  Consider the problem of Private Information Retrieval (PIR), where a user
wishes to retrieve a single message from $N$ non-communicating and
non-colluding databases (servers). All servers store the same set of $M$
messages and they respond to the user through a block fading Gaussian Multiple
Access Channel (MAC). The goal in this setting is to keep the index of the
required message private from the servers while minimizing the overall
communication overhead.
  This work provides joint privacy and channel coding retrieval schemes for the
Gaussian MAC with and without fading. The schemes exploit the linearity of the
channel while using the Compute and Forward (CF) coding scheme. Consequently,
single-user encoding and decoding are performed to retrieve the private
message. In the case of a channel without fading, the achievable retrieval rate
is shown to outperform a separation-based scheme, in which the retrieval and
the channel coding are designed separately. Moreover, this rate is
asymptotically optimal as the SNR grows, and are up to a constant gap of $2$
bits per channel use from the channel capacity without privacy constraints, for
all SNR values. When the channel suffers from fading, the asymmetry between the
servers' channels forces a more complicated solution, which involves a hard
optimization problem. Nevertheless, we provide coding scheme and lower bounds
on the expected achievable retrieval rate which are shown to have the same
scaling laws as the channel capacity, both in the number of servers and the
SNR.
