In classical cryptography, one-way functions are widely considered to be the minimal computational assumption. However, when taking quantum information into account, the situation is more nuanced. There are currently two major candidates for the minimal assumption: the search quantum generalization of one-way functions are one-way state generators (OWSG), whereas the decisional variant are EFI pairs. A well-known open problem in quantum cryptography is to understand how these two primitives are related. A recent breakthrough result of Khurana and Tomer (STOC'24) shows that OWSGs imply EFI pairs, for the restricted case of pure states. In this work, we make progress towards understanding the general case. To this end, we define the notion of inefficiently-verifiable one-way state generators (IV-OWSGs), where the verification algorithm is not required to be efficient, and show that these are precisely equivalent to EFI pairs, with an exponential loss in the reduction. Significantly, this equivalence holds also for mixed states. Thus our work establishes the following relations among these fundamental primitives of quantum cryptography: (mixed) OWSGs => (mixed) IV-OWSGs $\equiv_{\rm exp}$ EFI pairs, where $\equiv_{\rm exp}$ denotes equivalence up to exponential security of the primitives.

翻译：在经典密码学中，单向函数被广泛视为最小计算假设。然而，当引入量子信息后，情况更为复杂。目前最小假设有两个主要候选者：单向函数的搜索量子泛化形式是单向态生成器（OWSG），而决策变体则是EFI对。量子密码学中一个著名的开放问题是理解这两种原语之间的关系。Khurana和Tomer（STOC'24）的最新突破性成果表明，对于纯态的受限情况，OWSG蕴含EFI对。本文致力于推进对一般情况的理解。为此，我们定义了低效可验证单向态生成器（IV-OWSGs），其中验证算法不必高效，并证明它们恰好等价于EFI对，且归约过程中存在指数级损失。值得关注的是，这一等价关系对混合态同样成立。因此，我们的工作确立了量子密码学这些基本原语之间的如下关系：（混合）OWSG ⇒ （混合）IV-OWSGs $\equiv_{\rm exp}$ EFI对，其中$\equiv_{\rm exp}$表示原语在指数级安全性下的等价性。