We study the implications of the existence of weak Zero-Knowledge (ZK) protocols for worst-case hard languages. These are protocols that have completeness, soundness, and zero-knowledge errors (denoted $ε_c$, $ε_s$, and $ε_z$, respectively) that might not be negligible. Under the assumption that there are worst-case hard languages in NP, we show the following: 1. If all languages in NP have NIZK proofs or arguments satisfying $ ε_c+ε_s+ ε_z < 1 $, then One-Way Functions (OWFs) exist. This covers all possible non-trivial values for these error rates. It additionally implies that if all languages in NP have such NIZK proofs and $ε_c$ is negligible, then they also have NIZK proofs where all errors are negligible. Previously, these results were known under the more restrictive condition $ ε_c+\sqrt{ε_s}+ε_z < 1 $ [Chakraborty et al., CRYPTO 2025]. 2. If all languages in NP have $k$-round public-coin ZK proofs or arguments satisfying $ ε_c+ε_s+(2k-1).ε_z < 1 $, then OWFs exist. 3. If, for some constant $k$, all languages in NP have $k$-round public-coin ZK proofs or arguments satisfying $ ε_c+ε_s+k.ε_z < 1 $, then infinitely-often OWFs exist.

翻译：我们研究了针对最坏情况困难语言存在弱零知识（ZK）协议的含义。这些协议可能具有不可忽略的完备性误差、可靠性误差和零知识误差（分别记为 $ε_c$、$ε_s$ 和 $ε_z$）。在假设 NP 中存在最坏情况困难语言的前提下，我们证明了以下结论：1. 如果 NP 中的所有语言都具有满足 $ ε_c+ε_s+ ε_z < 1 $ 的非交互式零知识（NIZK）证明或论证，则单向函数（OWFs）存在。这涵盖了这些误差率所有可能的非平凡值。该结论还意味着，如果 NP 中的所有语言都具有此类 NIZK 证明且 $ε_c$ 可忽略，那么它们同样存在所有误差均可忽略的 NIZK 证明。此前，这些结果仅在更严格的条件 $ ε_c+\sqrt{ε_s}+ε_z < 1 $ 下已知 [Chakraborty et al., CRYPTO 2025]。2. 如果 NP 中的所有语言都具有满足 $ ε_c+ε_s+(2k-1).ε_z < 1 $ 的 $k$ 轮公开掷币零知识证明或论证，则单向函数存在。3. 如果对于某个常数 $k$，NP 中的所有语言都具有满足 $ ε_c+ε_s+k.ε_z < 1 $ 的 $k$ 轮公开掷币零知识证明或论证，则无限频繁单向函数存在。