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.

翻译：近期的一项突破性工作 [Hirahara and Nanashima, STOC'2024] 证明，若 $\mathsf{NP} \not \subseteq \mathsf{ioP/poly}$，则对于 $\mathsf{NP}$ 存在具有可忽略误差的零知识协议意味着单向函数（OWFs）的存在。在本工作中，我们基于{\em 高误差情形下}零知识的最坏情况复杂度，获得了单向函数的一个刻画。我们称一个零知识论证是{\em 非平凡的}，若其完备性误差、可靠性误差与零知识误差之和严格小于 $1$。我们的结果如下（假设 $\mathsf{NP} \not \subseteq \mathsf{ioP/poly}$）：1. 对于 $\mathsf{NP}$ 的{\em 非平凡}非交互式零知识（NIZK）论证蕴含 OWFs 的存在。利用已知的放大技术，该结果也为所有有意义的误差参数提供了从弱 NIZK 证明到标准 NIZK 证明的无条件转换。2. 我们还将结果推广到交互式情形：对于 $\mathsf{NP}$ 的{\em 非平凡}常数轮公开掷币零知识论证蕴含 OWFs 的存在，从而也蕴含（标准）四消息零知识论证的存在。在本工作之前，可以从具有常数零知识误差 $ε_{zk}$ 和可靠性误差 $ε_{s}$ 且满足 $ε_{zk} + \sqrt{ε_{s}} < 1$ 的 NIZK 协议构造单向函数 [Chakraborty, Hulett and Khurana, CRYPTO'2025]。然而，$ε_{zk} + \sqrt{ε_{s}} \geq 1$ 的情形仍然悬而未决。本工作填补了这一空白，并在交互式设定中获得了新的蕴含关系。我们的结果与技术可能成为从最坏情况硬度构造单向函数这一追求中的有用垫脚石。