  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.
