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Subspace Decomposition of Coset Codes

David Hunn, Willie Harrison

TL;DR

This work introduces subspace decomposition as a robust framework for analyzing coset codes over the BEWC, enabling efficient, continuous evaluation of secrecy performance via equivocation loss and the alternative metric $\text{χ}^2$ divergence. It derives a central, tractable expression for expected equivocation as a sum over subspaces with universal coefficient constants, and provides complexity improvements over brute-force erasure-pattern enumeration, with a break-even regime $n \gtrsim (\kappa^2+2\kappa)/4$. The paper also develops several code constructions (uniform vector fraction and subspace exclusion codes) and proves local and global optimality results under both equivocation and $\text{χ}^2$ divergence, including global optimality of UVF and first subspace exclusion codes. These results illuminate design principles for short-blocklength secrecy codes and establish a link between canonical secrecy metrics, suggesting practical benefits for secure communications in latency- and complexity-constrained settings.

Abstract

A new method is explored for analyzing the performance of coset codes over the binary erasure wiretap channel (BEWC) by decomposing the code over subspaces of the code space. This technique leads to an improved algorithm for calculating equivocation loss. It also provides a continuous-valued function for equivocation loss, permitting proofs of local optimality for certain finite-blocklength code constructions, including a code formed by excluding from the generator matrix all columns which lie within a particular subspace. Subspace decomposition is also used to explore the properties of an alternative secrecy code metric, the chi squared divergence. The chi squared divergence is shown to be far simpler to calculate than equivocation loss. Additionally, the codes which are shown to be locally optimal in terms of equivocation are also proved to be globally optimal in terms of chi squared divergence.

Subspace Decomposition of Coset Codes

TL;DR

This work introduces subspace decomposition as a robust framework for analyzing coset codes over the BEWC, enabling efficient, continuous evaluation of secrecy performance via equivocation loss and the alternative metric divergence. It derives a central, tractable expression for expected equivocation as a sum over subspaces with universal coefficient constants, and provides complexity improvements over brute-force erasure-pattern enumeration, with a break-even regime . The paper also develops several code constructions (uniform vector fraction and subspace exclusion codes) and proves local and global optimality results under both equivocation and divergence, including global optimality of UVF and first subspace exclusion codes. These results illuminate design principles for short-blocklength secrecy codes and establish a link between canonical secrecy metrics, suggesting practical benefits for secure communications in latency- and complexity-constrained settings.

Abstract

A new method is explored for analyzing the performance of coset codes over the binary erasure wiretap channel (BEWC) by decomposing the code over subspaces of the code space. This technique leads to an improved algorithm for calculating equivocation loss. It also provides a continuous-valued function for equivocation loss, permitting proofs of local optimality for certain finite-blocklength code constructions, including a code formed by excluding from the generator matrix all columns which lie within a particular subspace. Subspace decomposition is also used to explore the properties of an alternative secrecy code metric, the chi squared divergence. The chi squared divergence is shown to be far simpler to calculate than equivocation loss. Additionally, the codes which are shown to be locally optimal in terms of equivocation are also proved to be globally optimal in terms of chi squared divergence.
Paper Structure (35 sections, 18 theorems, 228 equations, 2 figures, 2 tables)

This paper contains 35 sections, 18 theorems, 228 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Coset Coding
  4. Code Performance Characterization
  5. Equivocation Properties
  6. Notation and Definitions
  7. Continuous Code Specification
  8. Subspace Structure Definitions
  9. Subspace Decomposition Example
  10. Analytical Results
  11. Supporting Lemmas
  12. Equivocation Loss Via Subspace Decomposition
  13. Complexity of Equivocation Computation
  14. All-Zero Column Exclusion
  15. Uniform Vector Fraction Construction
  16. ...and 20 more sections

Key Result

Lemma 1

For any subspace $S$ of $W^{\{\kappa\}}$, any positive integer $n$, any $0 \leq \epsilon < 1$, and any $\kappa$-dimensional code definition vector $q$ satisfying eqn:QConstraintPositive and eqn:QConstraintTotal,

Figures (2)

  • Figure 1: Binary erasure wiretap channel.
  • Figure 2: Calculation of $c(d,d')$ for $d=4$ and $d' \in [\![ 0,4]\!]$ by computing the function $\eta(d,d')$ via expansion of $\Psi(S)$ through the recursion paths $\theta \in \Theta_{d,d'}$.

Theorems & Definitions (31)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • proof
  • ...and 21 more