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.

翻译：本文提出了一种基于复数域的安全分布式矩阵乘法方案，该方案利用单位根的多项式插值，具有良好的数值稳定性和较小的互信息泄露。此外，我们通过一种称为"复化"的技术，先将实数矩阵编码为更小的复数矩阵，从而构建了基于实数域的构造方案。这些实数域方案继承了复数域方案的诸多优点，包括良好的数值稳定性，同时在计算上更为高效。为了分析数值稳定性和互信息泄露，我们给出了以单位根为求值点的范德蒙矩阵条件数的一些上界。