  We study the following private data transfer problem: Alice has a database of
files. Bob and Cathy want to access a file each from this database (which may
or may not be the same file), but each of them wants to ensure that their
choices of file do not get revealed even if Alice colludes with the other user.
Alice, on the other hand, wants to make sure that each of Bob and Cathy does
not learn any more information from the database than the files they demand
(the identities of which will be unknown to her). Moreover, they should not
learn any information about the other files even if they collude.
  It turns out that it is impossible to accomplish this if Alice, Bob, and
Cathy have access only to private randomness and noiseless communication links.
We consider this problem when a binary erasure broadcast channel with
independent erasures is available from Alice to Bob and Cathy in addition to a
noiseless public discussion channel. We study the
file-length-per-broadcast-channel-use rate in the honest-but-curious model. We
focus on the case when the database consists of two files, and obtain the
optimal rate. We then extend to the case of larger databases, and give upper
and lower bounds on the optimal rate.
