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Weak Zero-Knowledge and One-Way Functions

Rohit Chatterjee, Yunqi Li, Prashant Nalini Vasudevan

TL;DR

If all languages in NP have NIZK proofs or arguments satisfying $\epsilon_c+\epsilon_s+ \epsilon_z<1 $, then One-Way Functions (OWFs) exist.

Abstract

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.

Weak Zero-Knowledge and One-Way Functions

TL;DR

If all languages in NP have NIZK proofs or arguments satisfying , then One-Way Functions (OWFs) exist.

Abstract

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 , , and , 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 , 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 is negligible, then they also have NIZK proofs where all errors are negligible. Previously, these results were known under the more restrictive condition [Chakraborty et al., CRYPTO 2025]. 2. If all languages in NP have -round public-coin ZK proofs or arguments satisfying , then OWFs exist. 3. If, for some constant , all languages in NP have -round public-coin ZK proofs or arguments satisfying , then infinitely-often OWFs exist.
Paper Structure (39 sections, 27 theorems, 156 equations)

This paper contains 39 sections, 27 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Hardness from Zero-Knowledge
  3. Weak Zero-Knowledge
  4. Amplifying Weak NIZKs
  5. Hardness from Weak ZK
  6. Our Results
  7. Open problems
  8. Paper outline
  9. Technical Overview
  10. Non-interactive zero-knowledge
  11. The Ostrovsky-Wigderson approach
  12. Improving the condition on errors
  13. The distinguisher: an alternate view
  14. Multiple-round public-coin zero-knowledge
  15. Better bounds for constant rounds
  16. ...and 24 more sections

Key Result

Theorem 1.1

If $\NP \not \subseteq \mathsf{ioP/poly}$, and every language in $\NP$ has an $(\epsilon_c, \epsilon_s, \epsilon_z)$-NIZK proof (or argument) with $\epsilon_c+\epsilon_s+\epsilon_z < 1$, then one-way functions exist.

Theorems & Definitions (72)

  • Theorem 1.1: Informally, \ref{['the:nizk_owf']}
  • Corollary 1.2: NIZK Amplification
  • Theorem 1.3: Informally, \ref{['the:pczk_owf']}
  • Theorem 1.4: Informally, \ref{['the:cr_owf']}
  • Lemma 2.1: Hoeffding's inequality
  • Definition 2.1: Statistical Distance
  • Lemma 2.2: Triangle Inequality
  • Lemma 2.3: Data Processing Inequality
  • Definition 2.2: Statistical Indistinguishability
  • Definition 2.3: Computational Indistinguishability
  • ...and 62 more