Quantum key distribution (QKD) and quantum secret sharing play a pivotal role in securing con dential information. QKD allows legitimate parties to create a cryptographic key which they can use to communicate privately without being intercepted by a malicious eavesdropper. QKD exploits the laws of quantum mechanics such as the Heisenberg uncertainty principle, the no cloning theorem and the principle of entanglement to detect any eavesdropper who tries to gain the knowledge of the secret key. Another method of securing con dential information is quantum secret sharing (QSS). QSS is a cryptographic protocol aimed at distributing secret information to untrusted agents. In this method, a secret can be distributed to agents in a way that some subsets of agents can collaborate to fully recover the secret key or message while all other subsets have insu cient information to enable them to reconstruct the key or message even if in possession of unlimited computing power. The objective of this thesis is to review various security proof methods and to propose security proofs for two QKD protocols. In the rst protocol, a key is generated by using Greenberger-Horne-Zeilinger (GHZ) states in an environment of unknown and slowly varying reference frame. We also compute the secret key rate for this protocol. In the second protocol, a second security proof is derived for QKD protocol in which Eve's information is conditioned on a random variable which describes all projective measurements performed by communicating parties. Furthermore, we propose two quantum state sharing (QSTS) schemes which uses GHZ states and Einstein-Podolsky-Rosen (EPR) states to share an unknown three-particle state to n agents. Firstly, we introduce the ve party QSTS of an arbitrary three particle unknown state where Alice starts by sharing four GHZ entangled states with her four agents and performs three GHZ state measurements on her particles followed by two single particle measurements on the Hadamard basis. One of the agents, Bob1, performs single measurements on her particle and the three other agents perform unitary transformations on their particles to recover the unknown state. Subsequently, we propose the generalised multiparty QSTS of an arbitrary three particle state. Secondly, we present a scheme in which Alice shares an arbitrary three-particle unknown state with Bob1 and Bob2. Alice starts by sharing six EPR pairs with her agents and then performs joint three-particle GHZ state measurements on her particles. Bob1, who acts as controller performs a product measurement x x x whilst Bob2 retrieves the original state by performing three unitary operations on his particles. Thereafter, we propose the generalised multi-party QSTS of an arbitrary three particle state.

Theses (MSc-Physics)----Botswana International University of Science and Technology, 2017