Research On Super-star Access Structure With 3 Or 4 Super-edges And Its Information Rate

Secret sharing scheme is applied more and more widely in life, especially, which is an important application in information security and cryptography. Secret sharing scheme is a protocol that the the key can be shared among a group of participants, requiring authorized subset of participants can recover their information, whereas any unauthorized subset of participants cannot get it, the secret sharing scheme is prefect. The sect of all authorized subset is called the access structure. Which can be realized by multiple prefect secret sharing schemes. The efficiency can be measured by information.It is necessary to construct an efficient and practical secret sharing schemes, so many experts are studying the optimal information of the access structures, because the index can reflect the information storage capacity and the communication complexity. Calculating the optimal information has very important significance for the construction of secrect sharing schemes and application. Given an access structure, calculating its optimal information rate is very difficult. Giovanni Di Crescenzo et al. calculate the optimal information rate ρ*(E) being less than or equal to 2/3 of non-ideal hyprestar access structures and provide a depiction of a class of hyperstar access structures. However, this characterization is judged by using the regions, so it includes large amount of calculation, not intuitively. In this paper, we first character a kind of special hypergraph access structures using their hyperedges with specific form, then give the optimal information rates and upper and lower bound.The research results of this paper are showed in the following:1. We character and prove all the hyperstar access structures with three hyperedges and represent the ideal and ideal hyperstar access structures with 3 hyperedges as their hyperedges with specific form, then 78 hyperstar access structures with 5、6、7、8 vertices and 3 hyperedges are represented which are all access structures for these hyperstars. A new way can be fined to calculate the optimal information rates of the non-ideal hyperstar access structures2. The hyperstar access structures with four hyperedges have been in-depth researched. we character these hyperstar with the hypedges and the related theorem, and prove that a kind of access structures whose optimal information rates are 2/3 by using λ-decompostion method,3. Hyperstar access structures with 5、6、7、8 vertices and 4 hyperedges are given in the sense of isomorphism. Finally, this paper calculate optimal information rates of some non-ideal hyperstar access structures by the generalized vector space construction, which are 2/3.