Nonperfect Secret Sharing Schemes and Matroids

@inproceedings{Kurosawa1993NonperfectSS,
  title={Nonperfect Secret Sharing Schemes and Matroids},
  author={Kaoru Kurosawa and Koji Okada and Keiichi Sakano and Wakaha Ogata and Shigeo Tsujii},
  booktitle={EUROCRYPT},
  year={1993}
}
This paper shows that nonperfect secret sharing schemes (NSS) have matroid structures and presents a direct link between the secret sharing matroids and entropy for both perfect and nonperfect schemes. We define natural classes of NSS and derive a lower bound of |Vi| for those classes. "Ideal" nonperfect schemes are defined based on this lower bound. We prove that every such ideal secret sharing scheme has a matroid structure. The rank function of the matroid is given by the entropy divided by… 
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TLDR
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TLDR
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  • Computer Science, Mathematics
    IEEE Transactions on Information Theory
  • 2008
TLDR
This work proves that the fact that an access structure is induced by a matroid implies lower and upper bounds on the size of the domain of shares of subsets of participants even in nonideal schemes (as long as the shares are still relatively short) and generalized results of Brickell and Davenport for ideal schemes.
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TLDR
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TLDR
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TLDR
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Matroids Can Be Far from Ideal Secret Sharing
TLDR
The first proof that there exists an access structure induced by a matroid which is not nearly ideal is presented and a better lower bound is presented that applies only to linear secret-sharing schemes realizing the access structures induced by the Vamos matroid.
Lower Bound on the Size of Shares of Nonperfect Secret Sharing Schemes
TLDR
A general lower bound on ¦V i ¦ is presented, which includes the previous lower bounds for perfect SSs and nonperfect SSs as special cases and the optimum size of V i for a certain access hierarchy is determined.
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