Strong Secrecy for Erasure Wiretap Channels

TL;DR

This paper demonstrates that certain LDPC code duals, used in coset coding, achieve strong secrecy over the binary erasure wiretap channel by analyzing stopping sets and code ensemble properties.

Contribution

It introduces a new analysis linking LDPC code properties to strong secrecy guarantees over erasure wiretap channels.

Findings

Dual LDPC codes provide strong secrecy under certain conditions.

Expected block error probability decreases polynomially with block length.

Secrecy is achievable for erasure probabilities above a certain threshold.

Abstract

We show that duals of certain low-density parity-check (LDPC) codes, when used in a standard coset coding scheme, provide strong secrecy over the binary erasure wiretap channel (BEWC). This result hinges on a stopping set analysis of ensembles of LDPC codes with block length $n$ and girth $\geq 2k$, for some $k \geq 2$. We show that if the minimum left degree of the ensemble is $l_\mathrm{min}$, the expected probability of block error is $\calO(\frac{1}{n^{\lceil l_\mathrm{min} k /2 \rceil - k}})$ when the erasure probability $\epsilon < \epsilon_\mathrm{ef}$, where $\epsilon_\mathrm{ef}$ depends on the degree distribution of the ensemble. As long as $l_\mathrm{min} > 2$ and $k > 2$, the dual of this LDPC code provides strong secrecy over a BEWC of erasure probability greater than $1 - \epsilon_\mathrm{ef}$.