We consider the problem of threshold secret sharing in
groups with hierarchical structure: a subset of
participants is authorized if it has at least k_0 members from the highest level, as well as at least
k_1 members from the two highest levels and so forth. Such problems occur in settings where the
participants differ in their authority or level of confidence and the presence of higher-level participants
is imperative to allow the recovery of the common secret. Even though secret sharing in hierarchical groups has been
studied extensively in the past, none of the proposed solutions addresses the simple setting that was described above.
We present a secret sharing
scheme for this problem that is both perfect and ideal. As in Shamir's scheme,
the secret is represented as the free coefficient of some polynomial. The novelty of our scheme is the usage of
polynomial derivatives in order to generate lesser shares for participants of lower levels. Consequently,
our scheme uses Birkhoff interpolation, i.e., the construction of a polynomial according to an unstructured
set of point and derivative values. Unfortunately, Birkhoff interpolation problems are not always well posed.
Hence, a substantial part of our study is dedicated to strategies that guarantee well-posedness. Combining our
results with a duality result of A. Gal regarding monotone span programs, we infer that the closely related hierarchical
threshold access structures that were studied by Simmons and Brickell are also ideal. An explicit ideal scheme for
those problems is presented.