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Probabilistic Shaped Multilevel Polar Coding for Wiretap Channel

Li Shen, Yongpeng Wu, Peihong Yuan, Chengshan Xiao, Xiang-Gen Xia, Wenjun Zhang

TL;DR

A probabilistic shaped multilevel polar coding scheme integration with probabilistic shaping is proposed that can achieve the secrecy capacity of the Gaussian wiretap channel with discrete constellation input, and satisfies the reliability condition and weak security condition.

Abstract

A wiretap channel is served as the fundamental model of physical layer security techniques, where the secrecy capacity of the Gaussian wiretap channel is proven to be achieved by Gaussian input. However, there remains a gap between the Gaussian secrecy capacity and the secrecy rate with conventional uniformly distributed discrete constellation input, e.g. amplitude shift keying (ASK) and quadrature amplitude modulation (QAM). In this paper, we propose a probabilistic shaped multilevel polar coding scheme to bridge the gap. Specifically, the input distribution optimization problem for maximizing the secrecy rate with ASK/QAM input is solved. Numerical results show that the resulting sub-optimal solution can still approach the Gaussian secrecy capacity. Then, we investigate the polarization of multilevel polar codes for the asymmetric discrete memoryless wiretap channel, and thus propose a multilevel polar coding scheme integration with probabilistic shaping. It is proved that the scheme can achieve the secrecy capacity of the Gaussian wiretap channel with discrete constellation input, and satisfies the reliability condition and weak security condition. A security-oriented polar code construction method to natively satisfies the leakage-based security condition is also investigated. Simulation results show that the proposed scheme achieves more efficient and secure transmission than the uniform constellation input case over both the Gaussian wiretap channel and the Rayleigh fading wiretap channel.

Probabilistic Shaped Multilevel Polar Coding for Wiretap Channel

TL;DR

A probabilistic shaped multilevel polar coding scheme integration with probabilistic shaping is proposed that can achieve the secrecy capacity of the Gaussian wiretap channel with discrete constellation input, and satisfies the reliability condition and weak security condition.

Abstract

A wiretap channel is served as the fundamental model of physical layer security techniques, where the secrecy capacity of the Gaussian wiretap channel is proven to be achieved by Gaussian input. However, there remains a gap between the Gaussian secrecy capacity and the secrecy rate with conventional uniformly distributed discrete constellation input, e.g. amplitude shift keying (ASK) and quadrature amplitude modulation (QAM). In this paper, we propose a probabilistic shaped multilevel polar coding scheme to bridge the gap. Specifically, the input distribution optimization problem for maximizing the secrecy rate with ASK/QAM input is solved. Numerical results show that the resulting sub-optimal solution can still approach the Gaussian secrecy capacity. Then, we investigate the polarization of multilevel polar codes for the asymmetric discrete memoryless wiretap channel, and thus propose a multilevel polar coding scheme integration with probabilistic shaping. It is proved that the scheme can achieve the secrecy capacity of the Gaussian wiretap channel with discrete constellation input, and satisfies the reliability condition and weak security condition. A security-oriented polar code construction method to natively satisfies the leakage-based security condition is also investigated. Simulation results show that the proposed scheme achieves more efficient and secure transmission than the uniform constellation input case over both the Gaussian wiretap channel and the Rayleigh fading wiretap channel.

Paper Structure

This paper contains 27 sections, 8 theorems, 49 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wiretap Channel Model
  4. Secrecy Metrics
  5. Reliability and Security
  6. Secrecy Capacity
  7. Equivocation Rate
  8. Security Gap
  9. Polar Codes
  10. Optimization of the Input Distribution for the Gaussian Wiretap Channel
  11. Probabilistic Shaped Multilevel Polar Coding Scheme
  12. Polarization of Multilevel Polar Codes for the Asymmetric DMWC
  13. Proposed Coding Scheme
  14. Reliability Analysis
  15. Security Analysis
  16. ...and 12 more sections

Key Result

Lemma 1

Let $W_1 : X \to Y$ and $W_2 : X \to Z$ be two DMC with input alphabet $\mathcal{X}$ of cardinality $|\mathcal{X}|=2^q$. The input symbol $X$ is mapped by $f(\bm{B}) = f(B^1, B^2, \cdots, B^q)$. Consider the equivalent binary-input sub-channels $W_1^l: B^l \to \left(Y, \bm{B}^{[\![l-1]\!]}\right)$ a

Figures (10)

  • Figure 1: The wiretap channel model.
  • Figure 2: The achievable secrecy rate for the Gaussian wiretap channel with Gaussian input, shaped $Q$-ASK input and uniform $Q$-ASK input, where $Q \in \{2,4,8,16,32,64\}$ and ${\rm SNR}_B-{\rm SNR}_E=3$ dB.
  • Figure 3: The multilevel polar coding construction with $Q=8$ and $N=4$.
  • Figure 4: The MSD construction with $Q=8$.
  • Figure 5: Graphical representation of the polarization of $Z_B(U_i|\bm{V}_i^l,\bm{Z})$, $Z_B(U_i|\bm{V}_i^l,\bm{Y})$ and $Z_B(U_i|\bm{V}_i^l)$ as $N \to \infty$.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2: Degradation of $\bar{W}_N^{(i)}$ Korada2009polar
  • Lemma 3
  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Lemma 4
  • Theorem 3
  • Remark 1