Strong secrecy for erasure wiretap channels

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 ≥ 2k for some k ≥ 2. We show that if the minimum left degree of the ensemble is l_min, the expected probability of block error is O(1/n⌈^l min^k/2⌉ -k) when the erasure probability ϵ <;; ϵ_ef, where ϵ_ef depends on the degree distribution of the ensemble. As long as l_min and k > 2, the dual of this LDPC code provides strong secrecy over a BEWC of erasure probability greater than 1-ϵ_ef.