International Association
for Cryptologic Research

One important result in secret sharing is Brickell-Davenport Theorem: every ideal perfect secret sharing scheme defines a matroid that is uniquely determined by the access structure. Even though a few attempts have been made, there is no satisfactory definition of ideal secret sharing scheme for the general case, in which non-perfect schemes are considered as well. Without providing another unsatisfactory definition of ideal non-perfect secret sharing scheme, we present a generalization

of Brickell-Davenport Theorem to the general case. After analyzing that result under a new point of view and identifying its combinatorial nature, we present a characterization of the (not necessarily perfect)

secret sharing schemes that are associated to matroids. Some optimality properties of such schemes are discussed.