SNARGs and PPAD Hardness from the Decisional Diffie-Hellman Assumption.

We construct succinct non-interactive arguments (SNARGs) for bounded-depth computations assuming that the decisional Diffie-Hellman (DDH) problem is sub-exponentially hard. This is the first construction of such SNARGs from a Diffie-Hellman assumption. Our SNARG is also unambiguous : for every (true) statement x , it is computationally hard to find any accepting proof for x other than the proof produced by the prescribed prover strategy. We obtain our result by showing how to instantiate the Fiat-Shamir heuristic, under DDH, for a variant of the Goldwasser-Kalai-Rothblum (GKR) interactive proof system. Our new technical contributions are (1) giving a T C 0 circuit family for finding roots of cubic polynomials over a special family of characteristic-2 fields (Healy-Viola, STACS 2006) and (2) constructing a variant of the GKR protocol whose invocations of the sumcheck protocol (Lund-Fortnow-Karloff-Nisan, STOC 1990) only involve degree 3 polynomials over said fields. Along the way, since we can instantiate the Fiat-Shamir heuristic for certain variants of the sumcheck protocol, we also show the existence of (sub-exponentially) hard problems in the complexity class PPAD , assuming the sub-exponential hardness of DDH. Previous PPAD hardness results required either bilinear maps or the learning with errors assumption.