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Local Approximation of Secrecy Capacity

Emmanouil M. Athanasakos, Nicholas Kalouptsidis, Hariprasad Manjunath

TL;DR

EIT is used to analyze the wiretap channel and obtain the perturbed probability distributions such that secrecy is achievable, and an approximate estimate of the secrecy capacity is obtained by solving a generalized eigenvalue problem.

Abstract

This paper uses Euclidean Information Theory (EIT) to analyze the wiretap channel. We investigate a scenario of efficiently transmitting a small amount of information subject to compression rate and secrecy constraints. We transform the information-theoretic problem into a linear algebra problem and obtain the perturbed probability distributions such that secrecy is achievable. Local approximations are being used in order to obtain an estimate of the secrecy capacity by solving a generalized eigenvalue problem.

Local Approximation of Secrecy Capacity

TL;DR

EIT is used to analyze the wiretap channel and obtain the perturbed probability distributions such that secrecy is achievable, and an approximate estimate of the secrecy capacity is obtained by solving a generalized eigenvalue problem.

Abstract

This paper uses Euclidean Information Theory (EIT) to analyze the wiretap channel. We investigate a scenario of efficiently transmitting a small amount of information subject to compression rate and secrecy constraints. We transform the information-theoretic problem into a linear algebra problem and obtain the perturbed probability distributions such that secrecy is achievable. Local approximations are being used in order to obtain an estimate of the secrecy capacity by solving a generalized eigenvalue problem.
Paper Structure (8 sections, 2 theorems, 45 equations, 2 figures, 1 algorithm)

This paper contains 8 sections, 2 theorems, 45 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Preliminaries And Notations
  5. Problem Formulation
  6. Local Approximation of Secrecy Capacity
  7. Example: Binary Symmetric Wiretap Channel
  8. Discussion

Key Result

Lemma 1

Given a distribution $P_{X|U=u} \in \mathcal{N}_\epsilon^\mathcal{X}(P_X)$, for all $u \in \mathcal{U}$ and all $\epsilon>0$, it holds that the mutual information for the encoding satisfies where $L_u = [\sqrt{P_X}]^{-1}\cdot J_u$ is the spherical perturbation vector in $\mathbb{R}^{|\mathcal{X}|}$; $B_{Y|X}$ and $B_{Z|X}$ are the divergence transition matrices defined in DTM_1 of the legitimate

Figures (2)

  • Figure 1: The wiretap channel model.
  • Figure 2: The black line is the approximation of \ref{['sec_cap_app_bsc']} in comparison with $C_s = H_b(q)- H_b(p)$ (red line) for $\epsilon = 10^{-3}$, $\delta=0.085$ and $q=0.45$

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • proof