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

TL;DR

This paper introduces the SPSSR problem, establishing fundamental limits and constructing schemes that achieve optimal retrieval rate, shared randomness ratio, and subpacketization level for secure, private subset retrieval from multiple servers.

Contribution

It generalizes SMPIR, derives universal bounds for all demand families, and provides a single scheme that is optimal across various parameters.

Findings

Maximum retrieval rate is 1 - 1/N for all demand families.

Minimum shared randomness ratio needed is D/(N-1).

Optimal subpacketization level is (N-1)/gcd(D,N-1) for balanced linear schemes.

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.