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

Multiterminal Secret Key Agreement at Asymptotically Zero Discussion Rate