Secure and Private Structured-Subset Retrieval: Fundamental Limits and Achievable Schemes

2026-05-10

AI Summary

This work addresses the problem of structured private information retrieval (SPIR) in a multi-server setting, where a user wishes to privately retrieve a message subset of size $D$ from an arbitrary prescribed family of subsets, without revealing the identity of the chosen subset to the servers or gaining any knowledge beyond the desired messages. By introducing a unified linear coding framework and leveraging information-theoretic analysis, number-theoretic tools—particularly the greatest common divisor—and shared randomness, the study establishes, for the first time under the non-colluding server model, fundamental limits and achievable schemes applicable to any desired subset family. The proposed scheme achieves the optimal retrieval rate of $1 - 1/N$, requires shared randomness of size $D/(N-1)$, and entails a subpacketization level of $(N-1)/\gcd(D, N-1)$, thereby overcoming the limitation of prior approaches restricted to the full subset family and matching or outperforming existing results.

Abstract

This work introduces the \emph{Secure and Private Structured-Subset Retrieval (SPSSR)} problem. In SPSSR, a user wishes to retrieve one subset from an arbitrary family of size-$D$ subsets from $K$ messages replicated across $N$ non-colluding servers that share randomness unknown to the user. The privacy requirement ensures that no server learns which subset is requested, while the security requirement ensures that the user learns nothing about the messages outside the requested subset. This generalizes Symmetric Multi-message Private Information Retrieval (SMPIR), where the candidate demand sets consist of all size-$D$ subsets. We show that, for every candidate demand family, the maximum achievable retrieval rate is equal to ${1-1/N}$. We also show that the minimum ratio between the size of the shared randomness and the message size required to achieve this rate is ${D/(N-1)}$, and that, for balanced linear SPSSR schemes, the minimum required subpacketization level is ${(N-1)/\gcd(D,N-1)}$; both quantities are independent of the demand family. Our converse proof for the maximum achievable retrieval rate applies to arbitrary demand families, unlike the existing proof for SMPIR, which is tailored to the full demand family. For achievability, we construct a single SPSSR scheme that applies uniformly to every demand family, achieves the optimal retrieval rate with the optimal shared-randomness ratio, and requires the optimal subpacketization level among balanced linear schemes. This subpacketization level is no larger than that of known SMPIR schemes in any parameter regime and is smaller in some regimes.

Problem

• Secure and Private Structured-Subset Retrieval
• Private Information Retrieval
• Information-Theoretic Security
• Data Privacy
• Distributed Storage

Innovation

• Structured-Subset Retrieval
• Private Information Retrieval
• Information-Theoretic Security
• Subpacketization
• Shared Randomness