From Worst-Case Hardness of $\mathsf{NP}$ to Quantum Cryptography via Quantum Indistinguishability Obfuscation

TL;DR

This paper explores the potential of quantum indistinguishability obfuscation (iO) to construct various cryptographic primitives under quantum hardness assumptions, extending classical results and introducing new quantum cryptographic schemes.

Contribution

It defines variants of quantum iO, identifies cryptographic primitives achievable from quantum iO and quantum NP-hardness, and provides a new construction of one-way functions from classical iO.

Findings

Constructs pseudorandom unitaries and quantum encryption schemes from quantum iO.

Shows quantum iO implies several cryptographic primitives under quantum NP-hardness.

Provides a new simpler construction of one-way functions from classical iO and NP-hardness.

Abstract

Indistinguishability obfuscation (iO) has emerged as a powerful cryptographic primitive with many implications. While classical iO, combined with the infinitely-often worst-case hardness of $\mathsf{NP}$, is known to imply one-way functions (OWFs) and a range of advanced cryptographic primitives, the cryptographic implications of quantum iO remain poorly understood. In this work, we initiate a study of the power of quantum iO. We define several natural variants of quantum iO, distinguished by whether the obfuscation algorithm, evaluation algorithm, and description of obfuscated program are classical or quantum. For each variant, we identify quantum cryptographic primitives that can be constructed under the assumption of quantum iO and the infinitely-often quantum worst-case hardness of $\mathsf{NP}$ (i.e., $\mathsf{NP}\not\subseteq\mathsf{\text{i.o.} BQP}$). In particular, we construct pseudorandom unitaries, QCCC quantum public-key encryption and (QCCC) quantum symmetric-key encryption, and several primitives implied by them such as one-way state generators, (efficiently-verifiable) one-way puzzles, and EFI pairs, etc. While our main focus is on quantum iO, even in the classical setting, our techniques yield a new and arguably simpler construction of OWFs from classical (imperfect) iO and the infinitely-often worst-case hardness of $\mathsf{NP}$.