Entangled Polynomial Codes for Secure, Private, and Batch Distributed Matrix Multiplication: Breaking the "Cubic" Barrier

TL;DR

This paper introduces entangled polynomial codes that significantly reduce the computational complexity in distributed matrix multiplication, enabling secure, private, and batch processing with fewer computations than the traditional cubic approach.

Contribution

It extends entangled polynomial codes to secure, private, and batch matrix multiplication, achieving subcubic recovery thresholds and reducing overall computational costs.

Findings

Achieves subcubic recovery thresholds for distributed matrix multiplication.

Reduces total computational costs compared to existing cubic methods.

Unifies approaches for secure, private, and batch computation settings.

Abstract

In distributed matrix multiplication, a common scenario is to assign each worker a fraction of the multiplication task, by partitioning the input matrices into smaller submatrices. In particular, by dividing two input matrices into $m$-by-$p$ and $p$-by-$n$ subblocks, a single multiplication task can be viewed as computing linear combinations of $pmn$ submatrix products, which can be assigned to $pmn$ workers. Such block-partitioning based designs have been widely studied under the topics of secure, private, and batch computation, where the state of the arts all require computing at least "cubic" ($pmn$) number of submatrix multiplications. Entangled polynomial codes, first presented for straggler mitigation, provides a powerful method for breaking the cubic barrier. It achieves a subcubic recovery threshold, meaning that the final product can be recovered from \emph{any} subset of multiplication results with a size order-wise smaller than $pmn$. In this work, we show that entangled polynomial codes can be further extended to also include these three important settings, and provide a unified framework that order-wise reduces the total computational costs upon the state of the arts by achieving subcubic recovery thresholds.