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On the Secrecy Capacity of 1-2-1 Atomic Networks

Mohammad Milanian, Minoh Jeong, Martina Cardone

TL;DR

The paper studies secure communication over 1-2-1 atomic networks, modeling mmWave directivity constraints, and analyzes secrecy capacity under an eavesdropper tapping up to $K$ edges. It derives a novel, constructive lower bound based on path-grouping and MDS-coded keys, and a computable upper bound by distributing the adversary’s taps across vertex-disjoint subgraphs; these bounds tighten prior results in regimes where $M ext{ and } H_v$ satisfy certain relations. When $M geq H_v$, the authors show that their bounds can be tighter than existing ones in some regimes but may diverge in others, with examples illustrating both tighter and looser behavior. In the regime $M ext{ ext{≥}} H_v$, the lower and upper bounds match, thereby exactly characterizing the secrecy capacity $ ext{C}_s$ for 1-2-1 atomic networks, providing a complete capacity result for these structured networks.

Abstract

We consider the problem of secure communication over a noiseless 1-2-1 network, an abstract model introduced to capture the directivity characteristic of mmWave communications. We focus on structured networks, which we refer to as 1-2-1 atomic networks. Broadly speaking, these are characterized by a source, a destination, and three layers of intermediate nodes with sparse connections. The goal is for the source to securely communicate to the destination in the presence of an eavesdropper with unbounded computation capabilities, but limited network presence. We derive novel upper and lower bounds on the secrecy capacity of 1-2-1 atomic networks. These bounds are shown to be tighter than existing bounds in some regimes. Moreover, in such regimes, the bounds match and hence, they characterize the secrecy capacity of 1-2-1 atomic networks.

On the Secrecy Capacity of 1-2-1 Atomic Networks

TL;DR

The paper studies secure communication over 1-2-1 atomic networks, modeling mmWave directivity constraints, and analyzes secrecy capacity under an eavesdropper tapping up to edges. It derives a novel, constructive lower bound based on path-grouping and MDS-coded keys, and a computable upper bound by distributing the adversary’s taps across vertex-disjoint subgraphs; these bounds tighten prior results in regimes where satisfy certain relations. When , the authors show that their bounds can be tighter than existing ones in some regimes but may diverge in others, with examples illustrating both tighter and looser behavior. In the regime , the lower and upper bounds match, thereby exactly characterizing the secrecy capacity for 1-2-1 atomic networks, providing a complete capacity result for these structured networks.

Abstract

We consider the problem of secure communication over a noiseless 1-2-1 network, an abstract model introduced to capture the directivity characteristic of mmWave communications. We focus on structured networks, which we refer to as 1-2-1 atomic networks. Broadly speaking, these are characterized by a source, a destination, and three layers of intermediate nodes with sparse connections. The goal is for the source to securely communicate to the destination in the presence of an eavesdropper with unbounded computation capabilities, but limited network presence. We derive novel upper and lower bounds on the secrecy capacity of 1-2-1 atomic networks. These bounds are shown to be tighter than existing bounds in some regimes. Moreover, in such regimes, the bounds match and hence, they characterize the secrecy capacity of 1-2-1 atomic networks.
Paper Structure (8 sections, 4 theorems, 36 equations, 1 figure, 1 algorithm)

This paper contains 8 sections, 4 theorems, 36 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model and Known Results
  3. Main Results
  4. Proof of Main Results
  5. Proof of Theorem \ref{['thm:LB']}
  6. Proof of Theorem \ref{['thm:UB']}
  7. Proof of Theorem \ref{['thm:SecCap']}
  8. Proof of Lemma \ref{['lem:t']}

Key Result

Theorem 1

Define $\widehat{M}:= \min (M,H_{v})$ and, for all $c\in[0:\widehat{M}-1]$, let where $\mathsf{P}_{\widehat{M}-\eta}:=|\mathcal{P}_{\widehat{M}-\eta}|$ with $\mathcal{P}_{\widehat{M}-\eta}, \eta \in [0:c]$ being a setAlgorithm alg:algo1 will provide the detailed construction of $\mathcal{P}_{\widehat{M}-\eta}$. with groups of $\widehat{M}-\eta$ vertex-disjoint paths of $G\setmi

Figures (1)

  • Figure 1: A 1-2-1 atomic network with $H_e=7$ and $H_v=3$. The $H_e=7$ edge-disjoint paths are referred to as $p_i, i \in [7]$ from top to bottom, e.g., $p_4 = \mathsf{S} \to d \to v_2 \to k \to \mathsf{D}$.

Theorems & Definitions (10)

  • Definition 1
  • Theorem 1
  • Example 1
  • Remark 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Example 2
  • Example 3
  • Example 4