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 $1 - \epsilon_\mathrm{ ef}$.
Categories: cs.IT math.IT