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

2025-06-24

AI Summary

This work systematically investigates the cryptographic capabilities of quantum indistinguishability obfuscation (quantum iO), addressing the lack of formal definitions for quantum iO variants arising from classical–quantum combinations in obfuscation and evaluation algorithms. Method: We first formalize multiple quantum iO variants and, under the weak assumption that $mathsf{NP} otsubseteq mathsf{i.o.BQP}$—i.e., NP problems are infinitely often hard for quantum adversaries in the worst case—construct several foundational quantum cryptographic primitives. Our construction integrates quantum iO, quantum pseudorandom unitaries, the QCCC encryption framework, and an improved classical one-way function technique. Contribution/Results: We achieve the first systematic realization of broad quantum cryptographic functionality—including quantum public-key and symmetric-key encryption, quantum one-way state generators, verifiable quantum one-way puzzles, and efficiently samplable indistinguishable (EFI) states—under a non-black-box quantum assumption. This establishes quantum iO as a unifying foundation for quantum cryptography beyond generic quantum hardness assumptions.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Economic aspects of blockchain and cryptocurrencies
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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., $mathsf{NP} otsubseteq mathsf{i.o.BQP}$). 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 $mathsf{NP}$.

Problem

• Understanding cryptographic implications of quantum indistinguishability obfuscation
• Constructing quantum cryptographic primitives from quantum iO and NP hardness
• Simplifying classical one-way functions construction from imperfect iO

Innovation

• Quantum indistinguishability obfuscation variants defined
• Pseudorandom unitaries constructed via quantum iO
• Simpler OWF construction from classical iO