THE QUANTUM INTELLIGENCER

Non-Trivial Zero-Knowledge Arguments Imply One-Way Functions Under Worst-Case Hardness In Plain English: This research tackles a fundamental question in computer security: what basic ingredients are needed to build secure communication systems? The authors show that even very weak forms of privacy-preserving proofs—where a person can convince someone else of a fact without revealing any extra information—are enough to guarantee the existence of one-way functions, which are like digital locks that are easy to close but hard to open without a key. This means that if certain types of secure interactions are possible, then some essential tools for modern encryption must also exist. It matters because it helps us understand the bare minimum requirements for building secure digital systems. Summary: This paper establishes that the existence of non-trivial zero-knowledge (ZK) arguments for NP implies the existence of one-way functions (OWFs), assuming $\mathsf{NP} \not \subseteq \mathsf{ioP/poly}$. A zero-knowledge argument is deemed non-trivial if the sum of its completeness, soundness, and zero-knowledge errors is strictly less than 1. The authors prove two main results: (1) Non-Interactive Zero-Knowledge (NIZK) arguments with such non-trivial error parameters imply OWFs; this further enables an unconditional transformation from weak to standard NIZK proofs for all meaningful error settings. (2) The result extends to the interactive setting, showing that constant-round public-coin zero-knowledge arguments with non-trivial errors also imply OWFs—and therefore imply the existence of standard four-message ZK arguments for NP. Prior work required stricter error conditions (e.g., $\epsilon_{zk} + \sqrt{\epsilon_s} < 1$) to derive OWFs; this work closes the gap for the high-error regime where $\epsilon_{zk} + \sqrt{\epsilon_s} \geq 1$, thereby providing a more complete characterization of when cryptographic hardness emerges from ZK protocols [Chakraborty, Hulett and Khurana, CRYPTO'2025]. The findings advance the broader goal of constructing OWFs from worst-case complexity assumptions, linking minimal cryptographic functionality to foundational hardness primitives. Key Points: - 1. Non-trivial zero-knowledge—defined as the sum of completeness, soundness, and zero-knowledge errors being bounded away from 1—suffices to imply one-way functions under the assumption $\mathsf{NP} \not \subseteq \mathsf{ioP/poly}$. - 2. This applies to both Non-Interactive ZK (NIZK) and constant-round interactive public-coin ZK arguments for NP. - 3. The result generalizes prior work by covering previously open error regimes where $\epsilon_{zk} + \sqrt{\epsilon_s} \geq 1$. - 4. The existence of such weak NIZKs enables unconditional amplification to standard NIZK proofs. - 5. Therefore, even minimal forms of secure interaction imply the existence of fundamental cryptographic building blocks like OWFs. Notable Quotes: - "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)." - "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." Data Points: - 1. Reference to [Hirahara and Nanashima, STOC'2024] as prior breakthrough. - 2. Reference to [Chakraborty, Hulett and Khurana, CRYPTO'2025] for prior work requiring $\epsilon_{zk} + \sqrt{\epsilon_s} < 1$. - 3. The paper addresses the previously open regime where $\epsilon_{zk} + \sqrt{\epsilon_s} \geq 1$. - 4. Result applies to constant-round and four-message zero-knowledge arguments. - 5. Assumption: $\mathsf{NP} \not \subseteq \mathsf{ioP/poly}$. Controversial Claims: - The paper assumes $\mathsf{NP} \not \subseteq \mathsf{ioP/poly}$, a complexity-theoretic assumption that, while plausible, remains unproven and is stronger than P ≠ NP. The claim that non-trivial zero-knowledge in high-error regimes implies one-way functions depends entirely on this assumption, and if false, would invalidate the main results. Additionally, the practical relevance of 'non-trivial' zero-knowledge with high errors may be questioned, as real-world cryptographic systems typically require negligible error probabilities. Technical Terms: - zero-knowledge argument, one-way function (OWF), NP, ioP/poly, non-interactive zero-knowledge (NIZK), completeness error, soundness error, zero-knowledge error, public-coin protocol, worst-case hardness, cryptographic primitive, error amplification, complexity class, interactive proof system —Ada H. Pemberley Dispatch from The Prepared E0