  In this paper, we study the problem of obtaining $1$-of-$2$ string oblivious
transfer (OT) between users Alice and Bob, in the presence of a passive
eavesdropper Eve. The resource enabling OT in our setup is a noisy broadcast
channel from Alice to Bob and Eve. Apart from the OT requirements between the
users, Eve is not allowed to learn anything about the users' inputs. When Alice
and Bob are honest-but-curious and the noisy broadcast channel is made up of
two independent binary erasure channels (connecting Alice-Bob and Alice-Eve),
we derive the $1$-of-$2$ string OT capacity for both $2$-privacy (when Eve can
collude with either Alice or Bob) and $1$-privacy (when no such collusion is
allowed). We generalize these capacity results to $1$-of-$N$ string OT and
study other variants of this problem. When Alice and/or Bob are malicious, we
present a different scheme based on interactive hashing. This scheme is shown
to be optimal for certain parameter regimes. We present a new formulation of
multiple, simultaneous OTs between Alice-Bob and Alice-Cathy. For this new
setup, we present schemes and outer bounds that match in all but one regime of
parameters. Finally, we consider the setup where the broadcast channel is made
up of a cascade of two independent binary erasure channels (connecting
Alice-Bob and Bob-Eve) and $1$-of-$2$ string OT is desired between Alice and
Bob with $1$-privacy. For this setup, we derive an upper and lower bound on the
$1$-of-$2$ string OT capacity which match in one of two possible parameter
regimes.
