An algebraic framework for Diffie-Hellman assumptions

Abstract

We put forward a new algebraic framework to generalize and analyze Di_e-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 G` is linearly dependent of the columns of some matrix in G`_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 _nd new distributions for which the D`;k-MDDH-Assumption holds generically in m-linear groups. In particular, our new assumption 2-SCasc is generically hard in bilinear groups and, compared to 2-Lin, has shorter description size, which is a relevant parameter for e_ciency in many applications. These results support using our new assumption as a natural replacement 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 e_cient NIZK proofs for membership in a subgroup of G`, for validity of ciphertexts and for equality of plaintexts. The results imply very signi_cant e_ciency improvements for a large number of schemes, most notably Naor-Yung type of constructions.