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On the Extension of Private Distributed Matrix Multiplication Schemes to the Grid Partition

Christoph Hofmeister, Razane Tajeddine, Antonia Wachter-Zeh, Rawad Bitar

TL;DR

This work tackles private/secure distributed matrix multiplication under a collusion threat by using polynomial codes that encode block-partitioned matrices. It develops three extension operations to transform a wide class of OPP codes into GP codes, achieving reduced worker counts in many parameter regimes, while also revealing combinatorial constraints that can limit performance. In addition, it introduces a novel GP-CAT construction not tied to OPP extensions, which often outperforms prior schemes, and provides a comprehensive numerical comparison showing substantial gains across diverse parameter ranges. The results advance understanding of GP-based PDMM/SDMM schemes and highlight the trade-off between extension-based designs and GP-specific constructions for minimizing the number of workers.

Abstract

We consider polynomial codes for private distributed matrix multiplication (PDMM/SDMM). Existing codes for PDMM are either specialized for the outer product partitioning (OPP), or inner product partitioning (IPP), or are valid for the more general grid partitioning (GP). We design extension operations that can be applied to a large class of OPP code designs to extend them to the GP case. Applying them to existing codes improves upon the state-of-the-art for certain parameters. Additionally, we show that the GP schemes resulting from extension fulfill additional combinatorial constraints, potentially limiting their performance. We illustrate this point by presenting a new GP scheme that does not adhere to these constraints and outperforms the state-of-the-art for a range of parameters.

On the Extension of Private Distributed Matrix Multiplication Schemes to the Grid Partition

TL;DR

This work tackles private/secure distributed matrix multiplication under a collusion threat by using polynomial codes that encode block-partitioned matrices. It develops three extension operations to transform a wide class of OPP codes into GP codes, achieving reduced worker counts in many parameter regimes, while also revealing combinatorial constraints that can limit performance. In addition, it introduces a novel GP-CAT construction not tied to OPP extensions, which often outperforms prior schemes, and provides a comprehensive numerical comparison showing substantial gains across diverse parameter ranges. The results advance understanding of GP-based PDMM/SDMM schemes and highlight the trade-off between extension-based designs and GP-specific constructions for minimizing the number of workers.

Abstract

We consider polynomial codes for private distributed matrix multiplication (PDMM/SDMM). Existing codes for PDMM are either specialized for the outer product partitioning (OPP), or inner product partitioning (IPP), or are valid for the more general grid partitioning (GP). We design extension operations that can be applied to a large class of OPP code designs to extend them to the GP case. Applying them to existing codes improves upon the state-of-the-art for certain parameters. Additionally, we show that the GP schemes resulting from extension fulfill additional combinatorial constraints, potentially limiting their performance. We illustrate this point by presenting a new GP scheme that does not adhere to these constraints and outperforms the state-of-the-art for a range of parameters.
Paper Structure (9 sections, 6 theorems, 37 equations, 4 figures, 1 table)

This paper contains 9 sections, 6 theorems, 37 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. (Cyclic-Addition) Degree Tables for PDMM
  4. Grid Extension
  5. A New Construction of GP-CATs
  6. Numerical Comparison
  7. Conclusion
  8. Proof of \ref{['thm:ext']}
  9. Proof of \ref{['thm:newscheme']}

Key Result

Theorem 1

The extension operations according to Definitions def:dttdt, def:cattcat, and def:dttcat, result in valid DTs/CATs with $N \leq N^\prime + (M-1)(K+T)L$ workers where $N^\prime$ denotes the number of workers of the original OPP scheme. Further, $N\geq N^\prime$ for Definitions def:dttdt and def:cattc

Figures (4)

  • Figure 1: This graph shows which known scheme requires the fewest workers for a given $2\leq K=L \leq 40$, $2\leq M \leq 40$, and $T=20$. Circles mark GP schemes from the literature, namely ROU machado2022root, BGK byrne2023straggler, MP karpuk2024modular, and GGASPrkarpuk2024modular. Triangles mark a new extension of the existing OPP scheme DOGrshofmeister2025cat to the GP. Squares mark the scheme presented in \ref{['sec:newscheme']}. The x-axis represents how many blocks of $\bm{A}\xspace$ and $\bm{B}\xspace$ need to be multiplied. The y-axis indicates how wide or tall $\bm{A}\xspace$ and $\bm{B}\xspace$ are as block matrices. Points on the x-axis correspond to cases where $K=M=L$. Intuitively, the further up a point is, the closer it is to OPP rather than IPP.
  • Figure 2: The CATx scheme hofmeister2025cat for $K=6, M=1, L=3$ and $T=2$ using powers of the $29$th root of unity as evaluation points shown in (a) is extended to the GP with $K=2$, $M=3$, $L=3$ and $T=2$ shown in (b). The first column of each table corresponds to the vector $\boldsymbol{\alpha}$ and the first row to $\boldsymbol{\beta}$.
  • Figure 3: Illustration of the sets defined for an OPP-DT/CAT and a GP-DT/CAT for $K=M=L=3$ and $T=2$.
  • Figure 4: The cyclic-addition degree table of \ref{['constr:newscheme']} for $K=L=2$, $M=4$ and $T=5$. For this case, $x=5$, $z=3$, $y=15$ and $q=29$.

Theorems & Definitions (17)

  • Definition 1: GP Degree Table (GPDT) and GP Cyclic Addition DT (GPCAT)
  • Remark 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • ...and 7 more