Private Information Retrieval With Private Noisy Side Information

Consider Private Information Retrieval (PIR), where a client wants to retrieve one file out of
$K$
files that are replicated in
$N$
different servers and the client selection must remain private when up to
$T$
servers may collude. Additionally, suppose that the client has noisy side information about each of the
$K$
files, and the side information about a specific file is obtained by passing this file through one of
$D$
possible discrete memoryless test channels, where
$D\le K$
. While the statistics of the test channels are known by the client and by all the servers, the specific mapping
$\boldsymbol { \mathcal {M}}$
between the files and the test channels is unknown to the servers. We study this problem under two different privacy metrics. Under the first privacy metric, the client wants to preserve the privacy of its desired file selection and the mapping
$\boldsymbol { \mathcal {M}}$
. Under the second privacy metric, the client wants to preserve the privacy of its desired file and the mapping
$\boldsymbol { \mathcal {M}}$
but is willing to reveal the index of the test channel that is associated to its desired file. For both of these two privacy metrics, we derive the optimal normalized download cost. Our problem setup generalizes PIR with colluding servers, PIR with private noiseless side information, and PIR with private side information under storage constraints.