Private and Secure Distributed Matrix Multiplication With Flexible Communication Load

Abstract

Large matrix multiplications are central to largescale machine learning applications. These operations are often carried out on a distributed computing platform with a master server and multiple workers in the cloud operating in parallel. For such distributed platforms, it has been recently shown that coding over the input data matrices can reduce the computational delay, yielding a trade-off between recovery threshold, i.e., the number of workers required to recover the matrix product, and communication load, i.e., the total amount of data to be downloaded from the workers. In this paper, in addition to exact recovery requirements, we impose security and privacy constraints on the data matrices, and study the recovery threshold as a function of the communication load. We first assume that both matrices contain private information and that workers can collude to eavesdrop on the content of these data matrices. For this problem, we introduce a novel class of secure codes, referred to as secure generalized PolyDot (SGPD) codes, that generalize state-of-the-art non-secure codes for matrix multiplication. SGPD codes allow a flexible trade-off between recovery threshold and communication load for a fixed maximum number of colluding workers while providing perfect secrecy for the two data matrices. We then assume that one 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 private from the workers. For this model, we present a variant of generalized PolyDot codes that can guarantee both secrecy of one matrix and privacy for the identity of the other matrix for the case of no colluding servers.

Improved Private and Secure Distributed Matrix Multiplication

“…In particular, when p = 1, the recovery threshold of the new code is one smaller than that of Kim-Lee's code in [18] under the same upload cost. • The recovery threshold and the download cost of the new PSDMM code are smaller than those of Aliasgari et al's PSGPD code in [12] under the same upload cost 2 and the PSDMM code in [13] for some small parameters.…”

“…We considered the problem of PSDMM and proposed a scheme based on [12] and the entangled polynomial code in [13]. The performance of the scheme was characterized via a degree table.…”

“…• The new PSDMM code outperforms the ones in [12] and [18] in that it has a smaller recovery threshold as well as download cost under the same upload cost, and it provides a more flexible tradeoff between the upload and download costs than the one in [18]. • The new PSDMM code also has a smaller recovery threshold and download cost when compared with the ones in [13] and [35] for some regions.…”

“…In this section, we propose a new PSDMM code. The construction is largely similar to the one in [12], but the encoding phase of the new PSDMM scheme is more efficient. This is enabled by considering the problem of designing efficient encoding polynomials in the encoding phase and the server computation phase, in the view of designing a degree table, which was previously introduced for the problem of SDMM [7].…”

“…2 In [12], there was a small mistake in the recovery threshold of the PSGPD code. We give the correct value of the recovery threshold in Table II.…”