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 ℓ,k-MDDH assumption states that it is hard to decide whether a vector in Gℓ is linearly dependent of the columns of some matrix in Gℓxk sampled according to distribution Dℓ,k. It covers known assumptions such as DDH, Lin2 (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 Dℓ,k. We use the hardness results to find new distributions for which the Dℓ,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 Gℓ, for validity of ciphertexts and for equality of plaintexts. The results imply very significant efficiency improvements for a large number of schemes, most notably Naor-Yung type of constructions. © 2013 International Association for Cryptologic Research.
CITATION STYLE
Escala, A., Herold, G., Kiltz, E., Ràfols, C., & Villar, J. (2013). An algebraic framework for Diffie-Hellman assumptions. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 8043 LNCS, pp. 129–147). https://doi.org/10.1007/978-3-642-40084-1_8
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