Journal of Combinatorial Theory Series B
An explication of secret sharing schemes
Designs, Codes and Cryptography
Geometric secret sharing schemes and their duals
Designs, Codes and Cryptography
On the information rate of perfect secret sharing schemes
Designs, Codes and Cryptography
Perfect Secret Sharing Schemes on Five Participants
Designs, Codes and Cryptography
Tight Bounds on the Information Rate of Secret SharingSchemes
Designs, Codes and Cryptography
More information theoretical inequalities to be used in secret sharing?
Information Processing Letters
Communications of the ACM
Improved constructions of secret sharing schemes by applying (λ, ω)-decompositions
Information Processing Letters
Secret Sharing and Non-Shannon Information Inequalities
TCC '09 Proceedings of the 6th Theory of Cryptography Conference on Theory of Cryptography
Secret-sharing schemes: a survey
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
On characterization of entropy function via information inequalities
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
The Minimal Set of Ingleton Inequalities
IEEE Transactions on Information Theory
Decomposition constructions for secret-sharing schemes
IEEE Transactions on Information Theory
The complexity of the graph access structures on six participants
Designs, Codes and Cryptography
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Optimizing the maximum, or average, length of the shares in relation to the length of the secret for every given access structure is a difficult and long-standing open problem in cryptology. Most of the known lower bounds on these parameters have been obtained by implicitly or explicitly using that every secret sharing scheme defines a polymatroid related to the access structure. The best bounds that can be obtained by this combinatorial method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants. By applying this linear programming approach, we improve some of the known lower bounds for the access structures on five participants and the graph access structures on six participants for which these parameters were still undetermined. Nevertheless, the lower bounds that are obtained by this combinatorial method are not tight in general. For some access structures, they can be improved by adding to the linear program non-Shannon information inequalities as new constraints. We obtain in this way new separation results for some graph access structures on eight participants and for some ports of non-representable matroids. Finally, we prove that, for two access structures on five participants, the combinatorial lower bound cannot be attained by any linear secret sharing scheme.