We show that every \(\textrm{NP}\) relation that can be verified by a bounded-depth polynomial-sized circuit, or a bounded-space polynomial-time algorithm, has a computational zero-knowledge proof (with statistical soundness) with communication that is only additively larger than the witness length. Our construction relies only on the minimal assumption that one-way functions exist. In more detail, assuming one-way functions, we show that every \(\textrm{NP}\) relation that can be verified in \(\textrm{NC}\) has a zero-knowledge proof with communication \(|w|+\textrm{poly}(\lambda ,\log (|x|))\) and relations that can be verified in \(\textrm{SC}\) have a zero-knowledge proof with communication \(|w|+|x|^\varepsilon \cdot \textrm{poly}(\lambda )\) . Here \(\varepsilon >0\) is an arbitrarily small constant and \(\lambda \) denotes the security parameter. As an immediate corollary, we also get that any \(\textrm{NP}\) relation, with a size S verification circuit (using unbounded fan-in XOR, AND and OR gates), has a zero-knowledge proof with communication \(S+\textrm{poly}(\lambda ,\log (S))\) . Our result improves on a recent result of Nassar and Rothblum (Crypto, 2022), which achieves length \((1+\varepsilon ) \cdot |w|+|x|^\varepsilon \cdot \textrm{poly}(\lambda )\) for bounded-space computations, and is also considerably simpler. Building on a work of Hazay et al. (TCC 2023), we also give a more complicated version of our result in which the parties only make a black-box use of the one-way function, but in this case we achieve only an inverse polynomial soundness error.