Multilinear Maps from Obfuscation

We provide constructions of multilinear groups equipped with natural hard problems from indistinguishability obfuscation, homomorphic encryption, and NIZKs. This complements known results on the constructions of indistinguishability obfuscators from multilinear maps in the reverse direction. We provide two distinct, but closely related constructions and show that multilinear analogues of the DDH assumption hold for them. Our first construction is symmetric and comes with a \kappa-linear map e : G^\kappa \to G_T for prime-order groups G and G_T . To establish the hardness of the \kappa-linear DDH problem, we rely on the existence of a base group for which the \kappa-strong DDH assumption holds. Our second construction is for the asymmetric setting, where e : G_1 \times \dots \times G_\kappa \to G_T for a collection of \kappa + 1 prime-order groups G_i and G_T , and relies only on the 1-strong DDH assumption in its base group. In both constructions the linearity \kappa can be set to any arbitrary but a priori fixed polynomial value in the security parameter. We rely on a number of powerful tools in our constructions: probabilistic indistinguishability obfuscation, dual-mode NIZK proof systems (with perfect soundness, witness indistinguishability and zero knowledge), and additively homomorphic encryption for the group Z^+_N . At a high level, we enable “bootstrapping” multilinear assumptions from their simpler counterparts in standard cryptographic groups, and show the equivalence of PIO and multilinear maps under the existence of the aforementioned primitives