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. Maximizing the download rate 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. We also present lower bounds on N and show that $GASP_{r}$ achieves the lower bounds in the case of no server collusion.