Strong Secrecy for Erasure Wiretap Channels

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 $\ge 2k$, for some $k\ge 2$. We show that if the minimum left degree of the ensemble is $l_{\min }$, 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 $l_{\min }>2$ and $k>2$, the dual of this LDPC code provides strong secrecy over a BEWC of erasure probability greater than $1-{\epsilon }_{\mathrm{ef}}$.

Submitted to the Information Theory Workship (ITW) 2010, Dublin