Codes Trading Upload for Download Cost in Secure Distributed Matrix Multiplication
Jaber Kakar, Anton Khristoforov, Seyedhamed Ebadifar, Aydin Sezgin · IEEE Transactions on Communications · 2021
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$.