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Secrecy Coding for the Binary Symmetric Wiretap Channel via Linear Programming

Ali Nikkhah, Morteza Shoushtari, Bahareh Akhbari, Willie K. Harrison

TL;DR

This work addresses secrecy coding for a binary symmetric wiretap channel with a noiseless main link by introducing a finite-blocklength linear-programming framework to bound equivocation. It then develops the Ni coding scheme, with RASBA and RAHBA construction methods, to approach the LP-derived limit and demonstrates optimal performance for the l=1 case while closely approaching it for larger l. The authors also derive linear patterns for generator/parity-check matrices in linear forms and present a non-recursive AC+B formulation to generate Ni bins across all forms. The results show that the LP-derived limit tightens the classical Wyner bound under finite blocklength and that Ni codes outperform many random bin codes in the finite-length regime, offering practical guidance for secure coding in constrained wireless environments.

Abstract

In this paper, we use a linear programming (LP) optimization approach to evaluate the equivocation for a wiretap channel where the main channel is noiseless, and the wiretap channel is a binary symmetric channel (BSC). Using this technique, we present an analytical limit for the achievable secrecy rate in the finite blocklength regime that is tighter than traditional fundamental limits. We also propose a secrecy coding technique that outperforms random binning codes. When there is one overhead bit, this coding technique is optimum and achieves the analytical limit. For cases with additional bits of overhead, our coding scheme can achieve equivocation rates close to the new limit. Furthermore, we evaluate the patterns of the generator matrix and the parity-check matrix for linear codes and we present binning techniques for both linear and non-linear codes using two different approaches: recursive and non-recursive. To our knowledge, this is the first optimization solution for secrecy coding obtained through linear programming.

Secrecy Coding for the Binary Symmetric Wiretap Channel via Linear Programming

TL;DR

This work addresses secrecy coding for a binary symmetric wiretap channel with a noiseless main link by introducing a finite-blocklength linear-programming framework to bound equivocation. It then develops the Ni coding scheme, with RASBA and RAHBA construction methods, to approach the LP-derived limit and demonstrates optimal performance for the l=1 case while closely approaching it for larger l. The authors also derive linear patterns for generator/parity-check matrices in linear forms and present a non-recursive AC+B formulation to generate Ni bins across all forms. The results show that the LP-derived limit tightens the classical Wyner bound under finite blocklength and that Ni codes outperform many random bin codes in the finite-length regime, offering practical guidance for secure coding in constrained wireless environments.

Abstract

In this paper, we use a linear programming (LP) optimization approach to evaluate the equivocation for a wiretap channel where the main channel is noiseless, and the wiretap channel is a binary symmetric channel (BSC). Using this technique, we present an analytical limit for the achievable secrecy rate in the finite blocklength regime that is tighter than traditional fundamental limits. We also propose a secrecy coding technique that outperforms random binning codes. When there is one overhead bit, this coding technique is optimum and achieves the analytical limit. For cases with additional bits of overhead, our coding scheme can achieve equivocation rates close to the new limit. Furthermore, we evaluate the patterns of the generator matrix and the parity-check matrix for linear codes and we present binning techniques for both linear and non-linear codes using two different approaches: recursive and non-recursive. To our knowledge, this is the first optimization solution for secrecy coding obtained through linear programming.
Paper Structure (25 sections, 2 theorems, 53 equations, 8 figures, 2 algorithms)

This paper contains 25 sections, 2 theorems, 53 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Prior work
  3. Our Contributions
  4. Organization of the Paper
  5. Notation
  6. System Model and Preliminaries
  7. System Model
  8. Preliminaries
  9. Equivocation Calculation
  10. Optimization
  11. Ni Coding and Its Two Essential Parts
  12. RASBA Algorithm
  13. RAHBA Algorithm
  14. Our Standard Path and Some Examples
  15. The solutions of optimization for various cases
  16. ...and 10 more sections

Key Result

Lemma 1

For a specific form of $(l,k)$ and a given $p$, the value of optimum entropy for a conditioned equivocation, i.e., (Conditioned_Ent), is an upper bound for the total equivocation, i.e., (equivovation_t).

Figures (8)

  • Figure 1: Wiretap channel model.
  • Figure 2: Scheme map of our Ni coding scheme.
  • Figure 3: Generating different forms by RASBA method.
  • Figure 4: Generating different forms by RAHBA method.
  • Figure 5: Infinite blocklength limit, LP-derived limit, and equivocation results of the Ni code and different binning codes for $l=1$ and $n=5$.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof