2007 | OriginalPaper | Buchkapitel
On Secret Sharing Schemes, Matroids and Polymatroids
verfasst von : Jaume Martí-Farré, Carles Padró
Erschienen in: Theory of Cryptography
Verlag: Springer Berlin Heidelberg
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
One of the main open problems in secret sharing is the characterization of the access structures of ideal secret sharing schemes. As a consequence of the results by Brickell and Davenport, every one of those access structures is related in a certain way to a unique matroid.
Matroid ports are combinatorial objects that are almost equivalent to matroid-related access structures. They were introduced by Lehman in 1964 and a forbidden minor characterization was given by Seymour in 1976. These and other subsequent works on that topic have not been noticed until now by the researchers interested on secret sharing.
By combining those results with some techniques in secret sharing, we obtain new characterizations of matroid-related access structures. As a consequence, we generalize the result by Brickell and Davenport by proving that, if the information rate of a secret sharing scheme is greater than 2/3, then its access structure is matroid-related. This generalizes several results that were obtained for particular families of access structures.
In addition, we study the use of polymatroids for obtaining upper bounds on the optimal information rate of access structures. We prove that every bound that is obtained by this technique for an access structure applies to its dual structure as well.
Finally, we present lower bounds on the optimal information rate of the access structures that are related to two matroids that are not associated with any ideal secret sharing scheme: the Vamos matroid and the non-Desargues matroid.