Weak Zero-Knowledge and One-Way Functions

2026-02-17

AI Summary

This work investigates the connection between weak zero-knowledge protocols—permitting non-negligible completeness, soundness, and zero-knowledge errors—and the existence of one-way functions (OWFs). Assuming the existence of worst-case hard languages in NP, it relaxes the previously required stringent error condition $\varepsilon_c + \sqrt{\varepsilon_s} + \varepsilon_z < 1$ to the more permissive bound $\varepsilon_c + \varepsilon_s + \varepsilon_z < 1$, and extends the analysis to multi-round public-coin protocols. Through cryptographic complexity theory and reduction arguments, the paper establishes that if every language in NP admits such a weak zero-knowledge protocol under this relaxed error constraint, then one-way functions—or infinitely often one-way functions—must exist. This result strengthens the theoretical link between zero-knowledge proof systems and foundational cryptographic primitives.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Zero-Knowledge
• One-Way Functions
• NP
• Worst-case hardness
• Non-interactive proofs

Innovation

• Weak Zero-Knowledge
• One-Way Functions
• NIZK
• public-coin ZK
• worst-case hardness