We present Modular Polynomial (MP) Codes for Secure Distributed Matrix Multiplication (SDMM). The construction is based on the observation that one can decode certain proper subsets of the coefficients of a polynomial with fewer evaluations than is necessary to interpolate the entire polynomial. We also present Generalized Gap Additive Secure Polynomial (GGASP) codes. Both MP and GGASP codes are shown experimentally to perform favorably in terms of recovery threshold when compared to other comparable polynomials codes for SDMM which use the grid partition. Both MP and GGASP codes achieve the recovery threshold of Entangled Polynomial Codes for robustness against stragglers, but MP codes can decode below this recovery threshold depending on the set of worker nodes which fails. The decoding complexity of MP codes is shown to be lower than other approaches in the literature, due to the user not being tasked with interpolating an entire polynomial.

翻译：我们提出用于安全分布式矩阵乘法（SDMM）的模块化多项式（MP）编码。该构造基于以下观察：解码多项式的某些适当子集系数时，所需评估次数少于插值整个多项式所需的次数。我们还提出了广义间隙加性安全多项式（GGASP）编码。实验表明，相较于采用网格划分的其他同类SDMM多项式编码，MP和GGASP编码在恢复阈值方面表现更优。MP和GGASP编码均能达到纠缠多项式编码的恢复阈值以实现对落伍节点的鲁棒性，但MP编码可根据实际失效的工作节点集合，在低于该恢复阈值的情况下完成解码。由于用户无需承担插值整个多项式的任务，MP编码的译码复杂度低于文献中的其他方法。