| Summary: | 碩士 === 國立暨南國際大學 === 資訊工程學系 === 95 === A secret sharing scheme is a method to distribute a secret, also called master key, among a set of participants, such that only qualified subsets of the participants can recover the secret. A secret sharing scheme is perfect if any unqualified subset obtains no information regarding the master key. The collection of qualified subsets is called the access structure.
In a hypergraph, if the size of any edge is equal to $r$, the hypergraph is called an $r$-uniform hypergraph. Similarity, if the size of any edge is equal to either $r_1$ or $r_2$ in a hypergraph, the hypergraph is called an ($r_1, r_2$)-uniform hypergraph. And, if the possible size of any edge is $r_1, r_2$ or $r_3$ in a hypergraph,
the hypergraph is called an ($r_1, r_2, r_3$)-uniform hypergraph. Given any hypergraph $G$, a $G$-based access structure is an access structure which using $G$ present the access structure, where a vertex denote a participant and the edge set denote the minimal access structure of a secret sharing scheme.
In this thesis, we propose four perfect secret sharing schemes for $r$-uniform, ($r_1, r_2$)-uniform, ($r_1, r_2, r_3$)-uniform and general hypergraph-based access structures (called $r$-HA, ($r_1, r_2$)-HA, ($r_1, r_2, r_3$)-HA and G-HA scheme respectively). Moreover, we modify G-HA scheme such that it: 1. can verify the shares (verification) and detect the cheater (detection),
2. can be reused, that is, will be multi-use secret sharing scheme, and 3. will be a multi-use secret sharing scheme with verification and detection.
At last, this thesis shows that $r$-HA, ($r_1, r_2$)-HA, ($r_1, r_2, r_3$)-HA and G-HA schemes are all more efficient secret sharing scheme than the scheme be hold by Tochikubo, Uyematsu and Matsumoto in 2005 for the respective same access structure.
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