Achieving Secrecy Capacity of the Gaussian Wiretap Channel with Polar Lattices

Abstract

In this work, an explicit wiretap coding scheme based on polar lattices is proposed to achieve the secrecy capacity of the additive white Gaussian noise (AWGN) wiretap channel. Firstly, polar lattices are used to construct secrecy-good lattices for the mod-$Λ_s$ Gaussian wiretap channel. Then we propose an explicit shaping scheme to remove this mod-$Λ_s$ front end and extend polar lattices to the genuine Gaussian wiretap channel. The shaping technique is based on the lattice Gaussian distribution, which leads to a binary asymmetric channel at each level for the multilevel lattice codes. By employing the asymmetric polar coding technique, we construct an AWGN-good lattice and a secrecy-good lattice with optimal shaping simultaneously. As a result, the encoding complexity for the sender and the decoding complexity for the legitimate receiver are both O(N logN log(logN)). The proposed scheme is proven to be semantically secure.

Submitted to IEEE Trans. Information Theory, revised. This is the authors' own version of the paper