In this paper, we propose a new secure distributed matrix multiplication (SDMM) scheme using the inner product partitioning. We construct a scheme with a minimal number of workers and no redundancy, and another scheme with redundancy against stragglers. Unlike previous constructions in the literature, we do not utilize algebraic methods such as locally repairable codes or algebraic geometry codes. Our construction, which is based on generalized Reed-Solomon codes, improves the flexibility of the field size as it does not assume any divisibility constraints among the different parameters. We achieve a minimal number of workers by efficiently canceling all interference terms with a suitable orthogonal decoding vector. Finally, we discuss how the MDS conjecture impacts the smallest achievable field size for SDMM schemes and show that our construction almost achieves the bound given by the conjecture.

翻译：本文提出了一种基于内积分割的新型安全分布式矩阵乘法（SDMM）方案。我们构建了两种方案：一种使用最少工作节点且无冗余，另一种则具备针对落后节点的冗余能力。与现有文献中的方法不同，我们未采用局部可修复码或代数几何码等代数工具。本方案基于广义里德-所罗门码，通过消除不同参数间的可整除性约束，提升了域大小的灵活性。我们利用合适的正交解码向量高效消除所有干扰项，从而实现了最少工作节点数。最后，我们讨论了MDS猜想对SDMM方案最小可达域大小的影响，并证明本方案几乎达到了该猜想给出的界限。