  Among the most studied tasks in Quantum Cryptography one can find Bit
Commitment (BC) and Oblivious Transfer (OT), two central cryptographic
primitives. In this paper we propose for the first time protocols for these
tasks in the measurement-device independent (MDI) settings and analyze their
security. We analyze two different cases: first we assume the parties have
access to perfect single photon sources (but still in the presence of noise and
losses), and second we assume that they only have imperfect single photon
sources. In the first case we propose a protocol for both BC and OT and prove
their security in the Noisy Quantum Storage model. Interestingly, in the case
where honest parties do not have access to perfect single photon sources, we
find that BC is still possible, but that it is "more difficult" to get a secure
protocol for OT: We show that there is a whole class of protocols that cannot
be secure. All our security analyses are done in the finite round regime.
