Research On Two Kinds Of Access Structures And Their Information

Secret sharing scheme is proposed for leakage or loss of secret key for the safety of it, and it aims at the people who communicate in unsafe channel, which is an important research topic for information security and cryptography.Since the concept of threshold secret sharing scheme is proposed by Shamir and Blakley in 1979, fruitful results are achieved by many scholars after more in-depth research, which make it apply more widely. In a secret sharing scheme, the password designers divided master key into several sub-keys and share it secretly with a group of participants, so that the authorized subset of participants in the collection combine their secret key which can restore the master key. Any non-authorized subset can not get any information about the master key, which means that is the perfect secret sharing scheme. An efficient secret sharing scheme is measured by the information rate, if the information rate of a perfect secret sharing scheme peaked, then this is the ideal secret sharing scheme. It is very important to construct the ideal secret sharing scheme and to study the optimal information rate of access structure. This paper mainly studies the structure of a class of ideal access structures, and the hypergraph access structures and their optimal information rate of it have been studied.The research results of this paper are showed in the following:1. Aiming at the problem that although the tracks distributed to participants are different in Shamir’s type scheme their corresponding minimal access structures may be consistent in essence, the equivalent concept of two tracks is proposed, so we give a method for judging whether two minimal access structures are isomorphism in Shamir’s type scheme. Using a algorithm which can be used to calculate minimal privileged coalitions when they exist in the Shamir’s type schemes and relative combination technique, we can calculate all mutually equivalent track in the Shamir’s type schemes. Particularly, we calculate all the minimal privileged coalitions with 7 participants in finite field F13, and obtain all the tracks that are not mutually equivalent, and thus give all of the ideal minimal assess structures that are not mutually isomorphism by the above the judgment.2. The thesis studied the optimal information rates of a type of access structures on nine participants. First, based on the relationship between access structures and connected hypergraph,226 connected hypergraph access structures with 9 vertices,3 ranks and 4 or 5 hyperedges, which are not mutually isomorphism, are given. Then it is proved that a kind of hyperstar access structures with 4 hyperedges are ideal access structures by using threshold method, and shown that the lower bounds of the optimal information rates of the connected hypergraph with n(5≤n≤11)vertices and 3 ranks are 2/3 by using λ- decompostion method. Finally, using the above results and the relative hypergraphs theory, the optimal information rates for 16 ones of the 226 access structures are computed, the remaining 210 access structures are classified, and the bounds of the optimal information rates of these access structures are estimated.