  In this paper, we study the problem of secure and private distributed matrix
multiplication. Specifically, we focus on a scenario where a user wants to
compute the product of a confidential matrix $A$, with a matrix $B_\theta$,
where $\theta\in\{1,\dots,M\}$. The set of candidate matrices
$\{B_1,\dots,B_M\}$ are public, and available at all the $N$ servers. The goal
of the user is to distributedly compute $AB_{\theta}$, such that $(a)$ no
information is leaked about the matrix $A$ to any server; and $(b)$ the index
$\theta$ is kept private from each server. Our goal is to understand the
fundamental tradeoff between the upload vs download cost for this problem. Our
main contribution is to show that the lower convex hull of following (upload,
download) pairs: $(U,D)=(N/(K-1),(K/(K-1))(1+(K/N)+\dots+(K/N)^{M-1}))$ for
$K=2,\dots,N$ is achievable. The scheme improves upon state-of-the-art existing
schemes for this problem, and leverages ideas from secret sharing and coded
private information retrieval.
