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Perfect Secret Key Generation for a class of Hypergraphical Sources

Manuj Mukherjee, Sagnik Chatterjee, Alhad Sethi

TL;DR

The paper advances perfect secret key generation for hypergraphical sources by extending PIN concepts to complete $t$‑uniform hypergraphs through star hypergraph packing, achieving the capacity $C_{ ext{SK}}(X_{ ext{M}})= rac{t-1}{m-1}inom{m}{t}$. It also develops a 2‑bit per star hypergraph scheme for 3‑uniform hypergraphs with cycle projections and generalizes to generic 3‑uniform hypergraphs using star and Hamiltonian packings, establishing capacity results for select classes. The work clarifies the Type‑S structure of these sources and leverages linear communication and perfect omniscience as core tools. By combining edge‑disjoint packings and cycle/Hamiltonian decompositions, the authors unify combinatorial hypergraph properties with information‑theoretic capacity, yielding concrete capacity‑achieving schemes for families of hypergraphs. This framework has potential implications for scalable, secure key generation in distributed networks with higher‑order interactions.

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.

Perfect Secret Key Generation for a class of Hypergraphical Sources

TL;DR

The paper advances perfect secret key generation for hypergraphical sources by extending PIN concepts to complete ‑uniform hypergraphs through star hypergraph packing, achieving the capacity . It also develops a 2‑bit per star hypergraph scheme for 3‑uniform hypergraphs with cycle projections and generalizes to generic 3‑uniform hypergraphs using star and Hamiltonian packings, establishing capacity results for select classes. The work clarifies the Type‑S structure of these sources and leverages linear communication and perfect omniscience as core tools. By combining edge‑disjoint packings and cycle/Hamiltonian decompositions, the authors unify combinatorial hypergraph properties with information‑theoretic capacity, yielding concrete capacity‑achieving schemes for families of hypergraphs. This framework has potential implications for scalable, secure key generation in distributed networks with higher‑order interactions.

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 -uniform hypergraph on vertices by leveraging a packing of the complete -uniform hypergraphs by what we refer to as star hypergraphs, and designing a scheme that gives 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.
Paper Structure (9 sections, 18 theorems, 7 equations, 1 figure, 1 algorithm)

This paper contains 9 sections, 18 theorems, 7 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Technical Preliminaries
  3. Multiterminal Source Model
  4. Hypergraphical Source Model
  5. Secret Key Construction Scheme for complete $t$-Uniform Hypergraphical Sources
  6. Secret Key Construction Scheme for $3-$uniform Hypergraphs
  7. Ommitted Proofs
  8. Omitted Proofs of Section \ref{['sec:tunif']}
  9. Omitted Proofs of Section \ref{['sec:threeunif']}

Key Result

Lemma 1

Let $s\in\mathbb{N}$ and let $Y=(Y_1,Y_2,\ldots,Y_s)^T\sim\text{unif}\{\{0,1\}^s\}$. Let $M\in\{0,1\}^{l\times s}$ be a matrix whose rows we denote as $M_1,M_2,\ldots,M_l$. For any $1\leq l'<l$ if $\text{span}\{M_1,\ldots,M_{l'}\}\cap\text{span}\{M_{l'+1},\ldots,M_l\}=\{0\}$,$0$ here denotes the vec

Figures (1)

  • Figure 1: Hamiltonian cycle packing for $K_4$.

Theorems & Definitions (22)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Definition 2: Star Model $S_{m,i}$
  • Lemma 3: Global Source Decomposition
  • Lemma 3: Perfect Recoverability
  • Lemma 3: Perfect Secrecy and Uniformity
  • Theorem 4: Main Result
  • Definition 3: Induced cycle $3-$uniform hypergraphical source
  • ...and 12 more