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 ϵ, decays as O(exp(-(c₁)n(c₂))) with the block-length n for positive constants c₁ and c₂, as long as ϵ is less than the erasure threshold ϵ_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-ϵ_th.