Degree Tables for Secure Distributed Matrix Multiplication

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 minimizingN, 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 callGASPᵣ(Gap Additive Secure Polynomial codes) parameterized by an integerr.GASPᵣoutperforms all previously known polynomial codes for SDMM under an outer product partitioning. We also present lower bounds onNand 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 ofGASPᵣfor all the system parameters that we were able to test.