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与Narayan针对所谓的\emph{成对独立网络（PIN）模型}提出了一种完美密钥生成方案，该方案通过利用底层图的组合特性——即生成树填充率——来实现。本研究将PIN模型推广至底层图被超图替代的情形，并通过挖掘超图的组合特性，在设计类似的完美密钥生成方案方面取得进展。我们的贡献主要体现在两个方面。首先，针对具有$m$个顶点的完全$t$-均匀超图，我们利用完全$t$-均匀超图通过星形超图的填充结构，设计出一种方案，使得每个星形图能生成$\binom{m-2}{t-2}$比特的完美密钥，该方案达到了容量上限。其次，我们针对投影为环的3-均匀星形超图，提出了一种2比特完美密钥生成方案。该方案通过结合3-均匀超图的星形图填充与图的哈密顿填充，进一步推广至通用3-均匀超图，并证明其对特定超图类别能达到容量上限。