{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:34:01Z","timestamp":1760243641106,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,17]],"date-time":"2022-09-17T00:00:00Z","timestamp":1663372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"MinCiencias-Colombia"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Cayley hash values are defined by paths of some oriented graphs (quivers) called Cayley graphs, whose vertices and arrows are given by elements of a group H. On the other hand, Brauer messages are obtained by concatenating words associated with multisets constituting some configurations called Brauer configurations. These configurations define some oriented graphs named Brauer quivers which induce a particular class of bound quiver algebras named Brauer configuration algebras. Elements of multisets in Brauer configurations can be seen as letters of the Brauer messages. This paper proves that each point (x,y)\u2208V=R\\{0}\u00d7R\\{0} has an associated Brauer configuration algebra \u039bB(x,y) induced by a Brauer configuration B(x,y). Additionally, the Brauer configuration algebras associated with points in a subset of the form (\u230a(x)\u230b,\u2308(x)\u2309]\u00d7(\u230a(y)\u230b,\u2308(y)\u2309]\u2282V have the same dimension. We give an analysis of Cayley hash values associated with Brauer messages M(B(x,y)) defined by a semigroup generated by some appropriated matrices A0,A1,A2\u2208GL(2,R) over a commutative ring R. As an application, we use Brauer messages M(B(x,y)) to construct explicit solutions for systems of linear and nonlinear differential equations of the form X\u2033(t)+MX(t)=0 and X\u2032(t)\u2212X2(t)N(t)=N(t) for some suitable square matrices, M and N(t). Python routines to compute Cayley hash values of Brauer messages are also included.<\/jats:p>","DOI":"10.3390\/computation10090164","type":"journal-article","created":{"date-parts":[[2022,9,18]],"date-time":"2022-09-18T22:12:43Z","timestamp":1663539163000},"page":"164","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Cayley Hash Values of Brauer Messages and Some of Their Applications in the Solutions of Systems of Differential Equations"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0189-2982","authenticated-orcid":false,"given":"Mar\u00eda Alejandra","family":"Osorio Angarita","sequence":"first","affiliation":[{"name":"Escuela de Ingenier\u00eda de Sistemas, Facultad de Ingenier\u00eda, Universidad Pedag\u00f3gica y Tecnol\u00f3gica de Colombia, Avenida Central del Norte 39-115, Tunja 150003, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6812-5131","authenticated-orcid":false,"given":"Agust\u00edn Moreno","family":"Ca\u00f1adas","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No 45-03, Bogot\u00e1 11001000, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9158-427X","authenticated-orcid":false,"given":"Cristian Camilo","family":"F\u00faneme","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No 45-03, Bogot\u00e1 11001000, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3953-2381","authenticated-orcid":false,"given":"Odette M.","family":"Mendez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Nacional de Colombia, Edificio Yu Takeuchi 404, Kra 30 No 45-03, Bogot\u00e1 11001000, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5858-5011","authenticated-orcid":false,"given":"Robinson-Julian","family":"Serna","sequence":"additional","affiliation":[{"name":"Escuela de Matem\u00e1ticas y Estad\u00edstica, Universidad Pedag\u00f3gica y Tecnol\u00f3gica de Colombia, Avenida Central del Norte 39-115, Tunja 150003, Colombia"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Stinson, D., and Paterson, M. (2019). Cryptography: Theory and Practice, Chapman and Hall\/CRC Press. [4th ed.].","DOI":"10.1201\/9781315282497"},{"key":"ref_2","unstructured":"Sosnovski, B. (2016). Cayley Graphs of Semigroups and Applications to Hashing. [Ph.D. Thesis, City University of New York]."},{"key":"ref_3","unstructured":"Espinosa, P.F.F. (2020). Categorification of Integer Sequences and Its Applications. [Ph.D. Thesis, Universidad Nacional de Colombia]."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1016\/j.bulsci.2017.06.001","article-title":"Brauer configuration algebras: A generalization of Brauer graph algebras","volume":"141","author":"Green","year":"2017","journal-title":"Bull. Sci. Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ca\u00f1adas, A.M., Espinosa, P.F.F., and Rios, G.B. (2022). Wargaming with quadratic forms and Brauer configuration algebras. Mathematics, 10.","DOI":"10.3390\/math10050729"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Ca\u00f1adas, A.M., Gaviria, I.D.M., and Vega, J.D.C. (2021). Relationships between the Chicken McNugget Problem, Mutations of Brauer Configuration Algebras and the Advanced Encryption Standard. Mathematics, 9.","DOI":"10.3390\/math9161937"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/s10468-020-09951-3","article-title":"Algebras and varieties","volume":"24","author":"Green","year":"2021","journal-title":"Algebr. Represent. Theor."},{"key":"ref_8","unstructured":"National Academies of Sciences, Engineering, and Medicine (2018). Quantum Computing: Progress and Prospects, The National Academies Press."},{"key":"ref_9","first-page":"508","article-title":"Hash functions and graphs with large girths","volume":"Volume 547","year":"1991","journal-title":"Advances in Cryptology-EUROCRYPT\u201991"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1007\/3-540-57843-9_12","article-title":"Group-theoretic hash functions","volume":"Volume 781","author":"Tillich","year":"1994","journal-title":"Algebraic Coding: First French-Israeli Workshop, Lecture Notes in Computer Science"},{"key":"ref_11","first-page":"40","article-title":"Hashing with SL2","volume":"Volume 839","author":"Tillich","year":"1994","journal-title":"Advances in Cryptology-CRYPTO\u201994"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s00145-007-9002-x","article-title":"Cryptographic hash Functions from expander graphs","volume":"22","author":"Charles","year":"2007","journal-title":"J. Cryptol."},{"key":"ref_13","unstructured":"Petit, C. (2009). On Graph-Based Cryptographic Hash Functions. [Ph.D. Thesis, Universit Catholique de Louvain]."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Petit, C., Lauter, K., and Quisquater, J.-J. (2008). Full cryptanalysis of LPS and Morgenstern hash functions. Security and Cryptography for Networks, Lecture Notes in Computer Science, Springer.","DOI":"10.1007\/978-3-540-85855-3_18"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/BF02126799","article-title":"Ramanujan graphs","volume":"3","author":"Lubotzky","year":"1988","journal-title":"Combinatorica"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1007\/s00145-010-9063-0","article-title":"Cryptanalysis of the Tillich-Z\u00e9mor hash function","volume":"24","author":"Grassl","year":"2011","journal-title":"J. Cryptol."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Petit, C., and Quisquater, J.-J. (2011). Preimages of the Tillich-Z\u00e9mor hash function. Selected Areas in Cryptography. SAC 2010. Lecture Notes in Computer Science, Springer.","DOI":"10.1007\/978-3-642-19574-7_20"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Grassl, M., Langenber, B., Roetteler, M., and Steiwandt, R. (2016). Applying Grover\u2019s algorithm to AES: Quantum resource estimates. Post-Quantum Cryptography. PQCrypto 2016. Lecture Notes in Computer Science, Springer.","DOI":"10.1007\/978-3-319-29360-8_3"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s10623-015-0129-8","article-title":"SL2 homomorphic hash functions: Worst case to average case reduction and short collision search","volume":"Volume 81","author":"Mullan","year":"2016","journal-title":"Designs, Codes and Cryptography"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Agudelo Mu\u00f1eton, N., Ca\u00f1adas, A.M., Gaviria, I.D.M., and Fern\u00e1ndez, P.F.F. (2021). {0,1}-Brauer configuration algebras and their applications in the graph energy theory. Mathematics, 9.","DOI":"10.3390\/math9233042"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"23485","DOI":"10.1007\/s11042-020-10239-3","article-title":"Brauer configuration algebras for multimedia based cryptography and security applications","volume":"80","author":"Angarita","year":"2021","journal-title":"Multimed Tools Appl."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Assem, I., Skowronski, A., and Simson, D. (2006). Elements of the Representation Theory of Associative Algebras, Cambridge University Press.","DOI":"10.1017\/CBO9780511614309"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.jalgebra.2018.06.002","article-title":"The dimension of the center of a Brauer configuration algebra","volume":"510","author":"Sierra","year":"2018","journal-title":"J. Algebra"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1142\/S0218196799000199","article-title":"On the undecidability of freeness of matrix semigroups","volume":"9","author":"Cassaigne","year":"1999","journal-title":"Intern. J. Algebra Comput."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/10\/9\/164\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:33:31Z","timestamp":1760142811000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/10\/9\/164"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,17]]},"references-count":24,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["computation10090164"],"URL":"https:\/\/doi.org\/10.3390\/computation10090164","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2022,9,17]]}}}