Analog Secure Distributed Matrix Multiplication

2025-08-24

AI Summary

This work addresses the Secure Distributed Matrix Multiplication (SDMM) problem over both complex and real fields, proposing a unified framework that jointly ensures numerical stability and information-theoretic security. Methodologically, it employs polynomial interpolation based on roots of unity for encoding, augmented by complexification encoding to map real matrices into lower-dimensional complex matrices—thereby reducing computational and communication overhead. Numerical stability is rigorously guaranteed via theoretical bounds on the condition number of Vandermonde matrices induced by the encoding design. Information leakage during communication is quantified and strictly bounded using mutual information. The primary contribution is the first integration of root-of-unity interpolation with complexification encoding for SDMM, yielding a dual-field (real/complex) solution that achieves provably low information leakage, high numerical stability (with bounded condition number), and superior efficiency (significantly reduced computation and communication costs).

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Secure distributed matrix multiplication over complex numbers
• Encoding real matrices to complex matrices via complexification
• Analyzing numerical stability and mutual information leakage

Innovation

• Polynomial interpolation with roots of unity
• Complexification encoding for real matrices
• Bounded condition numbers for numerical stability