Entangled Polynomial Codes for Secure, Private, and Batch Distributed Matrix Multiplication: Breaking the ''Cubic'' Barrier

In distributed matrix multiplication, a common scenario is to assign each worker a fraction of the multiplication task, by partition the input matrices into smaller submatrices. In particular, by dividing two input matrices into $m$-by-$p$ and $p$-by-$n$ subblocks, a single multiplication task can be viewed as computing linear combinations of $pmn$ submatrix products, which can be assigned to $pmn$ workers. Such block-partitioning based designs have been widely studied under the topics of secure, private, and batch computation, where the state of the arts all require computing at least ``cubic'' ($pmn$) number of submatrix multiplications. Entangled polynomial codes, first presented for straggler mitigation, provides a powerful method for breaking the cubic barrier. It achieves a subcubic recovery threshold, meaning that the final product can be recovered from \emph{any} subset of multiplication results with a size order-wise smaller than $pmn$. In this work, we show that entangled polynomial codes can be further extended to also include these three important settings, and provide a unified framework that order-wise reduces the total computational costs upon the state of the arts by achieving subcubic recovery thresholds.