Degree Tables for Secure Distributed Matrix Multiplication

Abstract

We consider the problem of secure distributed matrix multiplication (SDMM) in which a user wishes to compute the product of two matrices with the assistance of honest but curious servers. We construct polynomial codes for SDMM by studying a recently introduced combinatorial tool called the degree table. For a fixed partitioning, minimizing the total communication cost of a polynomial code for SDMM is equivalent to minimizing N , the number of distinct elements in the corresponding degree table.We propose new constructions of degree tables with a low number of distinct elements. These new constructions lead to a general family of polynomial codes for SDMM, which we call GASP r (Gap Additive Secure Polynomial codes) parametrized by an integer r. GASP r outperforms all previously known polynomial codes for SDMM under an outer product partitioning. We also present lower bounds on N and prove the optimality or asymptotic optimality of our constructions for certain regimes. Moreover, we formulate the construction of optimal degree tables as an integer linear program and use it to prove the optimality of GASP r for all the system parameters that we were able to test.

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“…To assign the exponents for decodability while having as many zero coefficients as possible, the authors propose the degree table to solve the combinatorial problem. In [157], the authors further generalize the GASP codes to be applicable to different partitions of input matrices and security parameters. The GASP codes are shown to outperform the approaches in [141,153] in terms of download rate.…”