  In distributed matrix multiplication, a common scenario is to assign each
worker a fraction of the multiplication task, by partitioning 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.
