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. These codes are proven to outperform, in terms of recovery threshold, the currently best-known polynomial codes for the inner product partition. We also present Generalized Gap Additive Secure Polynomial (GGASP) codes for the grid partition. These two families of codes are shown experimentally to perform favorably in terms of recovery threshold when compared to other comparable polynomials codes for SDMM. 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：本文提出用于安全分布式矩阵乘法（SDMM）的模块化多项式（MP）码。该构造基于如下观察：通过较少的求值即可解码多项式的某些特定子集系数，而无需对整个多项式进行插值。实验证明，就恢复阈值而言，这类码在内积划分场景下优于当前已知最优的多项式码。我们还针对网格划分提出了广义间隙加性安全多项式（GGASP）码。实验表明，这两种码族在安全分布式矩阵乘法的恢复阈值性能上优于其他同类多项式码。MP码与GGASP码均能达到纠缠多项式码（Entangled Polynomial Codes）对落后节点的鲁棒性恢复阈值，但MP码可根据失效工作节点集合在低于该恢复阈值的条件下实现解码。由于用户无需对整个多项式进行插值，MP码的解码复杂度低于文献中其他方法。