Completeness theorems for non-cryptographic fault-tolerant distributed computation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Multiparty unconditionally secure protocols
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Elements of information theory
Elements of information theory
Journal of Combinatorial Theory Series B
An explication of secret sharing schemes
Designs, Codes and Cryptography
Nonperfect secret sharing schemes and matroids
EUROCRYPT '93 Workshop on the theory and application of cryptographic techniques on Advances in cryptology
On the information rate of perfect secret sharing schemes
Designs, Codes and Cryptography
On the information rate of secret sharing schemes
Theoretical Computer Science
Perfect Secret Sharing Schemes on Five Participants
Designs, Codes and Cryptography
A Linear Construction of Secret Sharing Schemes
Designs, Codes and Cryptography
Designs, Codes and Cryptography
Access Control and Signatures via Quorum Secret Sharing
IEEE Transactions on Parallel and Distributed Systems
Weighted threshold secret sharing schemes
Information Processing Letters
Probability of shares in secret sharing schemes
Information Processing Letters
Communications of the ACM
On the Composition of Matroids and Ideal Secret Sharing Schemes
Designs, Codes and Cryptography
Generalized Secret Sharing and Monotone Functions
CRYPTO '88 Proceedings of the 8th Annual International Cryptology Conference on Advances in Cryptology
Shared Generation of Authenticators and Signatures (Extended Abstract)
CRYPTO '91 Proceedings of the 11th Annual International Cryptology Conference on Advances in Cryptology
A Representation of a Family of Secret Sharing Matroids
Designs, Codes and Cryptography
Secret sharing schemes with three or four minimal qualified subsets
Designs, Codes and Cryptography
On the Power of Nonlinear Secret-Sharing
SIAM Journal on Discrete Mathematics
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
General secure multi-party computation from any linear secret-sharing scheme
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
Secret sharing schemes on access structures with intersection number equal to one
SCN'02 Proceedings of the 3rd international conference on Security in communication networks
Characterizing ideal weighted threshold secret sharing
TCC'05 Proceedings of the Second international conference on Theory of Cryptography
On codes, matroids and secure multi-party computation from linear secret sharing schemes
CRYPTO'05 Proceedings of the 25th annual international conference on Advances in Cryptology
Secret sharing schemes with bipartite access structure
IEEE Transactions on Information Theory
Ideal Multipartite Secret Sharing Schemes
EUROCRYPT '07 Proceedings of the 26th annual international conference on Advances in Cryptology
An impossibility result on graph secret sharing
Designs, Codes and Cryptography
Weakly-private secret sharing schemes
TCC'07 Proceedings of the 4th conference on Theory of cryptography
On secret sharing schemes, matroids and polymatroids
TCC'07 Proceedings of the 4th conference on Theory of cryptography
Matroids can be far from ideal secret sharing
TCC'08 Proceedings of the 5th conference on Theory of cryptography
On the optimization of bipartite secret sharing schemes
ICITS'09 Proceedings of the 4th international conference on Information theoretic security
Finding lower bounds on the complexity of secret sharing schemes by linear programming
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Ideal secret sharing schemes whose minimal qualified subsets have at most three participants
SCN'06 Proceedings of the 5th international conference on Security and Cryptography for Networks
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Secret-sharing schemes are a tool used in many cryptographic protocols. In these schemes, a dealer holding a secret string distributes shares to the parties such that only authorized subsets of participants can reconstruct the secret from their shares. The collection of authorized sets is called an access structure. An access structure is ideal if there is a secret-sharing scheme realizing it such that the shares are taken from the same domain as the secrets. Brickell and Davenport (J. of Cryptology, 1991) have shown that ideal access structures are closely related to matroids. They give a necessary condition for an access structure to be ideal – the access structure must be induced by a matroid. Seymour (J. of Combinatorial Theory B, 1992) showed that the necessary condition is not sufficient: There exists an access structure induced by a matroid that does not have an ideal scheme. In this work we continue the research on access structures induced by matroids. Our main result in this paper is strengthening the result of Seymour. We show that in any secret sharing scheme realizing the access structure induced by the Vamos matroid with domain of the secrets of size k, the size of the domain of the shares is at least $k + \Omega (\sqrt{k})$. Our second result considers non-ideal secret sharing schemes realizing access structures induced by matroids. We prove 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 (this generalized results of Brickell and Davenport for ideal schemes).