# 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…

## 87 Citations

Some Basic Properties of General Nonperfect Secret Sharing Schemes

- Computer Science, MathematicsJ. Univers. Comput. Sci.
- 1998

It is shown that a compact NSS has some special access hierarchy and it is closely related to a matroid, which means that it meets the equalities of both the bounds and the entropy type bound.

Extending Brickell–Davenport theorem to non-perfect secret sharing schemes

- Mathematics, Computer ScienceDes. Codes Cryptogr.
- 2012

This work presents a generalization of the Brickell–Davenport theorem to the general case, in which non-perfect schemes are also considered, and presents a characterization of the (not necessarily perfect) secret sharing schemes that are associated with matroids.

On Matroids and Non-ideal Secret Sharing

- Computer Science, MathematicsTCC
- 2006

It is proved 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 non-ideal schemes.

Recent Advances in Non-perfect Secret Sharing Schemes

- Computer Science, MathematicsCiE
- 2016

An overview of the techniques for constructing efficient non-perfect secret sharing schemes, bounds on the efficiency of these schemes, and results on the characterization of the ideal ones are provided.

On Matroids and Nonideal Secret Sharing

- Computer Science, MathematicsIEEE Transactions on Information Theory
- 2008

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.

On Ideal Non-perfect Secret Sharing Schemes

- MathematicsSecurity Protocols Workshop
- 1997

This paper first extends the result of Blakley and Kabatianski to general non-perfect SSS using information-theoretic arguments, and establishes that in the light of this generalization, ideal schemes do not always have a matroidal morphology.

On the Information Ratio of Non-perfect Secret Sharing Schemes

- Computer Science, MathematicsAlgorithmica
- 2016

It is proved that there exists a secret sharing scheme for every access function, and the known connections between matroids, polymatroids and perfect secret sharing schemes to the non-perfect case are extended.

Optimal Non-perfect Uniform Secret Sharing Schemes

- Computer Science, MathematicsCRYPTO
- 2014

This work extends the known connections between polymatroids and perfect secret sharing schemes to the non-perfect case and investigates the search of bounds on the information ratio of non- PerfectSecret sharing schemes.

Matroids Can Be Far from Ideal Secret Sharing

- Computer Science, MathematicsTCC
- 2008

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

- Economics, Computer ScienceASIACRYPT
- 1994

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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