Nitinawarat and Narayan proposed a perfect secret key generation scheme for the so-called \emph{pairwise independent network (PIN) model} by exploiting the combinatorial properties of the underlying graph, namely the spanning tree packing rate. This work considers a generalization of the PIN model where the underlying graph is replaced with a hypergraph, and makes progress towards designing similar perfect secret key generation schemes by exploiting the combinatorial properties of the hypergraph. Our contributions are two-fold. We first provide a capacity achieving scheme for a complete $t$-uniform hypergraph on $m$ vertices by leveraging a packing of the complete $t$-uniform hypergraphs by what we refer to as star hypergraphs, and designing a scheme that gives $\binom{m-2}{t-2}$ bits of perfect secret key per star graph. Our second contribution is a 2-bit perfect secret key generation scheme for 3-uniform star hypergraphs whose projections are cycles. This scheme is then extended to a perfect secret key generation scheme for generic 3-uniform hypergraphs by exploiting star graph packing of 3-uniform hypergraphs and Hamiltonian packings of graphs. The scheme is then shown to be capacity achieving for certain classes of hypergraphs.

翻译：Nitinawarat和Narayan通过利用底层图的组合性质（即生成树打包率），为所谓的“成对独立网络（PIN）模型”提出了一种完美密钥生成方案。本文考虑PIN模型的一种推广，其中底层图被替换为超图，并通过利用超图的组合性质，在设计类似的完美密钥生成方案方面取得了进展。我们的贡献有两个方面。首先，我们通过利用我们称为星形超图的完全t-均匀超图的打包，并为每个星形图设计一种方案以生成 \(\binom{m-2}{t-2}\) 比特的完美密钥，为 \(m\) 个顶点上的完全 \(t\)-均匀超图提供了一种可达容量的方案。其次，我们为投影为圈的3-均匀星形超图提出了一种2比特的完美密钥生成方案。该方案随后被扩展到一般3-均匀超图，通过利用3-均匀超图的星形图打包和图的哈密顿打包。进一步证明，该方案对于某些类别的超图是可达容量的。