Quantum State Group Actions

TL;DR

This paper investigates quantum state group actions, demonstrating their security limitations, constructing new schemes, and unifying existing quantum money and key distribution protocols, thus advancing post-quantum cryptography.

Contribution

It introduces quantum state group actions, analyzes their security, constructs new cryptographic schemes, and unifies existing quantum money and key distribution methods.

Findings

Statistical security is impossible in certain settings.

Constructed quantum state group actions with cryptographic properties.

Unified quantum money schemes and classical-quantum key distribution.

Abstract

Cryptographic group actions are a leading contender for post-quantum cryptography, and have also been used in the development of quantum cryptographic protocols. In this work, we explore quantum state group actions, which consist of a group acting on a set of quantum states. We show the following results: 1. In certain settings, statistical (even query bounded) security is impossible, analogously to post-quantum classical group actions. 2. We construct quantum state group actions and prove that many computational problems that have been proposed by cryptographers hold it. Depending on the construction, our proofs are either unconditional, rely on LWE, or rely on the quantum random oracle model. While our analysis does not directly apply to classical group actions, we argue it gives at least a sanity check that there are no obvious flaws in the post-quantum assumptions made by cryptographers. 3. Our quantum state group action allows for unifying two existing quantum money schemes: those based on group actions, and those based on non-collapsing hashes. We also explain how they can unify classical and quantum key distribution.