  The problem of securing data against eavesdropping in distributed storage
systems is studied. The focus is on systems that use linear codes and implement
exact repair to recover from node failures.The maximum file size that can be
stored securely is determined for systems in which all the available nodes help
in repair (i.e., repair degree $d=n-1$, where $n$ is the total number of nodes)
and for any number of compromised nodes. Similar results in the literature are
restricted to the case of at most two compromised nodes. Moreover, new explicit
upper bounds are given on the maximum secure file size for systems with
$d<n-1$. The key ingredients for the contribution of this paper are new results
on subspace intersection for the data downloaded during repair. The new bounds
imply the interesting fact that the maximum data that can be stored securely
decreases exponentially with the number of compromised nodes.
