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 ≥ 2k for some k ≥ 2. We show that if the minimum left degree of the ensemble is l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> , the expected probability of block error is O(1/n⌈ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sup> min <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k/2</sup> ⌉ -k) when the erasure probability ϵ <;; ϵ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ef</sub> , where ϵ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ef</sub> depends on the degree distribution of the ensemble. As long as l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> and k > 2, the dual of this LDPC code provides strong secrecy over a BEWC of erasure probability greater than 1-ϵ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ef</sub> .