Security of quantum key distribution and quantum state sharing

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.