Degree Tables for Secure Distributed Matrix Multiplication

TL;DR

This paper introduces new polynomial codes for secure distributed matrix multiplication that minimize communication costs by optimizing degree tables, outperforming previous codes and establishing optimality in many cases.

Contribution

It presents novel constructions of degree tables leading to the ASP_{r} codes, improving efficiency and providing bounds and optimality proofs for SDMM.

Findings

ASP_{r} codes outperform previous codes in communication efficiency.

New degree table constructions achieve near-optimal or optimal performance.

Integer linear programming confirms the optimality of the proposed codes in tested regimes.

Abstract

We consider the problem of secure distributed matrix multiplication (SDMM) in which a user wishes to compute the product of two matrices with the assistance of honest but curious servers. We construct polynomial codes for SDMM by studying a recently introduced combinatorial tool called the degree table. For a fixed partitioning, minimizing the total communication cost of a polynomial code for SDMM is equivalent to minimizing $N$, the number of distinct elements in the corresponding degree table. We propose new constructions of degree tables with a low number of distinct elements. These new constructions lead to a general family of polynomial codes for SDMM, which we call $\mathsf{GASP}_{r}$ (Gap Additive Secure Polynomial codes) parametrized by an integer $r$. $\mathsf{GASP}_{r}$ outperforms all previously known polynomial codes for SDMM under an outer product partitioning. We also present lower bounds on $N$ and prove the optimality or asymptotic optimality of our constructions for certain regimes. Moreover, we formulate the construction of optimal degree tables as an integer linear program and use it to prove the optimality of $\mathsf{GASP}_{r}$ for all the system parameters that we were able to test.