  In the multiterminal secret key agreement problem, a set of users want to
discuss with each other until they share a common secret key independent of
their discussion. We want to characterize the maximum secret key rate, called
the secrecy capacity, asymptotically when the total discussion rate goes to
zero. In the case of only two users, the capacity is equal to the
G\'acs-K\"orner common information. However, when there are more than two
users, the capacity is unknown. It is plausible that a multivariate extension
of the G\'acs-K\"orner common information is the capacity, however, proving the
converse is challenging. We resolved this for the hypergraphical sources and
finite linear sources, and provide efficiently computable characterizations. We
also give some ideas of extending the techniques to more general source models.
