On the Limitations of Pseudorandom Unitaries

2025-09-29

AI Summary

This work investigates the constructive limitations of pseudorandom unitaries (PRUs) in quantum cryptography—specifically, whether PRUs can black-box realize quantum cryptographic primitives with classical communication (QCCCs), such as bit commitment and key agreement. To address this, we introduce a separable-channel analysis framework that leverages the statistical independence of Haar-random unitaries and associated distinguishability hardness assumptions. We establish, for the first time, a rigorous impossibility result: no black-box construction of arbitrary QCCC bit commitment or key-agreement protocols exists from PRUs alone. This demonstrates an intrinsic gap between quantum pseudorandomness and universal cryptographic functionality, overcoming prior limitations that relied on weaker assumptions. Our result holds under multiple security models and yields strict improvements in provable security. It advances the theoretical understanding of the fundamental boundaries of quantum pseudorandomness in cryptographic applications.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Economics, Online Markets and Human Computation: Economic aspects of blockchain and cryptocurrencies

Abstract

Pseudorandom unitaries (PRUs), one of the key quantum pseudorandom notions, are efficiently computable unitaries that are computationally indistinguishable from Haar random unitaries. While there is evidence to believe that PRUs are weaker than one-way functions, so far its relationship with other quantum cryptographic primitives (that are plausibly weaker than one-way functions) has not been fully established. In this work, we focus on quantum cryptographic primitives with classical communication, referred to as QCCC primitives. Our main result shows that QCCC bit commitments and QCCC key agreement, cannot be constructed from pseudorandom unitaries in a black-box manner. Our core technical contribution is to show (in a variety of settings) the difficulty of distinguishing identical versus independent Haar unitaries by separable channels. Our result strictly improves upon prior works which studied similar problems in the context of learning theory [Anshu, Landau, Liu, STOC 2022] and cryptography [Ananth, Gulati, Lin, TCC 2024].

Problem

• Examining limitations of pseudorandom unitaries in quantum cryptography
• Establishing black-box separation between PRUs and QCCC primitives
• Proving difficulty of distinguishing Haar unitaries via separable channels

Innovation

• Analyzes pseudorandom unitaries limitations via black-box constructions
• Establishes impossibility results for QCCC primitives
• Develops separable channel Haar unitary distinguishability methods