Security in quantum key distribution protocols.
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Quantum theory forms one of the most studied theories of nature. It has inevitably led to a number of different research areas. One of the breakthroughs was the development of quantum cryptography which now forms one of the most advanced subjects in this field. One aspect of quantum cryptography known as quantum key distribution (QKD) is the art of generating a secure key which is used to encode a secret message between two legitimate parties conventionally known as Alice, the sender, and Bob, the receiver, in the presence of an eavesdropper, known as Eve. The goal of QKD is to guarantee security in the presence of an eavesdropper, who has access to the communication channel and unlimited technology, to ensure she is unable to obtain useful information about the message. Since the generated key is random and unknown to Eve, she is unable to learn anything about the encoded message. The most interesting and amazing phenomenon about a QKD scheme is that its security is based on the laws of physics rather than the computational or mathematical algorithms as in classical cryptography. The motivation behind this thesis is to provide a study of both theory and practical methods of security in QKD protocols. Concerning theory, we brie y clarify how the laws of physics allow the security of QKD protocols which are used for secret communication. Moreover, we give definitions, analysis and evaluation of tools used for proving the security of different classes of QKD protocols. On the practical side, we show an implementation of the Bennett 1992 (B92) protocol and a high dimensional mutually unbiased basis QKD protocol. In particular, we derive an irreducible lower bound of the uncertainty on the simultaneous measurement of observables when one use the Tsallis entropy to express the quantum uncertainty relation. This shows a possibility of using the Tsallis entropies for quantifying information in QKD protocols. We highlight that the Tsallis entropies have not been extensively investigated for this application. We will also demonstrate an implementation of the B92 QKD protocol by using the id3100 Clavis2 system at our laboratory at the University of KwaZulu-Natal. The id3100 Clavis2 system has been traditionally used for the implementation of the Bennett-Brassard 1984 (BB84) and Scarani-Acín-Ribordy-Gisin 2004 (SARG04) protocols. We investigate also the secure key rates in the B92 protocol by using the Rènyi entropies and the uncertainty relations which have been introduced recently. Lastly, we extend our work by showing an implementation of a high dimensional filter based QKD protocol. This QKD protocol is based on mutually unbiased bases (MUBs) which are implemented by means of photons carrying orbital angular momentum (OAM). In particular, we show that by encoding in high dimension of MUBs leads to an increase in the key generation rate per photon.