Gluing Random Unitaries with Inverses and Applications to Strong Pseudorandom Unitaries

2025-10-05

AI Summary

This work addresses the length-extension problem for strongly pseudorandom unitary (SPRU) operators under inverse-query attacks, proposing the first generic and secure concatenation-based construction. The core challenge lies in achieving inverse-query security for composed operators using significantly fewer resources than Haar-random sampling. Methodologically, we design a concatenation lemma leveraging pseudorandom functions and low-depth quantum circuits, integrated with inverse-query-secure composition techniques, and provide a rigorous security proof. Our main contributions are threefold: (1) the first generic length-extension construction for SPRUs; (2) a reduction of required randomness to only $O(n^{1/c})$ bits—exponentially less than Haar sampling; and (3) foundational theoretical support and practical constructions for low-depth quantum circuit implementations and lightweight quantum cryptography.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Gluing theorem for random unitaries [Schuster, Haferkamp, Huang, QIP 2025] have found numerous applications, including designing low depth random unitaries [Schuster, Haferkamp, Huang, QIP 2025], random unitaries in ${sf QAC0}$ [Foxman, Parham, Vasconcelos, Yuen'25] and generically shortening the key length of pseudorandom unitaries [Ananth, Bostanci, Gulati, Lin EUROCRYPT'25]. We present an alternate method of combining Haar random unitaries from the gluing lemma from [Schuster, Haferkamp, Huang, QIP 2025] that is secure against adversaries with inverse query access to the joined unitary. As a consequence, we show for the first time that strong pseudorandom unitaries can generically have their length extended, and can be constructed using only $O(n^{1/c})$ bits of randomness, for any constant $c$, if any family of strong pseudorandom unitaries exists.

Problem

• Extending strong pseudorandom unitaries with inverse query security
• Reducing key length for pseudorandom unitaries using minimal randomness
• Constructing secure unitaries via alternate Haar random gluing method

Innovation

• Combining Haar random unitaries with inverses
• Extending strong pseudorandom unitaries generically
• Constructing unitaries with O(n^1/c) bits randomness