Secure Partially Collaborative Codes

We study secrecy in distributed storage systems (DSS) in the presence of a passive eavesdropper. In particular, we are interested in the repair model called partially collaborative repair, in which
$t$
failures are repaired collaboratively and simultaneously, and each of these
$t$
failed nodes is only allowed to exchange data with a subset of
$t-s$
(
$1 \leq s \leq t$
) nodes. The eavesdropper is assumed to be able to gain access to the data stored in a subset of the storage nodes, and possibly also, to the data downloaded during repair of some nodes. The upper bound on the secrecy capacity of this setting was studied by Liu and Oggier. In this letter, we propose two code constructions, namely secure minimum storage partially collaborative repair (SMSPCR) codes and secure minimum bandwidth partially collaborative repair (SMBPCR) codes, that achieve the existing upper bounds at two extreme points, which implies that both codes are optimal in the sense of achieving the secrecy capacity.