  The problem of secure distributed batch matrix multiplication (SDBMM) studies
the communication efficiency of retrieving a sequence of desired matrix
products ${\bf AB}$ $=$ $({\bf A}_1{\bf B}_1,$ ${\bf A}_2{\bf B}_2,$ $\cdots,$
${\bf A}_S{\bf B}_S)$ from $N$ distributed servers where the constituent
matrices ${\bf A}=({\bf A}_1, {\bf A}_2, \cdots, {\bf A}_S)$ and ${\bf B}=({\bf
B}_1, {\bf B}_2,\cdots,{\bf B}_S)$ are stored in $X$-secure coded form, i.e.,
any group of up to $X$ colluding servers learn nothing about ${\bf A, B}$. It
is assumed that ${\bf A}_s\in\mathbb{F}_q^{L\times K}, {\bf
B}_s\in\mathbb{F}_q^{K\times M}, s\in\{1,2,\cdots, S\}$ are uniformly and
independently distributed and $\mathbb{F}_q$ is a large finite field. The rate
of an SDBMM 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 SDBMM. In
this work we explore the capacity of SDBMM, as well as several of its variants,
e.g., where the user may already have either ${\bf A}$ or ${\bf B}$ available
as side-information, and/or where the security constraint for either ${\bf A}$
or ${\bf B}$ may be relaxed. We obtain converse bounds, as well as achievable
schemes for various cases of SDBMM, 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 SDBMM 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 cross-subspace alignment and a
transformation argument that converts a scalar multiplication problem into a
scalar addition problem, allowing a surprisingly efficient solution.
