  The problem of $X$-secure $T$-private linear computation with graph based
replicated storage (GXSTPLC) is to enable the user to retrieve a linear
combination of messages privately from a set of $N$ distributed servers where
every message is only allowed to store among a subset of servers subject to an
$X$-security constraint, i.e., any groups of up to $X$ colluding servers must
reveal nothing about the messages. Besides, any groups of up to $T$ servers
cannot learn anything about the coefficients of the linear combination
retrieved by the user. In this work, we completely characterize the asymptotic
capacity of GXSTPLC, i.e., the supremum of average number of desired symbols
retrieved per downloaded symbol, in the limit as the number of messages $K$
approaches infinity. Specifically, it is shown that a prior linear programming
based upper bound on the asymptotic capacity of GXSTPLC due to Jia and Jafar is
tight by constructing achievability schemes. Notably, our achievability scheme
also settles the exact capacity (i.e., for finite $K$) of $X$-secure linear
combination with graph based replicated storage (GXSLC). Our achievability
proof builds upon an achievability scheme for a closely related problem named
asymmetric $\mathbf{X}$-secure $\mathbf{T}$-private linear computation with
graph based replicated storage (Asymm-GXSTPLC) that guarantees non-uniform
security and privacy levels across messages and coefficients. In particular, by
carefully designing Asymm-GXSTPLC settings for GXSTPLC problems, the
corresponding Asymm-GXSTPLC schemes can be reduced to asymptotic capacity
achieving schemes for GXSTPLC. In regard to the achievability scheme for
Asymm-GXSTPLC, interesting aspects of our construction include a novel query
and answer design which makes use of a Vandermonde decomposition of Cauchy
matrices, and a trade-off among message replication, security and privacy
thresholds.
