{
  "$type": "site.standard.document",
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  "path": "/report/2026/028",
  "publishedAt": "2026-02-22T14:26:49.000Z",
  "site": "https://eccc.weizmann.ac.il",
  "textContent": "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 $\\epsilon_c$, $\\epsilon_s$, and $\\epsilon_z$, 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 $ \\epsilon_c+\\epsilon_s+\\epsilon_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 $\\epsilon_c$ is negligible, then they also have NIZK proofs where all errors are negligible. Previously, these results were known under the more restrictive condition $ \\epsilon_c+\\sqrt{\\epsilon_s}+\\epsilon_z < 1 $ [Chakraborty et al., CRYPTO 2025]. 2. If all languages in NP have $k$-round public-coin ZK proofs or arguments satisfying $ \\epsilon_c+\\epsilon_s+(2k-1).\\epsilon_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 $ \\epsilon_c+\\epsilon_s+k.\\epsilon_z < 1 $, then infinitely-often OWFs exist.",
  "title": "TR26-028 |  Weak Zero-Knowledge and One-Way Functions | \n\n\tRohit Chatterjee, \n\n\tYunqi Li, \n\n\tPrashant Nalini Vasudevan"
}