugacy identification from powered products, and analyze how the enforced non-diagonalizable structure and Toeplitz constraints lead to coupled multivariate polynomial systems. These systems arise naturally from the algebraic design of the construction and do not admit immediate reductions to classical discrete logarithm settings. The framework illustrates how non-diagonalizable matrix structures and structured conjugacy relations can be used to define concrete authentication primitives in noncommutative algebraic settings, and provides a basis for further cryptanalytic and cryptographic investigation.<\/jats:p>","DOI":"10.3390\/cryptography10010007","type":"journal-article","created":{"date-parts":[[2026,1,20]],"date-time":"2026-01-20T08:36:33Z","timestamp":1768898193000},"page":"7","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Autopotency and Conjugacy of Non-Diagonalizable Matrices for Challenge\u2013Response Authentication"],"prefix":"10.3390","volume":"10","author":[{"given":"Daniel","family":"Alarc\u00f3n-Narv\u00e1ez","sequence":"first","affiliation":[{"name":"Computer Science, Autonomous University of Quer\u00e9taro (UAQ), Av. de las Ciencias, Quer\u00e9taro 76230, Quer\u00e9taro, Mexico"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5109-2927","authenticated-orcid":false,"given":"Luis Adri\u00e1n","family":"Lizama-P\u00e9rez","sequence":"additional","affiliation":[{"name":"Department of Electronics, Universidad T\u00e9cnica Federico Santa Mar\u00eda, Av. Vicu\u00f1a Mackenna, San Joaqu\u00edn 8940897, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8522-0368","authenticated-orcid":false,"given":"Fausto Abraham","family":"Jacques-Garc\u00eda","sequence":"additional","affiliation":[{"name":"Computer Science, Autonomous University of Quer\u00e9taro (UAQ), Av. de las Ciencias, Quer\u00e9taro 76230, Quer\u00e9taro, Mexico"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Chen, L., Chen, L., Jordan, S., Liu, Y.K., Moody, D., Peralta, R., Perlner, R.A., and Smith-Tone, D. (2016). Report on Post-Quantum Cryptography.","DOI":"10.6028\/NIST.IR.8105"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Dam, D., Tran, T., Hoang, V., Pham, C., and Hoang, T. (2023). A survey of post-quantum cryptography: Start of a new race. Cryptography, 7.","DOI":"10.3390\/cryptography7030040"},{"key":"ref_3","unstructured":"Shor, P.W. (1994, January 20\u201322). Algorithms for quantum computation: Discrete logarithms and factoring. Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, USA."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Alagic, G., Alagic, G., Apon, D., Cooper, D., Dang, Q., Dang, T., Kelsey, J., Lichtinger, J., Liu, Y.K., and Miller, C. (2022). Status Report on the Third Round of the NIST Post-Quantum Cryptography Standardization Process, National Institute of Standards and Technology.","DOI":"10.6028\/NIST.IR.8413-upd1"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Widodo, A.M., Pappachan, P., Sekti, B.A., Anwar, N., Widayanti, R., Rahaman, M., and Bansal, R. (2024). Quantum-resistant cryptography. Innovations in Modern Cryptography, IGI Global.","DOI":"10.4018\/979-8-3693-5330-1.ch005"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Bernstein, D.J. (2025). Post-quantum cryptography. Encyclopedia of Cryptography, Security and Privacy, Springer.","DOI":"10.1007\/978-3-030-71522-9_386"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Bos, J., Ducas, L., Kiltz, E., Lepoint, T., Lyubashevsky, V., Schanck, J.M., Schwabe, P., Seiler, G., and Stehl\u00e9, D. (2018, January 24\u201326). CRYSTALS-Kyber: A CCA-secure module-lattice-based KEM. Proceedings of the 2018 IEEE European Symposium on Security and Privacy (EuroS&P), London, UK.","DOI":"10.1109\/EuroSP.2018.00032"},{"key":"ref_8","first-page":"1","article-title":"CRYSTALS-Kyber algorithm specifications and supporting documentation","volume":"2","author":"Avanzi","year":"2019","journal-title":"NIST PQC Round"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"238","DOI":"10.46586\/tches.v2018.i1.238-268","article-title":"Crystals-dilithium: A lattice-based digital signature scheme","volume":"2018","author":"Ducas","year":"2018","journal-title":"Iacr Trans. Cryptogr. Hardw. Embed. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Soni, D., Basu, K., Nabeel, M., Aaraj, N., Manzano, M., and Karri, R. (2020). Crystals-dilithium. Hardware Architectures for Post-Quantum Digital Signature Schemes, Springer.","DOI":"10.1007\/978-3-030-57682-0"},{"key":"ref_11","unstructured":"Aumasson, J.P., Bernstein, D.J., Beullens, W., Dobraunig, C., Eichlseder, M., Fluhrer, S., Gazdag, S.L., H\u00fclsing, A., Kampanakis, P., and K\u00f6lbl, S. (2025, December 21). SPHINCS+\u2013Submission to the NIST Post-Quantum Project, Available online: https:\/\/csrc.nist.gov\/CSRC\/media\/Projects\/Post-Quantum-Cryptography\/documents\/round-2\/submissions\/SPHINCS-Round2.zip."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Alagic, G., Bros, M., Ciadoux, P., Cooper, D., Dang, Q., Dang, T., Kelsey, J., Lichtinger, J., Liu, Y.K., and Miller, C. (2025). Status Report on the Fourth Round of the Nist Post-Quantum Cryptography Standardization Process.","DOI":"10.6028\/NIST.IR.8545"},{"key":"ref_13","first-page":"1","article-title":"Classic McEliece: Conservative code-based cryptography","volume":"1","author":"Bernstein","year":"2017","journal-title":"NIST Submissions"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Matsumoto, T., and Imai, H. (1988, January 25\u201327). Public quadratic polynomial-tuples for efficient signature-verification and message-encryption. Proceedings of the Advances in Cryptology\u2014EUROCRYPT\u201988: Workshop on the Theory and Application of Cryptographic Techniques, Davos, Switzerland. Proceedings 7.","DOI":"10.1007\/3-540-45961-8_39"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Ding, J., and Schmidt, D. (2005, January 7\u201310). Rainbow, a new multivariable polynomial signature scheme. Proceedings of the International Conference on Applied Cryptography and Network Security, New York, NY, USA.","DOI":"10.1007\/11496137_12"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Lizama-P\u00e9rez, L.A. (2024). A Matrix Multiplication Approach to Quantum-Safe Cryptographic Systems. Cryptography, 8.","DOI":"10.3390\/cryptography8040056"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Menezes, A. (1993). The discrete logarithm problem. Elliptic Curve Public Key Cryptosystems, Springer.","DOI":"10.1007\/978-1-4615-3198-2"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Myasnikov, A.G., Shpilrain, V., and Ushakov, A. (2011). Non-Commutative Cryptography and Complexity of Group-Theoretic Problems, American Mathematical Society. Number 177.","DOI":"10.1090\/surv\/177"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1145\/358790.358797","article-title":"Password authentication with insecure communication","volume":"24","author":"Lamport","year":"1981","journal-title":"Commun. ACM"},{"key":"ref_20","unstructured":"Haller, N., Metz, C., Nesser, P., and Straw, M. (2025, December 21). A One-Time Password System. Technical Report. Available online: https:\/\/www.rfc-editor.org\/rfc\/rfc2289.html."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Ko, K.H., Lee, S.J., Cheon, J.H., Han, J.W., Kang, J.S., and Park, C. (2000, January 20\u201324). New public-key cryptosystem using braid groups. Proceedings of the Annual International Cryptology Conference, Santa Barbara, CA, USA.","DOI":"10.1007\/3-540-44598-6_10"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Horn, R.A., and Johnson, C.R. (2012). Matrix Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9781139020411"},{"key":"ref_23","unstructured":"Strang, G. (2022). Introduction to Linear Algebra, SIAM."}],"container-title":["Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2410-387X\/10\/1\/7\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,21]],"date-time":"2026-01-21T09:18:47Z","timestamp":1768987127000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2410-387X\/10\/1\/7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,18]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["cryptography10010007"],"URL":"https:\/\/doi.org\/10.3390\/cryptography10010007","relation":{},"ISSN":["2410-387X"],"issn-type":[{"value":"2410-387X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,18]]}}}