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.

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