O. Farrŕs, J.R. Metcalf-Burton, C. Padró, L. Vázquez.
On the Optimization of Bipartite Secret Sharing Schemes
Designs, Codes and Cryptography 63 (2012) 255-271. A previous version of this paper appeared in Fourth International Conference on Information Theoretic Security ICITS 2009, Lecture Notes in Computer Science 5973 (2010) 93-109. Shizuoka, Japan, 2009.
Optimizing the ratio between the maximum length of the shares and the length of the secret value in secret sharing schemes for general access structures is an extremely difficult and long-standing open problem. In this paper, we study it for bipartite access structures, in which the set of participants is divided in two parts, and all participants in each part play an equivalent role. We focus on the search of lower bounds by using a special class of polymatroids that is introduced here, the bipartite ones. We present a method based on linear programming to compute, for every given bipartite access structure, the best lower bound that can be obtained by this combinatorial method. In addition, we obtain some general lower bounds that improve the previously known ones, and we construct optimal secret sharing schemes for a family of bipartite access structures.
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