Lower Bound on SNARGs in the Random Oracle Model

Succinct non-interactive arguments (SNARGs) have become a fundamental primitive in the cryptographic community. The focus of this work is constructions of SNARGs in the Random Oracle Model (ROM). Such SNARGs enjoy post-quantum security and can be deployed using lightweight cryptography to heuristically instantiate the random oracle. A ROM-SNARG is (t, ε) -sound if no t -query malicious prover can convince the verifier to accept a false statement with probability larger than ε. Recently, Chiesa-Yogev (CRYPTO ’21) presented a ROM-SNARG of length Θ(log (t/ ε) · log t) (ignoring log n factors, for n being the instance size). This improvement, however, is still far from the (folklore) lower bound of Ω(log (t/ ε) ). Assuming the randomized exponential-time hypothesis, we prove a tight lower bound of Ω(log (t/ ε) · log t) for the length of (t, ε) -sound ROM-SNARGs. Our lower bound holds for constructions with non-adaptive verifiers and strong soundness notion called salted soundness, restrictions that hold for all known constructions (ignoring contrived counterexamples). We prove our lower bound by transforming any short ROM-SNARG (of the considered family) into a same length ROM-SNARG in which the verifier asks only a few oracles queries, and then apply the recent lower bound of Chiesa-Yogev (TCC ’20) for such SNARGs.