  Bidirectional relaying, where a relay helps two user nodes to exchange equal
length binary messages, has been an active area of recent research. A popular
strategy involves a modified Gaussian MAC, where the relay decodes the XOR of
the two messages using the naturally-occurring sum of symbols simultaneously
transmitted by user nodes. In this work, we consider the Gaussian MAC in
bidirectional relaying with an additional secrecy constraint for protection
against a honest but curious relay. The constraint is that, while the relay
should decode the XOR, it should be fully ignorant of the individual messages
of the users. We exploit the symbol addition that occurs in a Gaussian MAC to
design explicit strategies that achieve perfect independence between the
received symbols and individual transmitted messages. Our results actually hold
for a more general scenario where the messages at the two user nodes come from
a finite Abelian group, and the relay must decode the sum within the group of
the two messages. We provide a lattice coding strategy and study optimal rate
versus average power trade-offs for asymptotically large dimensions.
