CAT and DOG: Improved Codes for Private Distributed Matrix Multiplication

TL;DR

This paper introduces new polynomial coding schemes for private distributed matrix multiplication, utilizing cyclic-addition degree tables and roots of unity to improve efficiency and reduce worker requirements in various privacy regimes.

Contribution

It extends the degree table framework to cyclic-addition degree tables and presents explicit constructions that outperform existing schemes in efficiency and worker requirements.

Findings

CATx requires fewer workers than existing schemes in low-privacy regimes.

GASPrs and DOGrs outperform state-of-the-art schemes across various parameters.

The use of roots of unity enables modulo-addition in degree tables.

Abstract

We present novel constructions of polynomial codes for private distributed matrix multiplication (PDMM/SDMM) using outer product partitioning (OPP). We extend the degree table framework from the literature to cyclic-addition degree tables (CATs). By using roots of unity as evaluation points, we enable modulo-addition in the table. Based on CATs, we present an explicit construction, called CATx, that requires fewer workers than existing schemes in the low-privacy regime. Additionally, we present new families of schemes based on conventional degree tables, called GASPrs and DOGrs, that outperform the state-of-the-art for a wide range of parameters.