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Modular Polynomial Codes for Secure and Robust Distributed Matrix Multiplication

David Karpuk, Razane Tajeddine

TL;DR

This work tackles secure and robust distributed matrix multiplication under grid partition by introducing Modular Polynomial (MP) codes and Generalized GGASP (GGASP) codes. The core innovations are partial polynomial interpolation and the mod-$M$ transform, enabling decoding of targeted coefficient subsets with fewer evaluations, and explicit decodability and $T$-security guarantees. Concrete recovery-threshold formulas are derived for MP and GGASP, and the schemes are shown to achieve state-of-the-art robustness against stragglers in the absence of security, with competitive performance under security constraints. Empirically, MP and GGASP exhibit favorable recovery thresholds and rates across parameter regimes, while also offering reduced decoding complexity relative to prior polynomials-code approaches; the framework also enables practical evaluation with relatively small finite fields via field extensions.

Abstract

We present Modular Polynomial (MP) Codes for Secure Distributed Matrix Multiplication (SDMM). The construction is based on the observation that one can decode certain proper subsets of the coefficients of a polynomial with fewer evaluations than is necessary to interpolate the entire polynomial. We also present Generalized Gap Additive Secure Polynomial (GGASP) codes. Both MP and GGASP codes are shown experimentally to perform favorably in terms of recovery threshold when compared to other comparable polynomials codes for SDMM which use the grid partition. Both MP and GGASP codes achieve the recovery threshold of Entangled Polynomial Codes for robustness against stragglers, but MP codes can decode below this recovery threshold depending on the set of worker nodes which fails. The decoding complexity of MP codes is shown to be lower than other approaches in the literature, due to the user not being tasked with interpolating an entire polynomial.

Modular Polynomial Codes for Secure and Robust Distributed Matrix Multiplication

TL;DR

This work tackles secure and robust distributed matrix multiplication under grid partition by introducing Modular Polynomial (MP) codes and Generalized GGASP (GGASP) codes. The core innovations are partial polynomial interpolation and the mod- transform, enabling decoding of targeted coefficient subsets with fewer evaluations, and explicit decodability and -security guarantees. Concrete recovery-threshold formulas are derived for MP and GGASP, and the schemes are shown to achieve state-of-the-art robustness against stragglers in the absence of security, with competitive performance under security constraints. Empirically, MP and GGASP exhibit favorable recovery thresholds and rates across parameter regimes, while also offering reduced decoding complexity relative to prior polynomials-code approaches; the framework also enables practical evaluation with relatively small finite fields via field extensions.

Abstract

We present Modular Polynomial (MP) Codes for Secure Distributed Matrix Multiplication (SDMM). The construction is based on the observation that one can decode certain proper subsets of the coefficients of a polynomial with fewer evaluations than is necessary to interpolate the entire polynomial. We also present Generalized Gap Additive Secure Polynomial (GGASP) codes. Both MP and GGASP codes are shown experimentally to perform favorably in terms of recovery threshold when compared to other comparable polynomials codes for SDMM which use the grid partition. Both MP and GGASP codes achieve the recovery threshold of Entangled Polynomial Codes for robustness against stragglers, but MP codes can decode below this recovery threshold depending on the set of worker nodes which fails. The decoding complexity of MP codes is shown to be lower than other approaches in the literature, due to the user not being tasked with interpolating an entire polynomial.
Paper Structure (35 sections, 24 theorems, 59 equations, 4 figures)

This paper contains 35 sections, 24 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Related Work
  4. Outline and Summary of Main Contributions
  5. Notation
  6. Partial Polynomial Interpolation and the mod-$M$ Transform
  7. Generalized Vandermonde Matrices
  8. Partial Polynomial Interpolation
  9. Modular Polynomial Codes
  10. A General Polynomial Code Scheme
  11. Decodability and $T$-Security of Modular Polynomial Codes
  12. An Example: $K = L = 2$, $M = 3$, and $T = 3$
  13. Explicit Recovery Threshold of Modular Polynomial Codes
  14. Generalized GASP codes
  15. Definition of Generalized GASP Codes
  16. ...and 20 more sections

Key Result

Lemma 2.1

Let $\mathbb{F}$ be a field and let $\phi_1,\ldots,\phi_S$ be non-zero polynomials in $\mathbb{F}[X_1,\ldots,X_N]$. Then there exists a finite extension $\mathbb{K}/\mathbb{F}$ and a point $a\in \mathbb{K}^N$ such that $\phi_s(a)\neq 0$ for all $s$.

Figures (4)

  • Figure 1: Rates of various polynomial codes, with $K = L = 2$, $M = 10$ (left), and $K = L = 2$, $M = 40$ (right), plotted as a function of the security parameter $T$.
  • Figure 2: Rates of various polynomial codes, with $K = L = 10$, $M = 10$ (left), and $K = L = 40$, $M = 4$ (right), plotted as a function of the security parameter $T$.
  • Figure 3: Rates of various polynomial codes with $N = 200$, $K,L\geq 2$, $M\geq 4$ (left), and $N = 200$, $K,L\geq 2$, $M\geq 10$ (right), plotted as a function of the security parameter $T$.
  • Figure 4: The lower bound on the function $p(S)$ of Theorem \ref{['prob_recovery']} as a function of $N-S$, for $K=L=2$ and $M=5$ (left), and $K=L=2$ and $M=10$ (right). The different curves correspond to different numbers of total worker nodes.

Theorems & Definitions (59)

  • Lemma 2.1
  • proof
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 49 more