  The problem of $X$-secure $T$-private information retrieval from MDS coded
storage is studied in this paper, where the user wishes to privately retrieve
one out of $K$ independent messages that are distributed over $N$ servers
according to an MDS code. It is guaranteed that any group of up to $X$
colluding servers learn nothing about the messages and that any group of up to
$T$ colluding servers learn nothing about the identity of desired message. A
lower bound of achievable rates is proved by presenting a novel scheme based on
\emph{cross-subspace alignment} and a successive decoding with interference
cancellation strategy. For large number of messages $(K\rightarrow\infty)$ the
achieved rate, which we conjecture to be optimal, improves upon the best known
rates previously reported in the literature by Raviv and Karpuk, and
generalizes an achievable rate for MDS-TPIR previously found by Freij-Hollanti
et al. that is also conjectured to be asymptotically optimal. The setting is
then expanded to allow unresponsive and Byzantine servers. Finally, the scheme
is applied to find a new lower convex hull of (download, upload) pairs of
secure and private distributed matrix multiplication that generalizes, and in
certain asymptotic settings strictly improves upon the best known previous
results.
