Computationally efficient matrix multiplication is a fundamental requirement in various fields, including and particularly in data analytics. To do so, the computation task of large-scale matrix multiplication is typically outsourced to multiple servers. However, due to data misusage at the servers, security is typical of concern. In this paper, we first study the two-sided secure matrix multiplication problem, where a user is interested in the matrix product AB of two finite field private matrices A and B from an information-theoretic perspective. In this problem, the user exploits the computational resources of N servers to compute the matrix product but simultaneously tries to conceal the private matrices from the servers. Our goal is twofold: (i) to maximize the downlink communication rate, and (ii) to minimize the effective number of server observations needed to determine AB, while preserving security, where we allow for up to ≤ N servers to collude. To this end, we propose two schemes -an aligned secret sharing scheme (A3S) and a secure cross subspace alignment (SCSA) scheme. For A3S, we optimize the partitioning of matrices A and B in order to either optimize objective (i) or (ii) as a function of the system parameters (e.g., N and ). A proposed inductive approach gives us analytical, close-to-optimal solutions for both (i) and (ii). The SCSA, on the other hand, is shown to be (rate) capacity-optimal for the general J -sided distributed secure matrix multiplication problem J j=1 M j . We show this by developing a recursive information-theoretic upper bound (converse) on the downlink rate for the J -sided secure matrix multiplication problem. With respect to (i), both A3S and SCSA, significantly outperform the state-of-the-art in terms of (a) communication rate, (b) maximum tolerable number of colluding servers, and (c) computational complexity. Overall SCSA (A3S) is the preferred choice when the focus is on the downlink (uplink).

Computationally efficient matrix multiplication is a fundamental requirement in various fields, including and particularly in data analytics. To do so, the computation task of a large-scale matrix multiplication is typically outsourced to multiple servers. However, due to data misusage at the servers, security is typically of concern. In this paper, we study the two-sided secure matrix multiplication problem, where a user is interested in the matrix product AB of two finite field private matrices A and B from an information-theoretic perspective. In this problem, the user exploits the computational resources of N servers to compute the matrix product, but simultaneously tries to conceal the private matrices from the servers. Our goal is twofold:(i) to maximize the communication rate, and, (ii) to minimize the effective number of server observations needed to determine AB, while preserving security, where we allow for up to ℓ ≤ N servers to collude. To this end, we propose a general aligned secret sharing scheme for which we optimize the matrix partition of matrices A and B in order to either optimize objective (i) or (ii) as a function of the system parameters (e.g., N and ℓ). A proposed inductive approach gives us analytical, close-to-optimal solutions for both (i) and (ii). With respect to (i), our scheme significantly outperforms the existing scheme of Chang and Tandon in terms of (a) communication rate, (b) maximum tolerable number of colluding servers and (c) computational complexity.

No abstract

No abstract

In secure distributed matrix multiplication (SDMM) the multiplication AB from two private matrices A and B is outsourced by a user to N distributed servers. In -SDMM, the goal is to a design a joint communication-computation procedure that optimally balances conflicting communication and computation metrics without leaking any information on both A and B to any set of ≤ N servers. To this end, the user applies coding with Ãi and Bi representing encoded versions of A and B destined to the i-th server. Now, SDMM involves multiple tradeoffs. One such tradeoff is the tradeoff between uplink (UL) and downlink (DL) costs. To find a good balance between these two metrics, we propose two schemes which we term USCSA and GSCSA that are based on secure cross subspace alignment (SCSA). We show that there are various scenarios where they outperform existing SDMM schemes from the literature with respect to UL-DL efficiency.Next, we implement schemes from the literature, including USCSA and GSCSA, and test their performance on Amazon EC2. Our numerical results show that USCSA and GSCSA establish a good balance between the time spend on the communication and computation in SDMMs. This is because they combine advantages of polynomial codes, namely low time for the upload of Ãi , Bi N i=1 and the computation of O i = Ãi Bi , with those of SCSA, being a low timing overhead for the download of (O i ) N i=1 and the decoding of AB.