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In distributed matrix multiplication, a common scenario is to assign eachworker a fraction of the multiplication task, by partitioning the inputmatrices into smaller submatrices. In particular, by dividing two inputmatrices into $m$-by-$p$ and $p$-by-$n$ subblocks, a single multiplication taskcan be viewed as computing linear combinations of $pmn$ submatrix products,which can be assigned to $pmn$ workers. Such block-partitioning based designshave been widely studied under the topics of secure, private, and batchcomputation, where the state of the arts all require computing at least "cubic"($pmn$) number of submatrix multiplications. Entangled polynomial codes, firstpresented for straggler mitigation, provides a powerful method for breaking thecubic barrier. It achieves a subcubic recovery threshold, meaning that thefinal product can be recovered from \emph{any} subset of multiplication resultswith a size order-wise smaller than $pmn$. In this work, we show that entangledpolynomial codes can be further extended to also include these three importantsettings, and provide a unified framework that order-wise reduces the totalcomputational costs upon the state of the arts by achieving subcubic recoverythresholds.

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