Deterministic quantum key distribution protocols : development and security analysis

Abstract

The wonderful marriage between quantum mechanics and computation and information theory has brought about several interesting paradigmatic changes in matters related to computing and information security. In this regard, it results on one hand, the hope for true supercomputers, giving rise to the possibility of new algorithms and resources to solve computational issues which once were deemed nontractable or hard thus eventually spelling the obsoleteness of modern cryptographic systems relying on computationally hard problems. On the other hand, it promises cryptographic possibilities whose unconditional security relies on the laws of physics. The first quantum cryptography or more specifically quantum key distribution (QKD) protocol was introduced as far back as 1984, commonly referred to as BB84. The emergence of deterministic schemes, which stand in contrast to the nondeterministic nature of many earlier protocols, was noted later. We focus on two way deterministic schemes and emphasise the study in the context of key distribution. In consideration of the existing qubit based two way deterministic protocols, we develop in our thesis a nontrivial generalisation of such a protocol using only 2 mutually unbiased bases (MUB) to one with the maximal number, 3 MUB. This is despite the non existence of the universal-NOT which would forbid a trivial generalisation. We show that the protocol has a higher level of security reflected in both the probability of detecting an eavesdropper as well as its robustness against noise in the context of an eavesdropper executing an Intercept Resend attack. The protocol saturates the constraints of the Holevo bound and qualifies as a protocol in the Holevo limit. We later extend the protocol to exploiting an additional number of dimension of the Hilbert space, thus making use of the qutrit. This is in the same spirit as predecessing works which extend BB84 to 3 dimensional quantum systems. We proceed to prove a no-go theorem which has a similar effect to that of the universal-NOT. These protocols which do not resort to entanglement based setups, are envisioned to be well within the practical capacity of current technology. While the qubit and qutrit based two way deterministic protocols require specific recipes for generalisation to include the exhaustive number of MUB, the relevant arguments are made explicit when we consider the generalisation of such protocol to the use of qudits, quantum systems of arbitrary dimensions although in our work we consider only the case for prime numbered dimension. Referring to the common primitive in the above protocols in encoding onto unknown quantum states as ‘blind encoding’, we point out to a no-go theorem which properly generalises the inability to shift between orthogonal states of arbitrary qudits. We further study the necessary conditions for generalising to using the maximal number of MUB afforded as well as proposing a proper recipe to ensure a protocol of unitary efficiency.