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 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.
I. INTRODUCTIONLarge scale distributed computing faces several modern challenges, in particular, to provide resiliency against stragglers, robustness against computing errors, security against Byzantine and eavesdropping adversaries, privacy of sensitive information, and to efficiently handle repetitive computation [1]- [7]. Coded computing is an emerging field that resolves these issues by introducing and developing new coding theoretic concepts, started focusing on straggler mitigation [8]-[10], then later extended to secure and private computation [6], [7], [11]- [14].Coding for straggler mitigation is first studied in [8] for linear computation, where classical linear codes can be directly applied to achieve same performances. For computation beyond linear functions, new classes of coding designs are needed to achieve optimality. In [10], we studied matrix-by-matrix multiplication and introduced the polynomial coded computing (PCC) framework. The main coding idea is to jointly encode the input variables into single variate polynomials where the coefficients are functions of the inputs. By assigning each worker evaluations of these polynomials as coded variables, they essentially evaluate a new polynomial composed by each worker's computation and the encoding functions at the same point. As long as the needed final results can be recovered from the coefficients of the composed polynomial, the master can decode the final output when sufficiently many workers complete their computation. PCC significantly reduces the design problem of coded computation to finding polynomials satisfying the above decodability constraint while minimizing the degree of the decomposed polynomial. It has been shown that PCC achieves a great success in providing exact optimal coding constructions for large classes of computation tasks includi...