Summary:
The motivation for this paper is the need to develop a key management scheme
that is robust, secure, and well suited for a group of mutually suspicious
individuals with conflicting interests that need to operate together.  The key
sharing works as follows: We have a secret that we want to share among n
people such that any k (or more) of them can reconstruct the secret, but k-1
of them, even with full knowledge, cannot.  The scheme is based on polynomial
interpolation. For a (k,n)-threshold scheme, we generate a random polynomial
of degree (k-1) where the first coefficient, Ao = secret. We then assign to
each user a point (x,y) on the polynomial. Given any subset of size k of these
points, we can find the coefficients of the polynomial by interpolation and
then evaluate the polynomial at 0 and recover the secret. The contributions of
this paper include: (1) the size of each share of the key is not larger than
the key itself, (2) new shares can be dynamically added/deleted, (3) you can
change shares without having to change the original key, and (4) by using
tuples of polynomial values, we can enforce a hierarchical scheme of key
sharing.

Pros:
Cute, simple, and short paper.

Cons:
This still requires some trusted intermediary to perform the validation of
shares. If you trust k people too much, they can go out of band and
reconstruct the secret, and create new shares to distribute to others.  The
secret is only really shared before its used once because then everyone knows
it so we still need a trusted intermediary.