The problem of secure distributed matrix multiplication (SDMM) studies the communication efficiency of retrieving a sequence of desired L×M matrix products AB = ( are stored in X-secure coded form, i.e., any group of up to X colluding servers learn nothing about A, B. The rate of an SDMM scheme is defined as the ratio of the number of bits of desired information that is retrieved, to the total number of bits downloaded on average. The supremum of achievable rates is called the capacity of SDMM. In this work we explore the capacity of SDMM, as well as several of its variants, e.g., where the user may already have either A or B available as side-information, and/or where the security constraint for either A or B may be relaxed. As our main contribution, we obtain new converse bounds, as well as new achievable schemes for various cases of SDMM, depending on the L, K, M, N, X parameters, and identify parameter regimes where these bounds match. A remarkable aspect of our upper bounds is a connection between SDMM and a form of private information retrieval (PIR) problem, known as multi-message X-secure T -private information retrieval (MM-XSTPIR). Notable features of our achievable schemes include the use of crosssubspace alignment and a transformation argument that converts a scalar multiplication problem into a scalar addition problem, allowing a surprisingly efficient solution.
A new class of secure distributed matrix multiplication and its capacity is studied in REF .
On the Capacity of Secure Distributed Matrix Multiplication
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Computer Science Mathematics