An Algebraic Framework for Diffie–Hellman Assumptions

Abstract

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 Dℓ,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 Gℓ×k sampled according to distribution Dℓ,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 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. The results imply very significant efficiency improvements for a large number of schemes.