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Bipartite secret sharing schemes have a bipartite access structure in whichthe set of participants is divided into two parts and all participants in thesame part play an equivalent role. Such a bipartite scheme can be described bya \emph{staircase}: the collection of its minimal points. The complexity of ascheme is the maximal share size relative to the secret size; and the$\kappa$-complexity of an access structure is the best lower bound provided bythe entropy method. An access structure is $\kappa$-ideal if it has$\kappa$-complexity 1. Motivated by the abundance of open problems in thisarea, the main results can be summarized as follows. First, a newcharacterization of $\kappa$-ideal multipartite access structures is givenwhich offers a straightforward and simple approach to describe ideal bipartiteand tripartite access structures. Second, the $\kappa$-complexity is determinedfor a range of bipartite access structures, including those determined by twopoints, staircases with equal widths and heights, and staircases with allheights 1. Third, matching linear schemes are presented for some non-idealcases, including staircases where all heights are 1 and all widths are equal.Finally, finding the Shannon complexity of a bipartite access structure can beconsidered as a discrete submodular optimization problem. An interesting andintriguing continuous version is defined which might give further insight tothe large-scale behavior of these optimization problems.

Bipartite secret sharing and staircases