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 \emph{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 an access structure is the best lower bound provided by the entropy method. An access 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 describe ideal bipartite and tripartite access structures. Second, the $\kappa$-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.

翻译：二分秘密共享方案具有二分访问结构，其中参与者集合被划分为两部分，且同一部分内的所有参与者扮演等效角色。此类二分方案可通过"阶梯结构"（即其最小点集）来描述。方案的复杂度是指相对于秘密大小的最大份额尺寸；而访问结构的κ-复杂度是熵方法所提供的最优下界。若某访问结构的κ-复杂度为1，则称其为κ-理想结构。鉴于该领域存在大量未解决问题，主要研究成果可归纳如下。首先，给出了κ-理想多部访问结构的新刻画，为描述理想的二分和三分访问结构提供了直接简洁的方法。其次，确定了一类二分访问结构的κ-复杂度，包括由两点确定的阶梯结构、宽度与高度相等的阶梯结构，以及所有高度均为1的阶梯结构。第三，针对部分非理想情形（如所有高度为1且所有宽度相等的阶梯结构），给出了匹配的线性方案。最后，求取二分访问结构的香农复杂度可视为离散子模优化问题，并定义了有趣且引人入胜的连续版本，这或许能进一步揭示此类优化问题的大规模行为特征。