On Secret Sharing Schemes, Matroids and Polymatroids

Abstract. 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 ...

• Type: Conference
• Year: 2007
• Where: TCC
• Authors: Jaume Martí-Farré, Carles Padró
• Keywords: Access Structures, Cryptography, Secret Sharing, Secret Sharing Schemes, TCC 2007