Function Tables for Secure Distributed Matrix Multiplication

2026-09-08

AI Summary

该研究通过引入函数表来解决安全分布式矩阵乘法中的隐私和解码问题，提出了一种新的线性方案，并确定了在不同条件下的最优工作节点数量。

Abstract

We introduce function tables, an entrywise representation of the coefficient functions that appear in the worker responses of a secure distributed matrix multiplication (SDMM) scheme. We work under the outer-product partition, with $K$ row blocks, $L$ column blocks, and privacy against any $T$ colluding workers, in the general model of linear encoding and linear decoding. In this representation, privacy is a rank condition on the data and mask coefficients, and decodability is linear independence of the desired entries modulo the nuisance space. Degree tables, cyclic-addition tables, and algebraic-geometry constructions are the special cases obtained by restricting the coefficient functions to a structured family; we impose no such restriction, so our converses bind every linear scheme. For $T=1$, we determine the exact optimum over every finite field $\mathbb{F}_q$: it is $KL+K+L$ when $q\geq3$, and $KL+K+L+1$ over $\mathbb{F}_2$, where the identity $z^2=z$ forces one more worker. For arbitrary $T$, we prove $N\geq KL+K+L$ and $N\geq\max\{K,L\}+T$ with no MDS hypothesis on the masks; the first is stronger than the previously known bound $KL+\max\{K,L\}+2T-1$ whenever $\min\{K,L\}\geq2T$. We then reduce field feasibility exactly to MDS existence: a scheme exists over $\mathbb{F}_q$ if and only if an $[\max\{K,L\}+T,T]$ linear MDS code does, and whenever it does, a Cartesian construction attains $N=(K+T)(L+T)$ over that same field. For $T=2$ this makes $q\geq\max\{K,L\}+1$ necessary and sufficient, and we give a projective-line construction with $N=KL+K+L+2$ whenever $KL+K+L$ divides $q-1$; for $K,L\geq2$ it matches the best known worker count while requiring only an element of order $KL+K+L$.

Problem

• secure distributed matrix multiplication
• privacy
• decodability
• minimum number of workers
• rank condition

Innovation

• function tables
• secure distributed matrix multiplication (SDMM)
• outer-product partition
• linear MDS code
• projective-line construction