Perfect Secret Key Generation for a class of Hypergraphical Sources

2026-01-15

AI Summary

This study addresses the problem of perfect secrecy key generation in hypergraph-structured sources. It extends key generation techniques from conventional graph models to the hypergraph setting for the first time, proposing capacity-achieving schemes based on star hypergraph packing and Hamiltonian packing. For complete $t$-uniform hypergraphs, a general construction is devised that achieves an optimal key rate of $\binom{m-2}{t-2}$ bits per star in specific 3-uniform hypergraphs. The proposed scheme is further shown to attain the information-theoretic secret key capacity across several classes of hypergraphs, thereby establishing a foundational framework for secure key agreement in complex, higher-order network structures.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• perfect secret key generation
• hypergraphical sources
• pairwise independent network
• information-theoretic security
• combinatorial properties

Innovation

• perfect secret key generation
• hypergraphical sources
• star hypergraph packing
• capacity achieving scheme
• Hamiltonian packing