Elements of information theory
Elements of information theory
Some improved bounds on the information rate of perfect secret sharing schemes
Journal of Cryptology
Journal of Combinatorial Theory Series B
Nonperfect secret sharing schemes and matroids
EUROCRYPT '93 Workshop on the theory and application of cryptographic techniques on Advances in cryptology
Information Processing Letters
Designs, Codes and Cryptography
Communications of the ACM
Generalized Secret Sharing and Monotone Functions
CRYPTO '88 Proceedings of the 8th Annual International Cryptology Conference on Advances in Cryptology
Secret sharing schemes with three or four minimal qualified subsets
Designs, Codes and Cryptography
Robust computational secret sharing and a unified account of classical secret-sharing goals
Proceedings of the 14th ACM conference on Computer and communications security
On matroids and non-ideal secret sharing
TCC'06 Proceedings of the Third conference on Theory of Cryptography
On characterization of entropy function via information inequalities
IEEE Transactions on Information Theory
Two Constructions on Limits of Entropy Functions
IEEE Transactions on Information Theory
Networks, Matroids, and Non-Shannon Information Inequalities
IEEE Transactions on Information Theory
Secret Sharing and Non-Shannon Information Inequalities
TCC '09 Proceedings of the 6th Theory of Cryptography Conference on Theory of Cryptography
An impossibility result on graph secret sharing
Designs, Codes and Cryptography
On the optimization of bipartite secret sharing schemes
ICITS'09 Proceedings of the 4th international conference on Information theoretic security
Secret-sharing schemes: a survey
IWCC'11 Proceedings of the Third international conference on Coding and cryptology
Ideal secret sharing schemes for useful multipartite access structures
IWCC'11 Proceedings of the Third international conference on Coding and cryptology
Optimal complexity of secret sharing schemes with four minimal qualified subsets
Designs, Codes and Cryptography
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
On the optimization of bipartite secret sharing schemes
Designs, Codes and Cryptography
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In a secret-sharing scheme, a secret value is distributed among a set of parties by giving each party a share. The requirement is that only predefined subsets of parties can recover the secret from their shares. The family of the predefined authorized subsets is called the access structure. An access structure is ideal if there exists a secret-sharing scheme realizing it in which the shares have optimal length, that is, in which the shares are taken from the same domain as the secrets. Brickell and Davenport (J. of Cryptology, 1991) proved that ideal access structures are induced by matroids. Subsequently, ideal access structures and access structures induced by matroids have received a lot of attention. Seymour (J. of Combinatorial Theory, 1992) gave the first example of an access structure induced by a matroid, namely the Vamos matroid, that is non-ideal. Beimel and Livne (TCC 2006) presented the first non-trivial lower bounds on the size of the domain of the shares for secret-sharing schemes realizing an access structure induced by the Vamos matroid. In this work, we substantially improve those bounds by proving that the size of the domain of the shares in every secret-sharing scheme for those access structures is at least k1.1, where k is the size of the domain of the secrets (compared to k + Ω(√k) in previous works). Our bounds are obtained by using non-Shannon inequalities for the entropy function. The importance of our results are: (1) we present the first proof that there exists an access structure induced by a matroid which is not nearly ideal, and (2) we present the first proof that there is an access structure whose information rate is strictly between 2/3 and 1. In addition, we present a better lower bound that applies only to linear secret-sharing schemes realizing the access structures induced by the Vamos matroid.