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 DDHDDH assumption hold for them. Our first construction is symmetric and comes with a κκ-linear map e:Gκ⟶GTe:Gκ⟶GTfor prime-order groups GG and GTGT. To establish the hardness of the κκ-linear DDHDDH problem, we rely on the existence of a base group for which the (κ−1)(κ−1)-strong DDHDDH assumption holds. Our second construction is for the asymmetric setting, where e:G1×⋯×Gκ⟶GTe:G1×⋯×Gκ⟶GT for a collection of κ+1κ+1 prime-order groups GiGi and GTGT, and relies only on the standard DDHDDH assumption in its base group. In both constructions the linearity κκ 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+NZN+. At a high level, we enable “bootstrapping” multilinear assumptions from their simpler counterparts in standard cryptographic groups, and show the equivalence of IO and multilinear maps under the existence of the aforementioned primitives.
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