Exponential Quantum One-Wayness and EFI Pairs

TL;DR

This paper explores the relationships between key quantum cryptographic primitives, establishing that inefficiently-verifiable one-way state generators are equivalent to EFI pairs with exponential loss, extending previous results to mixed states.

Contribution

It introduces the notion of inefficiently-verifiable one-way state generators and proves their equivalence to EFI pairs, including for mixed states, with an exponential loss in the reduction.

Findings

IV-OWSGs are equivalent to EFI pairs with exponential loss.

This equivalence extends to mixed states, not just pure states.

The work clarifies the relationships among fundamental quantum cryptographic primitives.

Abstract

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.