  This paper studies a system of linear equations, denoted as $Ax = b$, which
is horizontally partitioned (rows in $A$ and $b$) and stored over a network of
$m$ devices connected in a fixed directed graph. We design a fast distributed
algorithm for solving such a partitioned system of linear equations, that
additionally, protects the privacy of local data against an honest-but-curious
adversary that corrupts at most $\tau$ nodes in the network. First, we present
TITAN, privaTe fInite Time Average coNsensus algorithm, for solving a general
average consensus problem over directed graphs, while protecting statistical
privacy of private local data against an honest-but-curious adversary. Second,
we propose a distributed linear system solver that involves each agent/devices
computing an update based on local private data, followed by private
aggregation using TITAN. Finally, we show convergence of our solver to the
least squares solution in finite rounds along with statistical privacy of local
linear equations against an honest-but-curious adversary provided the graph has
weak vertex-connectivity of at least $\tau+1$. We perform numerical experiments
to validate our claims and compare our solution to the state-of-the-art methods
by comparing computation, communication and memory costs.
