Applications of LDPC Codes to the Wiretap Channel

TLDR

Quantum key distribution achieves information‑theoretic security by employing classical error‑correcting codes for error correction and privacy amplification, and the wiretap channel model captures this setting. The paper investigates the fundamental limits and coding strategies for wiretap channels. The authors present an alternative proof of secrecy capacity, demonstrate that capacity‑achieving codes can attain it for any wiretap channel, and develop practical LDPC‑based linear‑time decodable codes for special cases such as the binary erasure and binary symmetric channels. Their designs reach secrecy capacity in some scenarios, provide sub‑capacity security in others, and for a noiseless main channel with a binary erasure eavesdropper, the LDPC codes achieve secrecy with linear‑time decoding.

Abstract

With the advent of quantum key distribution (QKD) systems, perfect (i.e., information-theoretic) security can now be achieved for distribution of a cryptographic key. QKD systems and similar protocols use classical error-correcting codes for both error correction (for the honest parties to correct errors) and privacy amplification (to make an eavesdropper fully ignorant). From a coding perspective, a good model that corresponds to such a setting is the wire tap channel introduced by Wyner in 1975. In this correspondence, we study fundamental limits and coding methods for wire tap channels. We provide an alternative view of the proof for secrecy capacity of wire tap channels and show how capacity achieving codes can be used to achieve the secrecy capacity for any wiretap channel. We also consider binary erasure channel and binary symmetric channel special cases for the wiretap channel and propose specific practical codes. In some cases our designs achieve the secrecy capacity and in others the codes provide security at rates below secrecy capacity. For the special case of a noiseless main channel and binary erasure channel, we consider encoder and decoder design for codes achieving secrecy on the wiretap channel; we show that it is possible to construct linear-time decodable secrecy codes based on low-density parity-check (LDPC) codes that achieve secrecy.