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

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 ℓ-SDMM, the goal is to 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 ℓ ≤ N servers. To this end, the user applies coding with Ãiand B̃irepresenting 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 implement schemes from the literature, in addition to 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, B̃i)i=1Nand the computation of Oi= Ãi, B̃i, with those of SCSA, being a low timing overhead for the download of (Oi)i=1Nand the decoding of AB.

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. 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 to maximize the communication rate while preserving security, where we allow for up to ℓ ≤ N servers to collude. To this end, we propose a general aligned secret sharing and matrix partition scheme for which we optimize the partition of matrices A and B as a function of N and ℓ in order to maximize the achievable rate. A proposed inductive approach gives us an analytical close-to-optimal solution which significantly outperforms the existing scheme of Chang and Tandon in terms of (i) communication rate, (ii) maximum tolerable number of colluding servers and (iii) computational complexity.