Fundamental Limits of Multi-Message Private Computation

TL;DR

This work addresses multi-message private computation (MM-PC), extending private information retrieval (PIR) and private computation (PC) to allow retrieval of $P$ linear combinations while preserving privacy. It introduces a novel MM-PC scheme that leverages a sign-assignment mechanism and maximum distance separable (MDS) coding to exploit dependencies among linear functions, achieving significant download reductions over baseline strategies. The baseline results establish an achievable rate $R_1$ and show order-optimality within a factor of $2$, while the proposed scheme delivers a higher rate $R_2$ with explicit construction and a low dependence on the library size $M$. The authors provide a detailed example and then generalize the construction, proving decodability and privacy and outlining the rate analysis, thereby advancing private distributed computation for multiple linear queries. Overall, the approach offers practical gains in downloading efficiency for MM-PC systems with non-colluding servers and shared linear dependencies.

Abstract

In a typical formulation of the private information retrieval (PIR) problem, a single user wishes to retrieve one out of $ K$ files from $N$ servers without revealing the demanded file index to any server. This paper formulates an extended model of PIR, referred to as multi-message private computation (MM-PC), where instead of retrieving a single file, the user wishes to retrieve $P>1$ linear combinations of files while preserving the privacy of the demand information. The MM-PC problem is a generalization of the private computation (PC) problem (where the user requests one linear combination of the files), and the multi-message private information retrieval (MM-PIR) problem (where the user requests $P>1$ files). A baseline achievable scheme repeats the optimal PC scheme by Sun and Jafar $P$ times, or treats each possible demanded linear combination as an independent file and then uses the near optimal MM-PIR scheme by Banawan and Ulukus. In this paper, we propose a new MM-PC scheme that significantly improves upon the baseline schemes. In doing so, we design the queries inspired by the structure in the cache-aided scalar linear function retrieval scheme by Wan {\it et al.}, which leverages the dependency between linear functions to reduce the amount of communications. To ensure the decodability of our scheme, we propose a new method to benefit from the existing dependency, referred to as the sign assignment step. In the end, we use Maximum Distance Separable matrices to code the queries, which allows the reduction of download from the servers, while preserving privacy. By the proposed schemes, we characterize the capacity within a multiplicative factor of $2$.

Figures (2)

• Figure 1: Comparison of the rates. The red lines are for our proposed scheme and the blue ones for the baseline scheme. The parameters are $K=7, N=2$. $P$ and $M$ changes as in the figure.
• Figure 2: Comparison of the rates. The red lines are for our proposed scheme and the blue ones for the baseline scheme. The parameters are $K=7, N=2$, $P=5$ and $M$ changes as in the figure. As evident, the dependency of the proposed scheme on $M$ is almost zero.

Theorems & Definitions (19)

• Theorem 1: Baseline scheme
• Theorem 2: order-optimality of the baseline scheme
• Remark 1: asymptotic optimality of the baseline scheme
• Theorem 3: Proposed scheme
• Remark 2: Rate calculation
• Remark 3: Privacy
• Remark 4: Outline of the proposed scheme
• Remark 5
• Remark 6
• Lemma 1: Index structure