Cryptography is used everywhere in our society, from simple internet searches to providing security and privacy for our banking systems. A key research area in this field is secure function evaluation, where multiple parties compute a function of their inputs, without revealing them to each other. However, as most cryptographic tasks, current implementations often rely on computational assumptions, which may become insecure in the future. Researchers have explored quantum communications to eliminate such computational assumptions since the 80s, but still challenges remain. For instance, many theoretical protocols using quantum communications rely on perfect single-photon sources. However, in practice, photon sources have a small probability of emitting multiple photons in one pulse, potentially leaking information to a dishonest party. Furthermore, a dishonest party could have secretly tampered the measurement device of the other party and potentially introduced a backdoor in it. The so-called “measurement device independent” setting has then been described, to provide security against this kind of attacks.
In this work, we investigate the possibility of achieving two secure function evaluation primitives - oblivious transfer and bit commitment - in the measurement device independent setting, using imperfect single-photon sources. We already know that under this setting, bit commitment is possible to achieve but oblivious transfer seems hard to achieve. It is still an open question if there exist a protocol to achieve oblivious transfer under this setting. If not, it would be interesting to find new achievable primitives other than oblivious transfer. Here, we study the existing protocols and provide some results towards answering these questions. We start by extending an existing result on the difficulty of achieving oblivious transfer by proving that even a weaker version of this primitive is difficult to achieve under such setting. We also shortly explore the possibility of using an existing reduction from oblivious transfer to bit commitment to see whether we can achieve oblivious transfer from the existing bit commitment protocol. Finally, we show that an existing bit commitment protocol can perform at a better rate by giving a slightly modified security proof.