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Finite-blocklength performance of polar wiretap codes under a total variation secrecy constraint

Laura Luzzi, Valerio Bioglio

Abstract

We study the performance of polarizing codes over a degraded symmetric wiretap channel under a total variation distance (TVD) secrecy constraint. We show that the leakage can be bounded by the sum of the TVDs of the bit-channels corresponding to the confidential and frozen bits. In the asymptotic regime, this gives a new criterion to design wiretap codes with vanishing TVD leakage. In finite blocklength, it allows us to compute lower bounds for the secrecy rate of different families of polarizing wiretap codes over a binary erasure wiretap channel.

Finite-blocklength performance of polar wiretap codes under a total variation secrecy constraint

Abstract

We study the performance of polarizing codes over a degraded symmetric wiretap channel under a total variation distance (TVD) secrecy constraint. We show that the leakage can be bounded by the sum of the TVDs of the bit-channels corresponding to the confidential and frozen bits. In the asymptotic regime, this gives a new criterion to design wiretap codes with vanishing TVD leakage. In finite blocklength, it allows us to compute lower bounds for the secrecy rate of different families of polarizing wiretap codes over a binary erasure wiretap channel.

Paper Structure

This paper contains 17 sections, 7 theorems, 28 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Properties of the TVD of symmetric channels
  4. Channel polarization
  5. Polarization of TVDs
  6. Secrecy bounds under a TVD metric
  7. Second order bounds for the secrecy rate
  8. Wiretap coding scheme
  9. Bound for the average error probability
  10. Bounds for the leakage
  11. Wiretap code design
  12. Wiretap code scheme with vanishing total variation leakage
  13. Wiretap code design in finite blocklength
  14. Numerical results and Discussion
  15. Computation of Bounds 1 and 2
  16. ...and 2 more sections

Key Result

Lemma 1

For a symmetric channel $W: \mathcal{X} \to \mathcal{Y}$ with transition probability $p_{\mathsf{Y}|\mathsf{X}}$, where $p_{\bar{\mathsf{X}}}$ is the uniform input distribution over $\mathcal{X}$ and $p_{\bar{\mathsf{Y}}}= p_{\mathsf{Y}|\mathsf{X}} \circ p_{\bar{\mathsf{X}}}$ is the corresponding output distribution.

Figures (2)

  • Figure 1: The degraded wiretap channel.
  • Figure 2: Lower secrecy-rate bounds for polarizing codes versus second-order bounds (\ref{['YSP_Theorem13_upper']}) and (\ref{['YSP_Theorem13_lower']}) over different degraded binary erasure wiretap channels with error probability constraint $\epsilon=0.001$ and secrecy constraint $\delta=0.01$.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Example 1
  • Definition 3
  • Lemma 2
  • Lemma 3
  • Proposition 1
  • Remark 1
  • Lemma 4
  • ...and 4 more