An algebraic approach to elliptic and hyperelliptic curve cryptography
During the past decades, cryptographic methods have radically improved as well as the mathematical tools employed in them. In this work we introduce elliptic curves as building blocks of cryptosystems and review their properties from a theoretical viewpoint. Moreover, we analyze the Discrete Logarit...
- Autores:
-
Garzón Mora, Sofía
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2019
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/45394
- Acceso en línea:
- http://hdl.handle.net/1992/45394
- Palabra clave:
- Curvas elípticas
Algebra abstracta
Jacobianos
Criptografía
Matemáticas
- Rights
- openAccess
- License
- http://creativecommons.org/licenses/by-nc-sa/4.0/
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Al consultar y hacer uso de este recurso, está aceptando las condiciones de uso establecidas por los autores.http://creativecommons.org/licenses/by-nc-sa/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Karpuk, David Antonc5f52f4c-7cf9-4ee9-ac8e-fd226ba6aee0500Garzón Mora, Sofíaad4b7f0f-0076-478e-9332-b39a4dd1e53f500Galindo Martínez, César Neyit2020-09-03T15:58:12Z2020-09-03T15:58:12Z2019http://hdl.handle.net/1992/45394u827439.pdfinstname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/During the past decades, cryptographic methods have radically improved as well as the mathematical tools employed in them. In this work we introduce elliptic curves as building blocks of cryptosystems and review their properties from a theoretical viewpoint. Moreover, we analyze the Discrete Logarithm Problem on groups of points on elliptic curves, and algorithms for its solution are implemented and compared. Furthermore, hyperelliptic curves are included as a generalization of elliptic curves for use in cryptography. The Jacobian is then described as an analogue of the group of points on elliptic curves for the case of higher genus curves. Two algorithms are also implemented and compared for the solution of the Discrete Logarithm Problem on the Jacobian of a general hyperelliptic curve. Finally, we find some conditions for curves to be employed in real system applications."Durante las décadas pasadas los métodos criptográficos han mejorado radicalmente, así como las herramientas matemáticas empleadas en ellos. En este trabajo se introducen las curvas elípticas como los bloques con que se construyen criptosistemas y se revisan sus propiedades desde un punto de vista teórico. Más aún, se analiza el Problema del Logaritmo Discreto en grupos de puntos de curvas elípticas y se implementan algoritmos para su solución que luego son comparados. Adicionalmente, se introducen las curvas hiperelípticas como una generalización de las curvas elípticas para uso en criptografía. El Jacobiano se describe entonces como el análogo del grupo de puntos sobre una curva elíptica para el caso de curvas de mayor género. Dos algoritmos se implementan y se comparan también en este caso para la solución del Problema del Logaritmo Discreto en el Jacobiano de una curva hiperelíptica general. Finalmente, se encuentran algunas condiciones para emplear ciertas curvas en aplicaciones a sistemas reales."--Tomado del Formato de Documento de Grado.MatemáticoPregrado72 hojasapplication/pdfengUniversidad de los AndesMatemáticasFacultad de CienciasDepartamento de Matemáticasinstname:Universidad de los Andesreponame:Repositorio Institucional SénecaAn algebraic approach to elliptic and hyperelliptic curve cryptographyTrabajo de grado - Pregradoinfo:eu-repo/semantics/bachelorThesishttp://purl.org/coar/resource_type/c_7a1fhttp://purl.org/coar/version/c_970fb48d4fbd8a85Texthttp://purl.org/redcol/resource_type/TPCurvas elípticasAlgebra abstractaJacobianosCriptografíaMatemáticasPublicationTEXTu827439.pdf.txtu827439.pdf.txtExtracted texttext/plain113648https://repositorio.uniandes.edu.co/bitstreams/d331e09c-2913-4eb0-8ffd-3a07102600a2/download42f9538f2fdc4ce0244d01fc8f5b1f56MD54ORIGINALu827439.pdfapplication/pdf465876https://repositorio.uniandes.edu.co/bitstreams/383e211c-fff5-4bac-aa7f-7c50c7e29107/downloadd04542ff2403734704776981e4733ef6MD51THUMBNAILu827439.pdf.jpgu827439.pdf.jpgIM Thumbnailimage/jpeg9318https://repositorio.uniandes.edu.co/bitstreams/5559f713-a6c6-4785-a10a-03665289c648/download71fff3b5f694934449023efc85b6fc0fMD551992/45394oai:repositorio.uniandes.edu.co:1992/453942023-10-10 17:47:12.676http://creativecommons.org/licenses/by-nc-sa/4.0/open.accesshttps://repositorio.uniandes.edu.coRepositorio institucional Sénecaadminrepositorio@uniandes.edu.co |
| dc.title.es_CO.fl_str_mv |
An algebraic approach to elliptic and hyperelliptic curve cryptography |
| title |
An algebraic approach to elliptic and hyperelliptic curve cryptography |
| spellingShingle |
An algebraic approach to elliptic and hyperelliptic curve cryptography Curvas elípticas Algebra abstracta Jacobianos Criptografía Matemáticas |
| title_short |
An algebraic approach to elliptic and hyperelliptic curve cryptography |
| title_full |
An algebraic approach to elliptic and hyperelliptic curve cryptography |
| title_fullStr |
An algebraic approach to elliptic and hyperelliptic curve cryptography |
| title_full_unstemmed |
An algebraic approach to elliptic and hyperelliptic curve cryptography |
| title_sort |
An algebraic approach to elliptic and hyperelliptic curve cryptography |
| dc.creator.fl_str_mv |
Garzón Mora, Sofía |
| dc.contributor.advisor.none.fl_str_mv |
Karpuk, David Anton |
| dc.contributor.author.none.fl_str_mv |
Garzón Mora, Sofía |
| dc.contributor.jury.none.fl_str_mv |
Galindo Martínez, César Neyit |
| dc.subject.armarc.es_CO.fl_str_mv |
Curvas elípticas Algebra abstracta Jacobianos Criptografía |
| topic |
Curvas elípticas Algebra abstracta Jacobianos Criptografía Matemáticas |
| dc.subject.themes.none.fl_str_mv |
Matemáticas |
| description |
During the past decades, cryptographic methods have radically improved as well as the mathematical tools employed in them. In this work we introduce elliptic curves as building blocks of cryptosystems and review their properties from a theoretical viewpoint. Moreover, we analyze the Discrete Logarithm Problem on groups of points on elliptic curves, and algorithms for its solution are implemented and compared. Furthermore, hyperelliptic curves are included as a generalization of elliptic curves for use in cryptography. The Jacobian is then described as an analogue of the group of points on elliptic curves for the case of higher genus curves. Two algorithms are also implemented and compared for the solution of the Discrete Logarithm Problem on the Jacobian of a general hyperelliptic curve. Finally, we find some conditions for curves to be employed in real system applications. |
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2019 |
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2019 |
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2020-09-03T15:58:12Z |
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2020-09-03T15:58:12Z |
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Trabajo de grado - Pregrado |
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http://purl.org/coar/version/c_970fb48d4fbd8a85 |
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info:eu-repo/semantics/bachelorThesis |
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http://purl.org/coar/resource_type/c_7a1f |
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Text |
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u827439.pdf |
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instname:Universidad de los Andes |
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reponame:Repositorio Institucional Séneca |
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repourl:https://repositorio.uniandes.edu.co/ |
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http://hdl.handle.net/1992/45394 |
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eng |
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72 hojas |
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Universidad de los Andes |
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Matemáticas |
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Facultad de Ciencias |
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