Secure Distributed Matrix Computation With Discrete Fourier Transform

Abstract

We consider the problem of secure distributed matrix computation (SDMC), where a user queries a function of data matrices generated at distributed source nodes. We assume the availability of N honest but curious computation servers, which are connected to the sources, the user, and each other through orthogonal and reliable communication links. Our goal is to minimize the amount of data that must be transmitted from the sources to the servers, called the upload cost, while guaranteeing that no T colluding servers can learn any information about the source matrices, and the user cannot learn any information beyond the computation result. We first focus on secure distributed matrix multiplication (SDMM), considering two matrices, and propose a novel polynomial coding scheme using the properties of finite field discrete Fourier transform, which achieves an upload cost significantly lower than the existing results in the literature. We then generalize the proposed scheme to include straggler mitigation, and to the multiplication of multiple matrices while keeping the input matrices, the intermediate computation results, as well as the final result secure against any T colluding servers. We also consider a special case, called computation with own data, where the data matrices used for computation belong to the user. In this case, we drop the security requirement against the user, and show that the proposed scheme achieves the minimal upload cost. We then propose methods for performing other common matrix computations securely on distributed servers, including changing the parameters of secret sharing, matrix transpose, matrix exponentiation, solving a linear system, and matrix inversion, which are then used to show how arbitrary matrix polynomials can be computed securely on distributed servers using the proposed procedure.

Improved Constructions for Secure Multi-Party Batch Matrix Multiplication

“…Both traditional SSMM and SMBMM mainly focus on the special case that the matrices A and B have the same security level (i.e., X = X A = X B ). For SSMM, the state of the art are reflected in Secure Generalized PolyDot (S-GPD) codes [1], GASP codes [6], A3S codes [13], USCSA codes [14], FFT-based Polynomial (FFT-P) codes [18], Polynomial Sharing (PS) codes [19] and Extended Entangled Polynomial (E-EP) codes [33]. Depending on the partitioning manners of the data matrices, these works are divided into two classes of row-by-column partition [6], [13], [14] (where the partitioning parameter p is set to be 1 and m, n are arbitrary) and arbitrary partition [1], [18], [19], [33] (where all the partitioning parameters m, p, n are arbitrary).…”

“…The problem of secure DMM was first launched by Tandon et al in [2] for Single Secure Matrix Multiplication (SSMM), which is shown to achieve the optimal download cost for one-sided SSMM (where only one of the data matrices is kept secure). Subsequently, many works [1], [6], [13], [14], [18], [19], [20], [30] focus on two-sided SSMM (both matrices are kept secure).…”

“…focuses on jointly computing an arbitrary polynomial function of some private datasets distributed at the users (parties) under the constraint that each user must not learn any additional information about the datasets beyond the function. For the secure DMM problem, references [19], [20], [18] expanded it under the framework of SMPC, which allows performing arbitrary polynomial functions on private massive matrices.…”

“…We will first focus on the SSMM problem (i.e., the case M = 1), which has received significant recent interest [1], [6], [13], [14], [18], [19], [20], [30]. Similar to these works, in this case the data security requirement against the user is not considered.…”

Analog Secure Distributed Matrix Multiplication over Complex Numbers

“…SDMM was first introduced by Chang and Tandon in [1]. Many different schemes have since been presented, such as the secure MatDot scheme in [2], the GASP scheme in [3] and the DFT scheme in [4]. These schemes compute the products over finite fields, but computations over complex numbers have also been considered in [5], [6], [7].…”