Cryptographic primitives such as Bit Commitment (BC) and Oblivious Transfer (OT) are foundational building blocks for two-party Secure Function Evaluations. While unconditional security for BC is impossible in the quantum setting, it can be realised under additional physical assumptions. In particular, the bounded- and noisy-storage models provide a framework where security is guaranteed against adversaries with limited quantum memory. Recent work by Ribeiro and Wehner [1] introduced the first Measurement-Device-Independent (MDI) protocols for BC and OT in the bounded storage model. For the BC protocols, they consider a variant of BC that is called Randomised String Commitment (RSC). They give two MDI-RSC protocols using polarisation-encoded photon sources: one with perfect single-photon emission and another with multi-photon emissions. They also give an MDI-OT protocol using sources with perfect single-photon emission. However, the MDI security for OT using sources with multi-photon emissions remains an open problem.
This thesis investigates the feasibility of MDI-RSC protocols using sources with multi-photon emissions, such as weak coherent pulses (WCP) and spontaneous parametric down-conversion (SPDC) sources. First, we correct a practical error in the existing MDI-RSC protocol by bounding the relevant parameters, ensuring the validity of the original security claims. Second, we analyse the achievable committed string rates while using WCP and SPDC sources. We further consider heralded SPDC sources, which in principle enable single-photon emission, and discuss the impact of imperfect local detectors on their performance and the consequences that has on the protocol implementation. Finally, motivated by techniques from Twin-Field Quantum Key Distribution (TF-QKD), we give a phase-encoded MDI-RSC protocol using coherent states and provide a sketch of the security proof in the bounded-storage model. We also investigate extending the approach to OT. However, this is still a challenge due to the basis-dependent information leakage inherent in phase-encoded coherent states.

This report reviews two quantum key distribution (QKD) protocols: the BB84 protocol and the measurement device independent (MDI) QKD protocol. The goal of this report is to recreate the security proof of the BB84 protocol, to generate the secret key rate for a practical application of the BB84 protocol, both with and without decoy states, and to review the MDI-QKD protocol and look at the advantages it has for practical QKD. The proof security of the BB84 protocol is done by designing an equivalent theoretical protocol and proving its security by bounding the information of the eavesdropper to an exponentially small number. The best distance over which secret key rate can be generated with the practical model of the BB84 protocol that is presented in this report, is 165 km. This is achieved by employing decoy states and is more than three times as far as the model can achieve without decoy states, which gave a distance of 52 km at best. The zero distance key rate of the model with decoy states is R = 1.21 · 10^-2 bits per pulse (bpp), at best. This is an order of magnitude larger than R =1.02 · 10^-3 bpp, which is what the model without decoy states could produce. The decoy state method is therefore an improvement for practical QKD. By reviewing the MDI-QKD protocol it can be concluded that because it eliminates the measurement device side channels, it is very useful for practical QKD, since it makes security analysis simpler and more precise. It also necessarily employs decoy states, which improves the secret key rate. MDI-QKD therefore is a promising protocol to use for future quantum communications.