Research Of Subspace Clustering Algorithms Towards The Security Analysis Of Quantum Key Distribution

Quantum secure communication is a new solution proposed by quantum infor-mation science to the requirements of modern secure communication. Because of its unconditional security, the rapid development of quantum secure communication in re-cent two decades has a very promising futrue and is of great significance in both theory and application. However, subject to the constraints of the current level of technology, the practical implementions of quantum cryptography today still faces security threats. In recent years, various quantum attack strategies were found efficient to eavesdrop on practical quantum secure communication systems, challenging the safety of modern quantum secure communication systems.One of the threatening attacks called collective attack utilizes joint measurements on multiple qubits to eavesdrop on the quantum secure communication protocol. The feasibility and possibility of success of joint measurements is the key to disign and analyze this kind of attack. The discovery of qubits that are feasible to be joint measured and the analysis of the possible attacks require us to segment the system state set into unambiguously discriminable pure state subsets. However, the theory of quantum state discrimination does not include a detailed method and discussion for such a purpose.In order to improve the security of quantum key distribution protocols, the con-struction, efficiency and defence of a kind of collective attacks based on quantum pure-state set discrimination are discussed both in theory and method in this dissertation. The main contents and innovations is summarized as follows.1. The quantum pure-state set partitioning problem is proposed and studied. The study of unambiguously discrimination of quantum pure-state set is crucial for the analysis and design of collective attacks. The existing work of unambiguously set discrimination is to discriminate among known quantum state sets. However, in a quantum key distribution protocol, often the subspace structure of the state set of qubits transmitted in the quantum channel is unknown. It is necessary to par-tition the state set into unambiguously discriminable subsets in order to discrim-inate among them, which satisfy a certain linearly independent condition. The structural feature of quantum pure-state set partitioning problem and the relation between number of subsets and successful probabiliy of unambiguous discrimi-nation are studied. Further, a recursive subspace clustering algorithm is proposed to solve the problem. Experiments show that the proposed algorithm is capa-ble to return all the solutions of the problem. Using the proposed algorithm, the construction and efficiency of a kind of collective attack is analyzed.2. The algorithm strategy is modified to improve the efficiency when solving the quantum pure-set partitioning problem of composite quantum system consisting of multiple independent subsystems. For the unambiguous discrimination be-tween the subsets of composite quantum system state sets, it is equivalent to solve the quantum pure-set partitioning problem for its independent subsystems respec-tively, and the subsystems result in the disjoint subspace structure in the compos-ite quantum system state set. Through clustering and dimension reduction, these subsystems can be extracted, resulting in a much smaller scale to solve. The modified strategy first clusters the disjoint subspaces based on sparse subspace clustering, then reduce the redundant dimensions in these subspaces through the standard pricipal component analyse, and apply subset segmentation to the re-duced data at last. The results of our simulation shows that when facing the prob-lem of quantum pure-set partitioning problem of this kind of composite quantum systems, the modified strategy has a very significant acceleration with compared to appling the subset segmentation algorithm directly.3. Hwang protocol is improved to defend a kind of collective attack based on unam-biguous set discrimination. As been proved, the real implementation of Hwang protocol using weak coherent pulses can be eavesdropped by two kinds of collec-tive attacks. From the view point of cipher modes, the defect of error propagation of a modified Hwang protocol utilizing cipher block chainning mode is analyzed. This defect will result in significant increase of error rate in the raw key. An-other cipher mode, namely the synchronous stream cipher mode is introduced to improve the Hwang protocol. It is proved capable to defend against collective attacks based on unambiguous set discrimination and has no error-propagation defect.