Private Information Retrieval for Graph-based Replication with Minimal Subpacketization
Vayur Shanbhag, Prasad Krishnan
TL;DR
The paper tackles private information retrieval for graph-based replicated databases under the constraint of minimal subpacketization ($L=1$). It introduces two unit-subpacketization PIR schemes: a star-graph scheme with $R \\ge \\frac{1}{2\\sqrt{N}-2+\\frac{1}{\\sqrt{N}+1}}$ and a general-graph scheme achieving $R \\ge \\max\\left(\\frac{2}{2N-\\alpha(G)},\\frac{1}{N-1}\\right)$, with an extension to $r$-multigraphs yielding $R=\\frac{1}{N-2^{1-r}}$ for the complete multigraph. The star-graph scheme uses a random subset of spoke servers and a hub-based XOR structure to recover the desired file while preserving privacy; the general-graph scheme employs a recursive independent-set decomposition to construct queries and achieve robustness across graph classes. The work shows that unit-subpacketization can still yield competitive rates for various graphs, including complete bipartite and complete multigraphs, and provides an extension framework to $r$-parallel-edge graphs. These results offer practical PIR protocols for graph-based storage with reduced subpacketization, highlighting pathways toward capacity in special graph families and guiding future improvements in query design and privacy analysis.
Abstract
We design new minimal-subpacketization schemes for information-theoretic private information retrieval on graph-based replicated databases. In graph-based replication, the system consists of $K$ files replicated across $N$ servers according to a graph with $N$ vertices and $K$ edges. The client wants to retrieve one desired file, while keeping the index of the desired file private from each server via a query-response protocol. We seek PIR protocols that have (a) high rate, which is the ratio of the file-size to the total download cost, and (b) low subpacketization, which acts as a constraint on the size of the files for executing the protocol. We report two new schemes which have unit-subpacketization (which is minimal): (i) for a special class of graphs known as star graphs, and (ii) for general graphs. Our star-graph scheme has a better rate than previously known schemes with low subpacketization for general star graphs. Our scheme for general graphs uses a decomposition of the graph via independent sets. This scheme achieves a rate lower than prior schemes for the complete graph, however it can achieve higher rates than known for some specific graph classes. An extension of our scheme to the case of multigraphs achieves a higher rate than previous schemes for the complete multi-graph.