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 ⋛ for some ⋛. We show that if the minimum left degree of the ensemble is l<inf>min</inf>, the expected probability of block error is O(1/n⌈<sup>l</sup> min<sup>k/2</sup>⌉ −k) when the erasure probability ∊ < ∊<inf>ef</inf>, where ∊<inf>ef</inf> depends on the degree distribution of the ensemble. As long as l<inf>min</inf> and k > 2, the dual of this LDPC code provides strong secrecy over a BEWC of erasure probability greater than 1–∊<inf>ef</inf>.

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