Optimal Eavesdropping in Quantum Cryptography

Abstract

Quantum key distribution (QKD) has raised some promise for more secured communica- tion than its classical counterpart. It allows the legitimate parties to detect eavesdropping which introduces error in the channel. If disturbed, there are ways to distill a secure key within some threshold error-rate. The amount of information gained by an attacker is generally quantified by (Shannon) mutual information. Knowing the maximum amount of information that an intruder can gain is important for post-processing purposes, and we mainly focus on that side in the thesis. Rényi information is also useful especially when post-processing is considered. The scope of this thesis is to first describe some relevant ingredients for QKD and then study some open-ended issues. We mostly focus on the BB84 protocol and some issues relating optimal eavesdropping on it when each information-carrying particles are attacked individually. However, our results and techniques can also be applied for other protocols and different eavesdropping strategies. We felt a few other eavesdropping tech- niques worthy to analyze in that line, despite limitations to achieve newer results. First we study the optimal eavesdropping technique on the BB84 protocol and show that the optimal information can be achieved in infinitely many different ways to interact and measure the information-carriers. Although they are mathematically equivalent in some sense, that variety may help when designing the eavesdropping setup. However, it was not clear whether more such optimal interactions exist or not. This has lead us to derive them through a chain of necessary and sufficient conditions (NSC), which are shown to be in a one-to-one correspondence with the earlier interactions. In this process we arrive at a new NSC restricting attackers particles to a specific orienta- tion, establishing the geometry of the attack more explicitly than earlier. Some explicit connections are shown with other modes of gleaning information like cloning. Nevertheless, for practical purposes all an attacker requires is the evolution that en- tangles her ancilla with the senders particle, and the corresponding measurement that will lead her to optimal information gain. This is generally neglected in the literature as they exhibit a specific interaction. In our case, having infinitely many options to interact, we felt it better to address the issue of findings optimal evolutions. Overall, we have added more mathematical structures in the framework of optimal eavesdropping. We wanted to analyse the more generalized ways to attack, where a whole chunk of information-carrying particles can be evolved and then measured at a go. The process becomes complex to tackle when the chunks go bigger. Yet, we have explained the mathematical details of some of the existing results to point out the difficulties.