Strong Secrecy on the Binary Erasure Wiretap Channel Using Large-Girth LDPC Codes

For an arbitrary degree distribution pair (DDP), we construct a sequence of low-density parity-check (LDPC) code ensembles with girth growing logarithmically in block-length using Ramanujan graphs. When the DDP has minimum left degree at least three, we show using density evolution analysis that the expected bit-error probability of these ensembles, when passed through a binary erasure channel with erasure probability epsilon, decays as O(exp(-c(1)n(c2))) with the block-length for positive constants c(1) and c(2), as long as epsilon is less than the erasure threshold epsilon(th) of the DDP. This guarantees that the coset coding scheme using the dual sequence provides strong secrecy over the binary erasure wiretap channel for erasure probabilities greater than 1 - epsilon(th).