Analog Secure Distributed Matrix Multiplication

TL;DR

This paper introduces secure distributed matrix multiplication schemes over complex and real numbers, leveraging polynomial interpolation with roots of unity to achieve numerical stability and minimal information leakage.

Contribution

It proposes novel SDMM schemes over complex and real numbers, using polynomial interpolation and complexification techniques for improved efficiency and security.

Findings

Schemes over complex numbers with good numerical stability.

Real number schemes that are more computationally efficient.

Bounds on condition numbers of Vandermonde matrices.

Abstract

In this paper, we present secure distributed matrix multiplication (SDMM) schemes over the complex numbers with good numerical stability and small mutual information leakage by utilizing polynomial interpolation with roots of unity. Furthermore, we give constructions utilizing the real numbers by first encoding the real matrices to smaller complex matrices using a technique we call complexification. These schemes over the real numbers enjoy many of the benefits of the schemes over the complex numbers, including good numerical stability, but are computationally more efficient. To analyze the numerical stability and the mutual information leakage, we give some bounds on the condition numbers of Vandermonde matrices whose evaluation points are roots of unity.