Non-Trivial Zero-Knowledge Implies One-Way Functions

2026-02-19

AI Summary

This work resolves the long-standing open question of whether non-trivial zero-knowledge argument systems—those whose completeness, soundness, and zero-knowledge errors sum to a value bounded away from 1—imply the existence of one-way functions. Under the complexity-theoretic assumption that NP ⊈ ioP/poly, the paper establishes for the first time an equivalence between such high-error zero-knowledge systems and one-way functions. The main contributions include proving that both non-trivial non-interactive zero-knowledge (NIZK) arguments and constant-round public-coin zero-knowledge arguments imply the existence of one-way functions, as well as providing an unconditional transformation from weak NIZK to standard NIZK. These results fill a critical theoretical gap in the regime where ε_zk + √ε_s ≥ 1.

Technology Category

• Knowledge Representation and Reasoning: Argumentation
• Multiagent Systems: Agreement, Argumentation & Negotiation
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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

A recent breakthrough [Hirahara and Nanashima, STOC'2024] established that if $\mathsf{NP} \not \subseteq \mathsf{ioP/poly}$, the existence of zero-knowledge with negligible errors for $\mathsf{NP}$ implies the existence of one-way functions (OWFs). In this work, we obtain a characterization of one-way functions from the worst-case complexity of zero-knowledge {\em in the high-error regime}. We say that a zero-knowledge argument is {\em non-trivial} if the sum of its completeness, soundness and zero-knowledge errors is bounded away from $1$. Our results are as follows, assuming $\mathsf{NP} \not \subseteq \mathsf{ioP/poly}$: 1. {\em Non-trivial} Non-Interactive ZK (NIZK) arguments for $\mathsf{NP}$ imply the existence of OWFs. Using known amplification techniques, this result also provides an unconditional transformation from weak to standard NIZK proofs for all meaningful error parameters. 2. We also generalize to the interactive setting: {\em Non-trivial} constant-round public-coin zero-knowledge arguments for $\mathsf{NP}$ imply the existence of OWFs, and therefore also (standard) four-message zero-knowledge arguments for $\mathsf{NP}$. Prior to this work, one-way functions could be obtained from NIZKs that had constant zero-knowledge error $ε_{zk}$ and soundness error $ε_{s}$ satisfying $ε_{zk} + \sqrt{ε_{s}} < 1$ [Chakraborty, Hulett and Khurana, CRYPTO'2025]. However, the regime where $ε_{zk} + \sqrt{ε_{s}} \geq 1$ remained open. This work closes the gap, and obtains new implications in the interactive setting. Our results and techniques could be useful stepping stones in the quest to construct one-way functions from worst-case hardness.

Problem

• zero-knowledge
• one-way functions
• non-trivial
• high-error regime
• NIZK

Innovation

• zero-knowledge
• one-way functions
• non-trivial error
• NIZK
• interactive proofs