An algebraic framework for Diffie–Hellman assumptions
We put forward a new algebraic framework to generalize and analyze Diffie-Hellman like Decisional Assumptions which allows us to argue about security and applications by considering only algebraic properties. Our D`,k-MDDH assumption states that it is hard to decide whether a vector in ¿ìs linearly...
| Autores: | , , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2015 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/91050 |
| Acceso en línea: | https://hdl.handle.net/2117/91050 https://dx.doi.org/10.1007/s00145-015-9220-6 |
| Access Level: | acceso abierto |
| Palabra clave: | Number theory Diffie-Hellman Assumption Generic Hardness Groth-Sahai proofs Hash Proof Systems Public-key Encryption Nombres, Teoria dels Classificació AMS::11 Number theory::11Y Computational number theory Àrees temàtiques de la UPC::Matemàtiques i estadística::Àlgebra::Teoria de nombres |
| Sumario: | We put forward a new algebraic framework to generalize and analyze Diffie-Hellman like Decisional Assumptions which allows us to argue about security and applications by considering only algebraic properties. Our D`,k-MDDH assumption states that it is hard to decide whether a vector in ¿ìs linearly dependent of the columns of some matrix in ¿`×k sampled according to distribution D`,k. It covers known assumptions such as DDH, 2-Lin (linear assumption), and k-Lin (the k-linear assumption). Using our algebraic viewpoint, we can relate the generic hardness of our assumptions in m-linear groups to the irreducibility of certain polynomials which describe the output of D`,k. We use the hardness results to find new distributions for which the D`,k-MDDH-Assumption holds generically in m-linear groups. In particular, our new assumptions 2-SCasc and 2-ILin are generically hard in bilinear groups and, compared to 2-Lin, have shorter description size, which is a relevant parameter for efficiency in many applications. These results support using our new assumptions as natural replacements for the 2-Lin Assumption which was already used in a large number of applications. To illustrate the conceptual advantages of our algebraic framework, we construct several fundamental primitives based on any MDDH-Assumption. In particular, we can give many instantiations of a primitive in a compact way, including public-key encryption, hash-proof systems, pseudo-random functions, and Groth-Sahai NIZK and NIWI proofs. As an independent contribution we give more efficient NIZK and NIWI proofs for membership in a subgroup of ¿` . The results imply very significant efficiency improvements for a large number of schemes. |
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