Research On Access Structure And Information Rate Of Several Kinds Of Secret Sharing Schemes

Secret sharing is an important branch of modern cryptography and an important tool for the information security and data privacy. It plays a key role in safely pre-serving, transmitting and legally using the important information and the secret data. Utilizing the secret sharing to keep a secret is not only prevented power from being kept centralized, but also kept security and integrity of informations. Thus, secret sharing has been widely used in politics, economy, military and diplomacy.The purpose of this paper is to study on the various of access structures of secret sharing schemes, including their determination, implementation and information rates, such as how to obtain the ideal access structures of the secret sharing schemes and construct multi-use and dynamic multi-secret sharing schemes for a type of access structure based on minimal linear codes; how to find bounds on the possible efficiency that any such scheme can achieve for a certain access structure; based on privileged coalitions, how to share multiple secrets for general access structures in an ideal multi-secret sharing scheme. Moreover, the last chapter discuss the optimal information rate of quantum access structures. Specifically, our main contributions are as follows:1. Access structures based on minimal linear codesThere are several methods to construct ideal access structures, one of which is based on coding theory. However, in general, determining the ideal access structures of the secret sharing schemes based on linear codes is very hard. This paper first proposes the concept of minimal linear codes so as to make it easier to determine the ideal access structures of the schemes based on the duals of minimal linear codes. It is proved that the shortening codes of minimal linear codes are also minimal ones. Then in order to study the conditions whether several types of irreducible cyclic codes are minimal, we investigate the weight enumerators of certain irreducible cyclic codes by means of cyclotomic classes and Gaussian periods. On the basis of our aforementioned studies, ideal access structures and present specific examples through programming are obtained. Furthermore, we propose a multi-secret sharing scheme with general access structures, which is proposed according to minimal linear codes. In this scheme, multiple secrets can be shared in one session, and each participant needs to keep only one reusable secret. The participant and the secret can be dynamically operated without updating any participant’s share. Moreover, the property of minimal linear codes makes it easier to determine the access structures of the schemes based on the duals of minimal linear codes. This work is described in Chapter 3.2. Access structures based on graph theoryThe information rate is an important metric of the performance of a secret-sharing scheme. Based on the relationship between these access structures and their connected graphs, we consider 493 non-isomorphic connected graph access structures with7,8,9vertices, and either determine or bound the optimal information rate in each case. We obtain exact values for the optimal information rate for 423 cases. Moreover, We first give a description of a class of graph access structures for which the infor-mation rate is at most 3/5 or 4/7 in terms of information theoretical and apply some of the constructions to determine lower bounds on the information rate. Regarding information rate, we conclude with a full listing (Table 4.1) of the known optimal infor-mation rate (or bounds on the optimal information rate) for all graph access structures we have investigated. This work is described in Chapter 4.3. Ideal access structures based on privileged coalitionsBased on privileged coalitions, we show that most of the existing multi-secret sharing schemes based on Shamir threshold secret sharing are not perfect. Then in accordance with a public problem about privileged coalitions we solved, we devise an ideal multi-secret sharing scheme for families of access structures, which possesses more vivid qualified sets than that of the threshold scheme. This work is described in Chapter 5.4. The optimal information rate of quantum access structuresThe information rate is an important metric of the performance of a quantum-secret-sharing scheme. We characterize the quantum access structures by means of the theory of hypergraph. Furthermore, we derive the optimal information rate and the construction of perfect quantum-secret-sharing schemes corresponding to 13 quantum access structures with at most four players which are given in terms of the relationship between certain access structures and hypergraphs. The exact values for the optimal information rate in 5 of the 13 access structures are computed and the relevant con-struction of perfect secret sharing schemes is discussed. At the same time, the upper bounds for the information rate of other 8 quantum access structures are computed. This work is described in Chapter 6.