Secret key agreement for hypergraphical sources with limited total discussion
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We consider the multiterminal secret key agreement problem under the total discussion rate constraint and characterized the secrecy capacity for the hypergraphical sources. Our result not only covers the existing characterization for pairwise independent network but also extends it to the case with helpers, where the existing tree-packing protocol is known to be suboptimal. We show that decremental secret key agreement is optimal instead, which resolves a previous conjecture in the affirmative. The converse is established by a single-letter upper bound on secrecy capacity for general sources that are not necessarily hypergraphical. The minimax optimization involved in the upper bound can be relaxed to give various existing bounds such as the lamination bounds for hypergraphical sources, helper-set bound for general sources, the bound at asymptotically zero discussion rate via multivariate Gács–Körner common information, and the lower bound on communication complexity via a multivariate extension of Wyner common information. Our result not only unifies existing bounding techniques but also reveals surprising connections between seemingly different information-theoretic notions. We point out further challenges by showing that the bound can be loose for a simple finite linear source.
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