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General Framework for Linear Secure Distributed Matrix Multiplication with Byzantine Servers

Okko Makkonen, Camilla Hollanti

TL;DR

The paper introduces a general linear SDMM framework anchored in star product codes to securely multiply matrices over distributed, potentially Byzantine, workers. It provides a rigorous connection between coding-theoretic objects (star-product and matrix codes, MDS properties, and interleaved codes) and SDMM decodability and security, deriving recovery-threshold bounds and showing that many existing schemes are special cases. A key contribution is the use of interleaved codes to robustly correct Byzantine errors and the introduction of a randomized variant to mitigate adversarial error patterns, with analysis under natural error distributions. The framework unifies prior results, offers optimality statements for certain parametric regimes, and suggests practical directions for robust, scalable SDMM, including extensions to AG codes and randomized strategies.

Abstract

In this paper, a general framework for linear secure distributed matrix multiplication (SDMM) is introduced. The model allows for a neat treatment of straggling and Byzantine servers via a star product interpretation as well as simplified security proofs. Known properties of star products also immediately yield a lower bound for the recovery threshold as well as an upper bound for the number of colluding workers the system can tolerate. Another bound on the recovery threshold is given by the decodability condition, which generalizes a bound for GASP codes. The framework produces many of the known SDMM schemes as special cases, thereby providing unification for the previous literature on the topic. Furthermore, error behavior specific to SDMM is discussed and interleaved codes are proposed as a suitable means for efficient error correction in the proposed model. Analysis of the error correction capability under natural assumptions about the error distribution is also provided, largely based on well-known results on interleaved codes. Error detection and other error distributions are also discussed.

General Framework for Linear Secure Distributed Matrix Multiplication with Byzantine Servers

TL;DR

The paper introduces a general linear SDMM framework anchored in star product codes to securely multiply matrices over distributed, potentially Byzantine, workers. It provides a rigorous connection between coding-theoretic objects (star-product and matrix codes, MDS properties, and interleaved codes) and SDMM decodability and security, deriving recovery-threshold bounds and showing that many existing schemes are special cases. A key contribution is the use of interleaved codes to robustly correct Byzantine errors and the introduction of a randomized variant to mitigate adversarial error patterns, with analysis under natural error distributions. The framework unifies prior results, offers optimality statements for certain parametric regimes, and suggests practical directions for robust, scalable SDMM, including extensions to AG codes and randomized strategies.

Abstract

In this paper, a general framework for linear secure distributed matrix multiplication (SDMM) is introduced. The model allows for a neat treatment of straggling and Byzantine servers via a star product interpretation as well as simplified security proofs. Known properties of star products also immediately yield a lower bound for the recovery threshold as well as an upper bound for the number of colluding workers the system can tolerate. Another bound on the recovery threshold is given by the decodability condition, which generalizes a bound for GASP codes. The framework produces many of the known SDMM schemes as special cases, thereby providing unification for the previous literature on the topic. Furthermore, error behavior specific to SDMM is discussed and interleaved codes are proposed as a suitable means for efficient error correction in the proposed model. Analysis of the error correction capability under natural assumptions about the error distribution is also provided, largely based on well-known results on interleaved codes. Error detection and other error distributions are also discussed.
Paper Structure (21 sections, 14 theorems, 89 equations, 2 figures)

This paper contains 21 sections, 14 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Contributions
  4. Organization
  5. Preliminaries
  6. Star Product Codes
  7. Algebraic Geometry Codes
  8. Interleaved Codes
  9. Matrix Codes
  10. Examples of SDMM Schemes
  11. Linear SDMM
  12. A General Linear SDMM Framework via Star Products
  13. Security of Linear SDMM Schemes
  14. Bounds for Linear SDMM
  15. Constructing SDMM Schemes Using the Framework
  16. ...and 6 more sections

Key Result

Proposition 1

The star product code $\mathcal{C} \star \mathcal{D}$ has minimum distance when $\mathcal{C}$ and $\mathcal{D}$ are linear codes of length $n$.

Figures (2)

  • Figure 1: System model of the linear SDMM framework. Worker 2 and 3 are a straggler and a Byzantine worker, respectively.
  • Figure 2: Diagram depicting the responses from the worker nodes. The Byzantine worker is depicted by the purple layer and the straggler by the blurred layer. Each response is a matrix, which is represented as a rectangular array in the figure. The codewords are the length $N$ vectors formed by stacking the responses and looking at the corresponding matrix entries. Hence, a Byzantine worker and stragglers can only affect their own position in the codewords.

Theorems & Definitions (29)

  • Definition 1: Star product code
  • Proposition 1: Product Singleton bound randriambololona2013upper
  • Proposition 2
  • Definition 2: Homogeneous interleaved codes
  • Definition 3: Matrix code
  • Lemma 1
  • Example 1: Secure MatDot aliasgari2020private
  • Example 2: GASP d2020gasp
  • Example 3: SDMM based on DFT mital2022secure
  • Definition 4
  • ...and 19 more