NIZK from SNARGs

We give a construction of a non-interactive zero-knowledge (NIZK) argument for all \({\textsf{NP}}\) languages based on a succinct non-interactive argument (SNARG) for all \({\textsf{NP}}\) languages and a one-way function. The succinctness requirement for the SNARG is rather mild: We only require that the proof size be \(|\pi |={\textsf{poly}}(\lambda )(|x|+|w|)^\delta \) for some constant \(\delta <1\), where |x| is the statement length, |w| is the witness length, and \(\lambda \) is the security parameter. Especially, we do not require the efficiency of the verification to be sublinear in |x| or |w|. As a corollary, we give a generic conversion from a SNARK to a zero-knowledge SNARG assuming the existence of one-way functions where SNARK is a SNARG with knowledge-extractability. For this conversion, we require the SNARK to be fully succinct, i.e., the proof size is \({\textsf{poly}}(\lambda )(|x|+|w|)^{o(1)}\). Before this work, such a conversion was only known if we additionally assume the existence of a NIZK. Along the way of obtaining our result, we give a generic compiler to upgrade a NIZK for all \({\textsf{NP}}\) languages with non-adaptive zero-knowledge to one with adaptive zero-knowledge. Though this can be shown by carefully combining known results, to the best of our knowledge, no explicit proof of this generic conversion has been presented.