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Large matrix multiplications are central to large-scale machine learningapplications. These operations are often carried out on a distributed computingplatform with a master server and multiple workers in the cloud operating inparallel. For such distributed platforms, it has been recently shown thatcoding over the input data matrices can reduce the computational delay,yielding a trade-off between recovery threshold, i.e., the number of workersrequired to recover the matrix product, and communication load, i.e., the totalamount of data to be downloaded from the workers. In this paper, in addition toexact recovery requirements, we impose security and privacy constraints on thedata matrices, and study the recovery threshold as a function of thecommunication load. We first assume that both matrices contain privateinformation and that workers can collude to eavesdrop on the content of thesedata matrices. For this problem, we introduce a novel class of secure codes,referred to as secure generalized PolyDot (SGPD) codes, that generalizestate-of-the-art non-secure codes for matrix multiplication. SGPD codes allow aflexible trade-off between recovery threshold and communication load for afixed maximum number of colluding workers while providing perfect secrecy forthe two data matrices. We then study a connection between secure matrixmultiplication and private information retrieval. We specifically assume thatone of the data matrices is taken from a public set known to all the workers.In this setup, the identity of the matrix of interest should be kept privatefrom the workers. For this model, we present a variant of generalized PolyDotcodes that can guarantee both secrecy of one matrix and privacy for theidentity of the other matrix for the case of no colluding servers.

Private and Secure Distributed Matrix Multiplication with Flexible Communication Load