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

TL;DR

This paper constructs LDPC codes with large girth using Ramanujan graphs to achieve strong secrecy over the binary erasure wiretap channel, with error probability decaying exponentially in block-length.

Contribution

It introduces a method to construct LDPC codes with logarithmic girth growth for arbitrary degree distributions, ensuring strong secrecy in wiretap channels.

Findings

Expected bit-error probability decays exponentially with block-length.

Codes achieve strong secrecy for erasure probabilities above a certain threshold.

Girth growth is logarithmic in block-length using Ramanujan graphs.

Abstract

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 $\mathcal{O}(\exp(-c_1 n^{c_2}))$ with the block-length $n$ for positive constants $c_1$ and $c_2$, as long as $\epsilon$ is lesser than the erasure threshold $\epsilon_\mathrm{th}$ 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 $1 - \epsilon_\mathrm{th}$.