Achieving Capacity of PIR with Private Side Information with Low Sub-packetization and without MDS Codes

TL;DR

This work resolves capacity-achieving PIR-PSI with low sub-packetization by introducing a scheme that attains the known converse bound for all $N\ge M+1$ under the condition $N-1\mid L$. The scheme partitions each message into exactly $N-1$ sub-packets, and each server returns a single coded sub-packet, enabling recovery of the demand from a distinct server, while preserving privacy of both the demand and the side information. Unlike prior approaches, the construction does not rely on MDS codes and operates over any finite field, including binary, with sub-packetization independent of the number of messages $K$. The main contribution is the combination of a randomized, nonuniform query design with carefully balanced probabilities to ensure privacy, together with a recoverability mechanism that hinges on subspace sums and side-information, yielding a practical capacity-achieving PIR-PSI scheme.

Abstract

This paper revisits the problem of multi-server Private Information Retrieval with Private Side Information (PIR-PSI). In this problem, $N$ non-colluding servers store identical copies of $K$ messages, each comprising $L$ symbols from $\mathbb{F}_q$, and a user, who knows $M$ of these messages, wants to retrieve one of the remaining $K-M$ messages. The user's goal is to retrieve the desired message by downloading the minimum amount of information from the servers while revealing no information about the identities of the desired message and side information messages to any server. The capacity of PIR-PSI, defined as the maximum achievable download rate, was previously characterized for all $N$, $K$, and $M$ when $L$ and $q$ are sufficiently large -- specifically, growing exponentially with $K$, to ensure the divisibility of each message into $N^K$ sub-packets and to guarantee the existence of an MDS code with its length and dimension being exponential in $K$. In this work, we propose a new capacity-achieving PIR-PSI scheme that is applicable to all $N$, $K$, $M$, $L$, and $q$ where $N\geq M+1$ and $N-1\mid L$. The proposed scheme operates with a sub-packetization level of $N-1$, independent of $K$, and works over any finite field without requiring an MDS code.

Theorems & Definitions (10)

• Theorem 1
• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6