Bipartite secret sharing and staircases

Bipartite secret sharing schemes have a bipartite access structure in which the set of participants is divided into two parts and all participants in the same part play an equivalent role. Such a bipartite scheme can be described by a "staircase": the collection of its minimal points. The complexity of a scheme is the maximal share size relative to the secret size; and the $\kappa$-complexity of a structure is the best lower bound provided by the entropy method. A structure is $\kappa$-ideal if it has $\kappa$-complexity 1. Motivated by the abundance of open problems in this area, the main results can be summarized as follows. First, a new characterization of $\kappa$-ideal multipartite access structures is given which offers a straightforward and simple approach to specify ideal bipartite and tripartite structures. Second, $\kappa$-complexity is determined for a range of bipartite access structures, including those determined by two points, staircases with equal widths and heights, and staircases with all heights 1. Third, matching linear schemes are presented for some non-ideal cases, including staircases where all heights are 1 and all widths are equal. Finally, finding the Shannon complexity of a bipartite access structure can be considered as a discrete submodular optimization problem. An interesting and intriguing continuous version is defined which might give further insight to the large-scale behavior of these optimization problems.