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- Type of Document: M.Sc. Thesis
- Language: Farsi
- Document No: 39731 (05)
- University: Sharif University of Technology
- Department: Electrical Engineering
- Advisor(s): Eghlidos, Taraneh
- Abstract:
- Secret sharing scheme is a method for distributing the secret (secret information) among a set of participants in such a way that only the authorized sets can recover the secret and the unauthorized sets could not. In a perfect secret sharing scheme, unauthorized sets cannot get any additional (i.e. a posteriori) information about the possible value of the secret. In these schemes, to prevent information leakage and increase communication efficiency, the size of the share should be as close to the secret size as possible. In other words, finding the shares with reasonable size which results in an optimal information rate for a given access structure, improves the efficiency of the scheme. Because of comprehensiveness of general access structures, including threshold ones, we study the optimal information rate for these access structures in this thesis. For this purpose, the access structures are modelled with graphs (for those consisting subsets with cardinality 2) and hypergraphs (for general case). Computing the information rate of access structures, modelled by graphs and the hypergraphs have widely been considered in the literatures and some bounds have been proposed. In this thesis, we aim to improve these bounds. For this purpose, we obtain the optimal information rate of access structures based on trees and then propose a lower bound for the information rate of access structures based on hypertrees. After investigating the information rate of graphs with one cycle and presenting the upper and lower bounds, we achieve the optimal information rates for cycles and graphs with one cycle, having length three, as the shortest cycle. Also, upper and lower bounds for the information rate of access structures representable by a special class of hypergraphs, are given. The main objective of this thesis is to construct secret sharing schemes, having access structure with high or possibly optimal information rate. It has already been shown that most of the existing secret sharing schemes have geometric interpretations, that is most of such schemes are covered by geometric ones. Hence, we utilize geometric schemes, as a class of
multilinear ones.
- Keywords:
- Information Rate ; Hypergraph ; Secret Sharing Scheme ; Graphs ; Geometric Schemes ; Subspace Configuration
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