In this paper, we propose a novel construction for secure distributed matrix multiplication (SDMM) based on algebraic geometry (AG) codes, which we call the PoleGap SDMM scheme. The proposed construction is inspired by the GASP code, where so-called gaps in a certain polynomial are utilized to achieve higher communication rates. Our construction considers the gaps in a Weierstrass semigroup of a rational place in an algebraic function field to achieve a similar increase in the rate. This construction shows that there is potential in utilizing AG codes and their subcodes in SDMM since we demonstrate a better performance compared to state-of-the-art schemes in some parameter regimes.

翻译：本文提出了一种基于代数几何码的安全分布式矩阵乘法新构造方案，称为PoleGap SDMM方案。该构造受GASP码启发，利用特定多项式中的所谓"间隙"实现更高通信速率。我们的构造考虑了代数函数域中有理点的Weierstrass半群中的间隙，以类似方式提升速率。该构造表明，在AG码及其子码应用于SDMM领域具有潜力——我们在某些参数区间内展现出优于现有方案的性能表现。