On the Upload versus Download Cost for Secure and Private Matrix Multiplication

TL;DR

This paper investigates the fundamental tradeoff between upload and download costs in secure, private distributed matrix multiplication, proposing a scheme that improves upon existing methods by leveraging secret sharing and private information retrieval techniques.

Contribution

It introduces a new achievable scheme characterizing the upload-download tradeoff for secure, private matrix multiplication, surpassing prior approaches.

Findings

Achieves the lower convex hull of (upload, download) pairs for the problem.

Provides a scheme that improves upon state-of-the-art methods.

Leverages secret sharing and coded private information retrieval techniques.

Abstract

In this paper, we study the problem of secure and private distributed matrix multiplication. Specifically, we focus on a scenario where a user wants to compute the product of a confidential matrix $A$, with a matrix $B_\theta$, where $\theta\in\{1,\dots,M\}$. The set of candidate matrices $\{B_1,\dots,B_M\}$ are public, and available at all the $N$ servers. The goal of the user is to distributedly compute $AB_{\theta}$, such that $(a)$ no information is leaked about the matrix $A$ to any server; and $(b)$ the index $\theta$ is kept private from each server. Our goal is to understand the fundamental tradeoff between the upload vs download cost for this problem. Our main contribution is to show that the lower convex hull of following (upload, download) pairs: $(U,D)=(N/(K-1),(K/(K-1))(1+(K/N)+\dots+(K/N)^{M-1}))$ for $K=2,\dots,N$ is achievable. The scheme improves upon state-of-the-art existing schemes for this problem, and leverages ideas from secret sharing and coded private information retrieval.