*15:17*[Pub][ePrint] An Algebraic Framework for Diffie-Hellman Assumptions, by Alex Escala and Gottfried Herold and Eike Kiltz and Carla R\\`afols and Jorge Villar

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 $D_{\\ell,k}-MDDH$ assumption states that it is hard to decide whether

a vector in $G^\\ell$ is linearly dependent of the columns of some matrix in $G^{\\ell\\times k}$ sampled according to distribution $D_{\\ell,k}$.

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 $D_{\\ell,k}$.

We use the hardness results to find new distributions for which the $D_{\\ell,k}-MDDH$-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 $G^\\ell$, 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.