Secure Distributed Matrix Computation with Discrete Fourier Transform

Abstract

We consider the problem of secure distributed matrix computation (SDMC), where a user can query 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, as well as 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.

Degree Tables for Secure Distributed Matrix Multiplication

“…Next, (10) provides a sufficient criterion for the application of Lemma 11, which completes the proof.…”

“…The literature on SDMM has also studied different variations on the model we focus on here. For instance, in [10], [15], [26], [27] the encoder and decoder are considered to be separate, in [26] servers are allowed to cooperate, and in [28] they consider a hybrid between SDMM and private information retrieval where the user has a matrix A and wants to privately multiply it with a matrix B belonging to some public list. The scheme we present here can be readily used or adapted to many of these settings (e.g., [11], [29]).…”

“…Theorem 7 allows us to perform an exhaustive search of all normal degree tables (α * , β * ) meeting (10) with N(α * , β * ) = N(K, L, T ). For example, consider the case where K = L = 2 and T = 5.…”

“…The following degree table with the same amount of distinct entries can be found by applying the greedy algorithm. α p = (0, 1,2,3,4,5,6,7,8,9,10,11,12,13,14), (225,226,227,229,240,241,242,244,255,256,257,259,270,271,272), 15,30,45,60,75,90,105,120,135,150,165,180,195,210), (225,226,227,228,229,230,231,232,233,234,235,236,237,238,239).…”

Bivariate Polynomial Codes for Secure Distributed Matrix Multiplication

“…In [15]- [17], lower recovery threshold values than [14] are obtained by using different matrix partitioning techniques and different choices of encoding polynomials, but this is achieved at the expense of a considerable increase in the upload cost. In [18], a novel coding approach for distributed matrix multiplication is proposed based on polynomial evaluation at the roots of unity in a finite field. It has constant time decoding complexity and a low recovery threshold compared to traditional polynomial-type coding approaches, but the sub-tasks are not one-to-any replaceable and its straggler mitigation capability is limited.…”