  We introduce the problem of random symmetric private information retrieval
(RSPIR). In canonical PIR, a user downloads a message out of $K$ messages from
$N$ non-colluding and replicated databases in such a way that no database can
know which message the user has downloaded (user privacy). In SPIR, the privacy
is symmetric, in that, not only that the databases cannot know which message
the user has downloaded, the user itself cannot learn anything further than the
particular message it has downloaded (database privacy). In RSPIR, different
from SPIR, the user does not have an input to the databases, i.e., the user
does not pick a specific message to download, instead is content with any one
of the messages. In RSPIR, the databases need to send symbols to the user in
such a way that the user is guaranteed to download a message correctly (random
reliability), the databases do not know which message the user has received
(user privacy), and the user does not learn anything further than the one
message it has received (database privacy). This is the digital version of a
blind box, also known as gachapon, which implements the above specified setting
with physical objects for entertainment. This is also the blind version of
$1$-out-of-$K$ oblivious transfer (OT), an important cryptographic primitive.
We study the information-theoretic capacity of RSPIR for the case of $N=2$
databases. We determine its exact capacity for the cases of $K = 2, 3, 4$
messages. While we provide a general achievable scheme that is applicable to
any number of messages, the capacity for $K\geq 5$ remains open.
