Abstract
One important result in secret sharing is the Brickell---Davenport theorem: every ideal perfect secret sharing scheme defines a matroid that is uniquely determined by the access structure. We present a generalization of the Brickell---Davenport theorem to the general case, in which non-perfect schemes are also considered. 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 with matroids. Some optimality properties of such schemes are discussed.
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