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 $\mathcal{D}_{\ell,k}\mathsf{MDDH}$ assumption states that it is hard to decide whether a vector in $\mathbb{G}^\ell$ is linearly dependent of the columns of some matrix in $\mathbb{G}^{\ell\times k}$ sampled according to distribution $\mathcal{D}_{\ell,k}$. 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 $\mathcal{D}_{\ell,k}$. We use the hardness results to find new distributions for which the $\mathcal{D}_{\ell,k}\mathsf{MDDH}$-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 $\mathbb{G}^\ell$, 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.