An explication of secret sharing schemes
Designs, Codes and Cryptography
Mathematical Programming: Series A and B
Weighted threshold secret sharing schemes
Information Processing Letters
Communications of the ACM
On the Composition of Matroids and Ideal Secret Sharing Schemes
Designs, Codes and Cryptography
Journal of Algebraic Combinatorics: An International Journal
How to (Really) Share a Secret
CRYPTO '88 Proceedings of the 8th Annual International Cryptology Conference on Advances in Cryptology
Generalized Secret Sharing and Monotone Functions
CRYPTO '88 Proceedings of the 8th Annual International Cryptology Conference on Advances in Cryptology
A Representation of a Family of Secret Sharing Matroids
Designs, Codes and Cryptography
Discrete Convex Analysis: Monographs on Discrete Mathematics and Applications 10
Discrete Convex Analysis: Monographs on Discrete Mathematics and Applications 10
Hierarchical Threshold Secret Sharing
Journal of Cryptology
Characterizing Ideal Weighted Threshold Secret Sharing
SIAM Journal on Discrete Mathematics
Ideal Multipartite Secret Sharing Schemes
EUROCRYPT '07 Proceedings of the 26th annual international conference on Advances in Cryptology
Ideal secret sharing schemes whose minimal qualified subsets have at most three participants
Designs, Codes and Cryptography
Multipartite Secret Sharing by Bivariate Interpolation
Journal of Cryptology
Monotone circuits for monotone weighted threshold functions
Information Processing Letters
On secret sharing schemes, matroids and polymatroids
TCC'07 Proceedings of the 4th conference on Theory of cryptography
Secret sharing schemes with bipartite access structure
IEEE Transactions on Information Theory
Correction to “Secret Sharing Schemes With Bipartite Access Structure”
IEEE Transactions on Information Theory
Two Constructions on Limits of Entropy Functions
IEEE Transactions on Information Theory
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
On the optimization of bipartite secret sharing schemes
Designs, Codes and Cryptography
Natural generalizations of threshold secret sharing
ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
Secured hierarchical secret sharing using ECC based signcryption
Security and Communication Networks
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Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention from the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization deals with the properties of the hierarchically minimal sets of the access structure, which are the minimal qualified sets whose participants are in the lowest possible levels in the hierarchy. By using our characterization, it can be efficiently checked whether any given hierarchical access structure that is defined by its hierarchically minimal sets is ideal. We use the well known connection between ideal secret sharing and matroids and, in particular, the fact that every ideal access structure is a matroid port. In addition, we use recent results on ideal multipartite access structures and the connection between multipartite matroids and integer polymatroids. We prove that every ideal hierarchical access structure is the port of a representable matroid and, more specifically, we prove that every ideal structure in this family admits ideal linear secret sharing schemes over fields of all characteristics. In addition, methods to construct such ideal schemes can be derived from the results in this paper and the aforementioned ones on ideal multipartite secret sharing. Finally, we use our results to find a new proof for the characterization of the ideal weighted threshold access structures that is simpler than the existing one.