Zero-knowledge proofs allow to encode a computation so that it can be verified without revealing any additional information beyond its correctness. In this work we focus on proofs that are statistically sound meaning that even an unbounded prover cannot make the verifier accept a false statement, except with negligible probability, and computationally zero-knowledge. The seminal result of Goldreich, Micali and Wigderson (CRYPTO 1986) shows that, assuming the existence of a one-way function, such zero-knowledge proofs exist for all languages in NP.

Some of the early protocols, such as that of GMW, have a large polynomial overhead in communication compared to the original NP witness. A line of works has shown that in many cases this communication overhead can be avoided.
Most recently, Athamnah et al. (TCC 2024) constructed zero-knowledge proofs for all bounded-depth NP relations, where the communication complexity is only larger by an additive factor than the original NP witness. The main caveat of their result is that the protocol makes a non-blackbox use of the one-way function.

In this work we show that such succinct zero-knowledge proofs exist for the same class of NP relations, where the protocol makes only a blackbox use of a one-way function. Our protocol achieves a negligible soundness error, in contrast to recent works which can achieve, at best, an inverse polynomial error.