New characteristic dependent linear rank inequalities

En este trabajo estudiamos como construir desigualdades rango lineales dependientes de la característica y sus aplicaciones a la Teoría de Codificación de Redes y a la Teoría de Repartición de Secretos en protocolos criptográficos. Proponemos dos métodos que aprovechan la existencia de ciertas matri...

Full description

Autores:
Peña Macias, Victor Bryallan
Tipo de recurso:
Work document
Fecha de publicación:
2020
Institución:
Universidad Nacional de Colombia
Repositorio:
Universidad Nacional de Colombia
Idioma:
eng
OAI Identifier:
oai:repositorio.unal.edu.co:unal/77841
Acceso en línea:
https://repositorio.unal.edu.co/handle/unal/77841
Palabra clave:
510 - Matemáticas
000 - Ciencias de la computación, información y obras generales
500 - Ciencias naturales y matemáticas
Teoría de la Información
linear rank inequality
matroid
network coding
secret sharing
index coding
complementary vector space
binary matrix
desigualdad rango lineal
matroide
codificación de redes
repartición de secretos
codificación de índices
espacio vectorial complementario
matriz binaria
Rights
openAccess
License
Atribución-NoComercial 4.0 Internacional
id UNACIONAL2_40f66ff4650135c603edf9b1606d8fc3
oai_identifier_str oai:repositorio.unal.edu.co:unal/77841
network_acronym_str UNACIONAL2
network_name_str Universidad Nacional de Colombia
repository_id_str
dc.title.spa.fl_str_mv New characteristic dependent linear rank inequalities
title New characteristic dependent linear rank inequalities
spellingShingle New characteristic dependent linear rank inequalities
510 - Matemáticas
000 - Ciencias de la computación, información y obras generales
500 - Ciencias naturales y matemáticas
Teoría de la Información
linear rank inequality
matroid
network coding
secret sharing
index coding
complementary vector space
binary matrix
desigualdad rango lineal
matroide
codificación de redes
repartición de secretos
codificación de índices
espacio vectorial complementario
matriz binaria
title_short New characteristic dependent linear rank inequalities
title_full New characteristic dependent linear rank inequalities
title_fullStr New characteristic dependent linear rank inequalities
title_full_unstemmed New characteristic dependent linear rank inequalities
title_sort New characteristic dependent linear rank inequalities
dc.creator.fl_str_mv Peña Macias, Victor Bryallan
dc.contributor.advisor.spa.fl_str_mv Sarria Zapata, Humberto
dc.contributor.author.spa.fl_str_mv Peña Macias, Victor Bryallan
dc.contributor.corporatename.spa.fl_str_mv Universidad Nacional de Colombia
dc.contributor.researchgroup.spa.fl_str_mv Teoría de Matrices
dc.subject.ddc.spa.fl_str_mv 510 - Matemáticas
000 - Ciencias de la computación, información y obras generales
500 - Ciencias naturales y matemáticas
Teoría de la Información
topic 510 - Matemáticas
000 - Ciencias de la computación, información y obras generales
500 - Ciencias naturales y matemáticas
Teoría de la Información
linear rank inequality
matroid
network coding
secret sharing
index coding
complementary vector space
binary matrix
desigualdad rango lineal
matroide
codificación de redes
repartición de secretos
codificación de índices
espacio vectorial complementario
matriz binaria
dc.subject.proposal.eng.fl_str_mv linear rank inequality
matroid
network coding
secret sharing
index coding
complementary vector space
binary matrix
dc.subject.proposal.spa.fl_str_mv desigualdad rango lineal
matroide
codificación de redes
repartición de secretos
codificación de índices
espacio vectorial complementario
matriz binaria
description En este trabajo estudiamos como construir desigualdades rango lineales dependientes de la característica y sus aplicaciones a la Teoría de Codificación de Redes y a la Teoría de Repartición de Secretos en protocolos criptográficos. Proponemos dos métodos que aprovechan la existencia de ciertas matrices binarias. El primer método está basado en la construcción de ciertos espacios vectoriales complementarios y tiene aplicaciones directas a la Teoría de Codificación de Redes. Presentando así, entre las aplicaciones y usando problemas de programación lineal, que para cada conjunto finito o cofinito de números primos P, existe una sucesión de redes (N(t)), en la cual cada miembro es soluble linealmente sobre un cuerpo finito si, y sólo si, la característica del cuerpo está en P; además, la capacidad lineal sobre cuerpos cuya característica no está en P, tiende a 0, cuando t tiende a infinito. El segundo método está basado en la construcción de ciertos espacios que se comportan en cierta forma como un esquema de repartición de secretos y tiene aplicaciones directas en la Teoría de Repartición de Secretos; calculamos cotas inferiores de radios de información lineal de algunas estructuras. Adicionalmente, proponemos una extensión del problema de solubilidad de un operador de clausura. Estudiamos la capacidad de un operador de clausura y una serie de problemas de programación lineal cuyas soluciones son cotas superiores sobre esta capacidad; este problema está relacionado al cálculo de capacidades de redes de uniemisión múltiple.
publishDate 2020
dc.date.accessioned.spa.fl_str_mv 2020-07-24T05:11:36Z
dc.date.available.spa.fl_str_mv 2020-07-24T05:11:36Z
dc.date.issued.spa.fl_str_mv 2020-02-18
dc.type.spa.fl_str_mv Documento de trabajo
dc.type.driver.spa.fl_str_mv info:eu-repo/semantics/workingPaper
dc.type.version.spa.fl_str_mv info:eu-repo/semantics/acceptedVersion
dc.type.coar.spa.fl_str_mv http://purl.org/coar/resource_type/c_8042
dc.type.content.spa.fl_str_mv Text
dc.type.redcol.spa.fl_str_mv http://purl.org/redcol/resource_type/WP
format http://purl.org/coar/resource_type/c_8042
status_str acceptedVersion
dc.identifier.citation.spa.fl_str_mv V. Peña-Macias. New Characteristic Dependent Linear Rank Inequalities, Dissertation for the Doctoral Degree, Universidad Nacional de Colombia, Bogotá, 2020.
dc.identifier.uri.none.fl_str_mv https://repositorio.unal.edu.co/handle/unal/77841
identifier_str_mv V. Peña-Macias. New Characteristic Dependent Linear Rank Inequalities, Dissertation for the Doctoral Degree, Universidad Nacional de Colombia, Bogotá, 2020.
url https://repositorio.unal.edu.co/handle/unal/77841
dc.language.iso.spa.fl_str_mv eng
language eng
dc.relation.references.spa.fl_str_mv R. Ahlswede, N. Cai, S.-Y. R. Li and R. W. Yeung. Network Information Flow. IEEE Transactions on Information Theory, 46(4): 1204-1216, 2000.
N. Alon, E. Lubetzky, U. Stav, A. Weinstein, and A. Hassidim. Broadcasting with Side Information. IEEE Symposium on Foundations of Computer Science, pp. 823-832, 2008.
M. Bamiloshin, A. Ben-Efraim, O. Farràs and C. Padró. Common Information, Matroid Representation, and Secret Sharing for Matroid Ports, 2019.
Z. Bar-Yossef, Y. Birk, T. S. Jayram & T. Kol. Index Coding with Side Information. IEEE Symposium on Foundations of Computer Science, pp. 197–206, 2006. Also in IEEE Transactions on Information Theory, 57(3): 1479-1494, 2011.
A. Blasiak, R. Kleinberg and E. Lubetzky. Lexicographic Products and the Power of non-Linear Network Coding. IEEE Symposium on Foundations of Computer Science, pp. 609-618, 2011.
E. F. Brickell and D. M. Davenport. On the Classification of Ideal Secret Sharing, J. of Cryptology, 4:123-134, 1991.
S. Burris and H. P. Sankappanavar. A Course in Universal Algebra, Springer-Verlag, 1981.
J. Cannons, R. Dougherty, C. Freiling and K. Zeger. Network Routing Capacity. IEEE Transactions on Information Theory, 52(3): 7877-7888, 2006.
T. H. Chan. Balanced Information Inequalities. IEEE Transactions of Information Theory, 49(12): 3261-3267, 2003.
N. Das and B. K. Rai. On the Dependence of Linear Coding Rates on the Characteristic of the Finite Field, 2017.
R. Dougherty, C. Freiling and K. Zeger. Insufficiency of Linear Coding in Network Information Flow. IEEE Transactions on Information Theory, 51(8): 2745-2759, 2005.
R. Dougherty, C. Freiling and K. Zeger. Networks, Matroids, and non-Shannon Information Inequalities. IEEE Transactions on Information Theory, 53(6): 1949-1969, 2007.
R. Dougherty, C. Freiling and K. Zeger. Linear Rank Inequalities on Five or More Variables. ArXiv 0910.0284, 2010.
R. Dougherty, C. Freiling and K. Zeger. Achievable Rate Regions for Network Coding. IEEE Transactions on Information Theory, 61(5): 2488–2509, 2015. Also in ArXiv 1311.4601, 2013.
R. Dougherty and K. Zeger. Non-Reversibility and Equivalent Constructions of Multiple-Unicast Networks, IEEE Transactions on Information Theory, 52(11):5067-5077, 2006.
R. Dougherty. Computations of Linear Rank Inequalities on Six Variables. IEEE International Symposium on Information Theory, pp. 2819–2823, Hawaii, 2014.
O. Farràs, T. Kaced, S. Martín and C. Padró. Improving the Linear Programming Technique in the Search for Lower Bounds in Secret Sharing, 2018.
E. F. Freiling. Characteristic Dependent Linear Rank Inequalities and Applications to Network Coding. Dissertation for the Doctoral Degree. University of California, San Diego, 2014.
M. Gadouleau, Closure solvability for Network Coding and Secret Sharing, IEEE Trans. Inform. Theory, UK , 59(12): 7858-7869, 2013.
M. Gadouleau. Entropy of Closure Operators and Network Coding Solvability. Entropy, 16(9): 5122-5143, 2014.
A. Gomez, C. Mejia, and J. A. Montoya. Linear Network Coding and the Model Theory of Linear Rank Inequalities. IEEE International Symposium on Network Coding, pp. 1-6, Aalborg, Denmark, 2014.
A. W. Ingleton. Representation of Matroids. Combinatorial Mathematics and its Applications, pp. 149-167, Oxford, 1969.
A. Jafari and S. Khazaei. On Abelian Secret Sharing: duality and separation, Cryptology ePrint Archive: Report 2019/575, 2019.
R. Kinser. New Inequalities for Subspace Arrangements. Journal Combinatorial Theory Serie A, 118(1): 152-161, 2011.
B. Lindström. On The Algebraic Characteristic Set for a Class of Matroids. Transactions of the American Mathematical Society, 95(1): 147–151, 1985.
J. Martí-Farre, C. Padró. On Secret Sharing Schemes, Matroids and Polymatroids. J. Math. Cryptol. 4: 95-120, 2010.
S. Martín, C. Padró and A. Yang. Secret Sharing, Rank Inequalities, and Information Inequalities, 2015.
F. Matús. Matroid Representations by Partitions, Discrete Math., 203: 169-194, 1999.
F. Matús. Two Constructions on Limits of Entropy Functions. IEEE Transactions on Information Theory, 53(1): 320-330, 2007.
C. Mejia, Linear Secret Sharing and the Automatic Search of Linear Rank Inequalities. Applied Mathematical Science, 107(9): 5305-5324, DOI: 10.12988/ams.2015.57478, 2015.
C. Mejia. On the Theory of Linear Rank Inequalities, Dissertation for the Doctoral Degree, Universidad Nacional de Colombia, Bogotá, 2016.
C. Mejia and J. A. Montoya. On the Information Rates of Homomorphic Secret Sharing Schemes, Journal of Information and Optimization Sciences, 39(7): 1463-1482, DOI: 10.1080/02522667.2017.1367513, 2018.
C. Ngai and R. Yeung. Network Coding Gain of Combination Networks. IEEE Information Theory Workshop, pp. 283–287, 2004.
J. G. Oxley. Matroid Theory. Oxford University Press, New York, 1992.
C. Padró. Lecture Notes in Secret Sharing. Cryptology ePrint Archive: Report 674, 2012.
V. B. Peña Macias. Conexiones entre Codificación en Red, Operadores de Clausura y Matroides de Secreto Compartido, Dissertation for the Master Degree, Universidad Nacional de Colombia, Bogotá, 2015.
V. Peña-Macias and H. Sarria. Characteristic-Dependent Linear Rank Inequalities via Complementary Vector Spaces, Journal of Information and Optimization Sciences, DOI: 10.1080/02522667.2019.1668157, 2020.
V. Peña-Macias and H. Sarria. How to Find New Characteristic-Dependent Linear Rank Inequalities using Binary Matrices as a Guide, arXiv:1905.00003, 2019.
V. Peña-Macias and H. Sarria. Characteristic-Dependent Linear Rank Inequalities in 21 variables, Revista Academia Colombiana de Ciencias Exactas, Físicas y Naturales, 43(169): 765-770, 2019.
V. Peña-Macias and H. Sarria. Linear Programming Problems in Network Coding and Closure Operators via Partitions, Revista Selecciones Matemáticas, Universidad Nacional de trujillo, 6(2): 269-274, 2019.
V. Peña-Macias. How to Find New Characteristic-Dependent Linear Rank Inequalities using Secret Sharing, Pre-print, 2019.
S. Riis and M. Gadouleau. Graph-Theoretical Constructions for Graph Entropy and Network Coding based Communications. IEEE Transactions on Information Theory, 57(10): 6703–6717, 2011.
P. D. Seymour. On Secret-Sharing Matroids, Journal of Combinatorial Theory, Series B, 56: 69-73, 1992.
A. Shamir. How to Share a Secret, Communications of the ACM, 22(11): 612-613, 1979.
A. Shen, D. Hammer, A. E. Romashchenko and N.K. Vereshchagin. Inequalities for Shannon Entropy and Kolmogorov Complexity. Journal of Computer and Systems Sciences, 60: 442-464, 2000.
J. Simonis and A. Ashikhmin. Almost Affine Codes. Designs, Codes Cryptography, 14: 179-197, 1998.
D. R. Stinson. Decomposition Constructions for Secret-Sharing Schemes. IEEE Transactions on Information Theory, 40(1): 118-125, 1994.
M. Van-Dijk, W. A. Jackson, and K. M. Martin. A General Decomposition Construction for Incomplete Secret Sharing Schemes. Designs, Codes and Cryptography, 15(3): 301-321, 1998.
R. Yeung. A First Course in Information Theory, Springer, Berlin, 2002.
C. Yuan, H. Kan, W. Xin and I. Hideki. A Construction Method of Matroidal Networks, 2012.
dc.rights.spa.fl_str_mv Derechos reservados - Universidad Nacional de Colombia
dc.rights.coar.fl_str_mv http://purl.org/coar/access_right/c_abf2
dc.rights.license.spa.fl_str_mv Atribución-NoComercial 4.0 Internacional
dc.rights.spa.spa.fl_str_mv Acceso abierto
dc.rights.uri.spa.fl_str_mv http://creativecommons.org/licenses/by-nc/4.0/
dc.rights.accessrights.spa.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv Atribución-NoComercial 4.0 Internacional
Derechos reservados - Universidad Nacional de Colombia
Acceso abierto
http://creativecommons.org/licenses/by-nc/4.0/
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.extent.spa.fl_str_mv 110
dc.format.mimetype.spa.fl_str_mv application/pdf
dc.publisher.program.spa.fl_str_mv Bogotá - Ciencias - Doctorado en Ciencias - Matemáticas
dc.publisher.department.spa.fl_str_mv Departamento de Matemáticas
dc.publisher.branch.spa.fl_str_mv Universidad Nacional de Colombia - Sede Bogotá
institution Universidad Nacional de Colombia
bitstream.url.fl_str_mv https://repositorio.unal.edu.co/bitstream/unal/77841/1/Tesis%20version%20final%202020%20victor%20pe%c3%b1a.pdf
https://repositorio.unal.edu.co/bitstream/unal/77841/2/license.txt
https://repositorio.unal.edu.co/bitstream/unal/77841/3/license_rdf
https://repositorio.unal.edu.co/bitstream/unal/77841/4/Tesis%20version%20final%202020%20victor%20pe%c3%b1a.pdf.jpg
bitstream.checksum.fl_str_mv 35e34c0ebd8792110eab4434c5d07ebf
6f3f13b02594d02ad110b3ad534cd5df
42fd4ad1e89814f5e4a476b409eb708c
d9279b61e84a7a3fab8a5faa21fb94ba
bitstream.checksumAlgorithm.fl_str_mv MD5
MD5
MD5
MD5
repository.name.fl_str_mv Repositorio Institucional Universidad Nacional de Colombia
repository.mail.fl_str_mv repositorio_nal@unal.edu.co
_version_ 1814090126205124608
spelling Atribución-NoComercial 4.0 InternacionalDerechos reservados - Universidad Nacional de ColombiaAcceso abiertohttp://creativecommons.org/licenses/by-nc/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Sarria Zapata, Humberto66b0e396-8651-44fb-b651-31d280716fc0-1Peña Macias, Victor Bryallan39b3583b-8392-476b-a893-978b070f860dUniversidad Nacional de ColombiaTeoría de Matrices2020-07-24T05:11:36Z2020-07-24T05:11:36Z2020-02-18V. Peña-Macias. New Characteristic Dependent Linear Rank Inequalities, Dissertation for the Doctoral Degree, Universidad Nacional de Colombia, Bogotá, 2020.https://repositorio.unal.edu.co/handle/unal/77841En este trabajo estudiamos como construir desigualdades rango lineales dependientes de la característica y sus aplicaciones a la Teoría de Codificación de Redes y a la Teoría de Repartición de Secretos en protocolos criptográficos. Proponemos dos métodos que aprovechan la existencia de ciertas matrices binarias. El primer método está basado en la construcción de ciertos espacios vectoriales complementarios y tiene aplicaciones directas a la Teoría de Codificación de Redes. Presentando así, entre las aplicaciones y usando problemas de programación lineal, que para cada conjunto finito o cofinito de números primos P, existe una sucesión de redes (N(t)), en la cual cada miembro es soluble linealmente sobre un cuerpo finito si, y sólo si, la característica del cuerpo está en P; además, la capacidad lineal sobre cuerpos cuya característica no está en P, tiende a 0, cuando t tiende a infinito. El segundo método está basado en la construcción de ciertos espacios que se comportan en cierta forma como un esquema de repartición de secretos y tiene aplicaciones directas en la Teoría de Repartición de Secretos; calculamos cotas inferiores de radios de información lineal de algunas estructuras. Adicionalmente, proponemos una extensión del problema de solubilidad de un operador de clausura. Estudiamos la capacidad de un operador de clausura y una serie de problemas de programación lineal cuyas soluciones son cotas superiores sobre esta capacidad; este problema está relacionado al cálculo de capacidades de redes de uniemisión múltiple.In this work, we develop some methods for producing characteristic-dependent linear rank inequalities and show some applications to Network Coding and Secrets Sharing. We propose two methods that take advantage of the existence of certain binary matrices. The first method is based on the construction of certain complementary vector spaces and has direct applications to Network Coding. Using linear programming problems, for each finite or co-finite set of primes P, we show as application that there exists a sequence of networks (N(t)) in which each member is linearly solvable over a finite field if and only if the characteristic of the field is in P; and the linear capacity over fields whose characteristic is not in P, tends to 0 as t tends to infinity. The second method is based on the construction of certain spaces that behave in a certain way as a linear secret sharing scheme and has direct applications in Secret Sharing; we calculate lower bounds on the linear information ratios of some access structures. Additionally, we propose an extension of the solubility problem of a closure operator. We study the capacity of a closure operator and a class of linear programming problems whose optimal solutions are upper bounds on this capacity; this problem is related to the calculation of capacities of multiple-unicast networks.COLCIENCIASConvocatoria 727Doctor in Mathematics. Research Topic: Information Theory .Doctorado110application/pdfeng510 - Matemáticas000 - Ciencias de la computación, información y obras generales500 - Ciencias naturales y matemáticasTeoría de la Informaciónlinear rank inequalitymatroidnetwork codingsecret sharingindex codingcomplementary vector spacebinary matrixdesigualdad rango linealmatroidecodificación de redesrepartición de secretoscodificación de índicesespacio vectorial complementariomatriz binariaNew characteristic dependent linear rank inequalitiesDocumento de trabajoinfo:eu-repo/semantics/workingPaperinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_8042Texthttp://purl.org/redcol/resource_type/WPBogotá - Ciencias - Doctorado en Ciencias - MatemáticasDepartamento de MatemáticasUniversidad Nacional de Colombia - Sede BogotáR. Ahlswede, N. Cai, S.-Y. R. Li and R. W. Yeung. Network Information Flow. IEEE Transactions on Information Theory, 46(4): 1204-1216, 2000.N. Alon, E. Lubetzky, U. Stav, A. Weinstein, and A. Hassidim. Broadcasting with Side Information. IEEE Symposium on Foundations of Computer Science, pp. 823-832, 2008.M. Bamiloshin, A. Ben-Efraim, O. Farràs and C. Padró. Common Information, Matroid Representation, and Secret Sharing for Matroid Ports, 2019.Z. Bar-Yossef, Y. Birk, T. S. Jayram & T. Kol. Index Coding with Side Information. IEEE Symposium on Foundations of Computer Science, pp. 197–206, 2006. Also in IEEE Transactions on Information Theory, 57(3): 1479-1494, 2011.A. Blasiak, R. Kleinberg and E. Lubetzky. Lexicographic Products and the Power of non-Linear Network Coding. IEEE Symposium on Foundations of Computer Science, pp. 609-618, 2011.E. F. Brickell and D. M. Davenport. On the Classification of Ideal Secret Sharing, J. of Cryptology, 4:123-134, 1991.S. Burris and H. P. Sankappanavar. A Course in Universal Algebra, Springer-Verlag, 1981.J. Cannons, R. Dougherty, C. Freiling and K. Zeger. Network Routing Capacity. IEEE Transactions on Information Theory, 52(3): 7877-7888, 2006.T. H. Chan. Balanced Information Inequalities. IEEE Transactions of Information Theory, 49(12): 3261-3267, 2003.N. Das and B. K. Rai. On the Dependence of Linear Coding Rates on the Characteristic of the Finite Field, 2017.R. Dougherty, C. Freiling and K. Zeger. Insufficiency of Linear Coding in Network Information Flow. IEEE Transactions on Information Theory, 51(8): 2745-2759, 2005.R. Dougherty, C. Freiling and K. Zeger. Networks, Matroids, and non-Shannon Information Inequalities. IEEE Transactions on Information Theory, 53(6): 1949-1969, 2007.R. Dougherty, C. Freiling and K. Zeger. Linear Rank Inequalities on Five or More Variables. ArXiv 0910.0284, 2010.R. Dougherty, C. Freiling and K. Zeger. Achievable Rate Regions for Network Coding. IEEE Transactions on Information Theory, 61(5): 2488–2509, 2015. Also in ArXiv 1311.4601, 2013.R. Dougherty and K. Zeger. Non-Reversibility and Equivalent Constructions of Multiple-Unicast Networks, IEEE Transactions on Information Theory, 52(11):5067-5077, 2006.R. Dougherty. Computations of Linear Rank Inequalities on Six Variables. IEEE International Symposium on Information Theory, pp. 2819–2823, Hawaii, 2014.O. Farràs, T. Kaced, S. Martín and C. Padró. Improving the Linear Programming Technique in the Search for Lower Bounds in Secret Sharing, 2018.E. F. Freiling. Characteristic Dependent Linear Rank Inequalities and Applications to Network Coding. Dissertation for the Doctoral Degree. University of California, San Diego, 2014.M. Gadouleau, Closure solvability for Network Coding and Secret Sharing, IEEE Trans. Inform. Theory, UK , 59(12): 7858-7869, 2013.M. Gadouleau. Entropy of Closure Operators and Network Coding Solvability. Entropy, 16(9): 5122-5143, 2014.A. Gomez, C. Mejia, and J. A. Montoya. Linear Network Coding and the Model Theory of Linear Rank Inequalities. IEEE International Symposium on Network Coding, pp. 1-6, Aalborg, Denmark, 2014.A. W. Ingleton. Representation of Matroids. Combinatorial Mathematics and its Applications, pp. 149-167, Oxford, 1969.A. Jafari and S. Khazaei. On Abelian Secret Sharing: duality and separation, Cryptology ePrint Archive: Report 2019/575, 2019.R. Kinser. New Inequalities for Subspace Arrangements. Journal Combinatorial Theory Serie A, 118(1): 152-161, 2011.B. Lindström. On The Algebraic Characteristic Set for a Class of Matroids. Transactions of the American Mathematical Society, 95(1): 147–151, 1985.J. Martí-Farre, C. Padró. On Secret Sharing Schemes, Matroids and Polymatroids. J. Math. Cryptol. 4: 95-120, 2010.S. Martín, C. Padró and A. Yang. Secret Sharing, Rank Inequalities, and Information Inequalities, 2015.F. Matús. Matroid Representations by Partitions, Discrete Math., 203: 169-194, 1999.F. Matús. Two Constructions on Limits of Entropy Functions. IEEE Transactions on Information Theory, 53(1): 320-330, 2007.C. Mejia, Linear Secret Sharing and the Automatic Search of Linear Rank Inequalities. Applied Mathematical Science, 107(9): 5305-5324, DOI: 10.12988/ams.2015.57478, 2015.C. Mejia. On the Theory of Linear Rank Inequalities, Dissertation for the Doctoral Degree, Universidad Nacional de Colombia, Bogotá, 2016.C. Mejia and J. A. Montoya. On the Information Rates of Homomorphic Secret Sharing Schemes, Journal of Information and Optimization Sciences, 39(7): 1463-1482, DOI: 10.1080/02522667.2017.1367513, 2018.C. Ngai and R. Yeung. Network Coding Gain of Combination Networks. IEEE Information Theory Workshop, pp. 283–287, 2004.J. G. Oxley. Matroid Theory. Oxford University Press, New York, 1992.C. Padró. Lecture Notes in Secret Sharing. Cryptology ePrint Archive: Report 674, 2012.V. B. Peña Macias. Conexiones entre Codificación en Red, Operadores de Clausura y Matroides de Secreto Compartido, Dissertation for the Master Degree, Universidad Nacional de Colombia, Bogotá, 2015.V. Peña-Macias and H. Sarria. Characteristic-Dependent Linear Rank Inequalities via Complementary Vector Spaces, Journal of Information and Optimization Sciences, DOI: 10.1080/02522667.2019.1668157, 2020.V. Peña-Macias and H. Sarria. How to Find New Characteristic-Dependent Linear Rank Inequalities using Binary Matrices as a Guide, arXiv:1905.00003, 2019.V. Peña-Macias and H. Sarria. Characteristic-Dependent Linear Rank Inequalities in 21 variables, Revista Academia Colombiana de Ciencias Exactas, Físicas y Naturales, 43(169): 765-770, 2019.V. Peña-Macias and H. Sarria. Linear Programming Problems in Network Coding and Closure Operators via Partitions, Revista Selecciones Matemáticas, Universidad Nacional de trujillo, 6(2): 269-274, 2019.V. Peña-Macias. How to Find New Characteristic-Dependent Linear Rank Inequalities using Secret Sharing, Pre-print, 2019.S. Riis and M. Gadouleau. Graph-Theoretical Constructions for Graph Entropy and Network Coding based Communications. IEEE Transactions on Information Theory, 57(10): 6703–6717, 2011.P. D. Seymour. On Secret-Sharing Matroids, Journal of Combinatorial Theory, Series B, 56: 69-73, 1992.A. Shamir. How to Share a Secret, Communications of the ACM, 22(11): 612-613, 1979.A. Shen, D. Hammer, A. E. Romashchenko and N.K. Vereshchagin. Inequalities for Shannon Entropy and Kolmogorov Complexity. Journal of Computer and Systems Sciences, 60: 442-464, 2000.J. Simonis and A. Ashikhmin. Almost Affine Codes. Designs, Codes Cryptography, 14: 179-197, 1998.D. R. Stinson. Decomposition Constructions for Secret-Sharing Schemes. IEEE Transactions on Information Theory, 40(1): 118-125, 1994.M. Van-Dijk, W. A. Jackson, and K. M. Martin. A General Decomposition Construction for Incomplete Secret Sharing Schemes. Designs, Codes and Cryptography, 15(3): 301-321, 1998.R. Yeung. A First Course in Information Theory, Springer, Berlin, 2002.C. Yuan, H. Kan, W. Xin and I. Hideki. A Construction Method of Matroidal Networks, 2012.ORIGINALTesis version final 2020 victor peña.pdfTesis version final 2020 victor peña.pdfapplication/pdf1364757https://repositorio.unal.edu.co/bitstream/unal/77841/1/Tesis%20version%20final%202020%20victor%20pe%c3%b1a.pdf35e34c0ebd8792110eab4434c5d07ebfMD51LICENSElicense.txtlicense.txttext/plain; charset=utf-83991https://repositorio.unal.edu.co/bitstream/unal/77841/2/license.txt6f3f13b02594d02ad110b3ad534cd5dfMD52CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8701https://repositorio.unal.edu.co/bitstream/unal/77841/3/license_rdf42fd4ad1e89814f5e4a476b409eb708cMD53THUMBNAILTesis version final 2020 victor peña.pdf.jpgTesis version final 2020 victor peña.pdf.jpgGenerated Thumbnailimage/jpeg4289https://repositorio.unal.edu.co/bitstream/unal/77841/4/Tesis%20version%20final%202020%20victor%20pe%c3%b1a.pdf.jpgd9279b61e84a7a3fab8a5faa21fb94baMD54unal/77841oai:repositorio.unal.edu.co:unal/778412024-07-22 00:39:15.581Repositorio Institucional Universidad Nacional de Colombiarepositorio_nal@unal.edu.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

Publicaciones similares