Achieving Secrecy Capacity of the Gaussian Wiretap Channel With Polar Lattices

In this paper, an explicit scheme of wiretap coding based on polar lattices is proposed to achieve the secrecy capacity of the additive white Gaussian noise (AWGN) wiretap channel. First, polar lattices are used to construct secrecy-good lattices for the mod- $\Lambda _{s}$ Gaussian wiretap channel (GWC). Then, we propose an explicit shaping scheme to remove this mod- $\Lambda _{s}$ front end and extend polar lattices to the genuine GWC. 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 $O(N\log N\log (\log N))$ . The proposed scheme is proven to be semantically secure.