Perfect Secret Key Generation for a class of Hypergraphical Sources

TL;DR

This paper generalizes secret key generation schemes from graphs to hypergraphs, providing capacity-achieving methods for complete hypergraphs and schemes for specific hypergraph classes.

Contribution

It introduces new secret key generation schemes for hypergraphical sources, extending prior graph-based methods to hypergraphs with capacity-achieving results.

Findings

Capacity achieving scheme for complete t-uniform hypergraphs.

2-bit secret key scheme for 3-uniform star hypergraphs.

Extension of schemes to generic 3-uniform hypergraphs with certain optimality.

Abstract

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.