A Classification of Unimodular Lattice Wiretap Codes in Small Dimensions

TL;DR

This paper classifies unimodular lattice codes for Gaussian wiretap channels by computing their secrecy gains, providing a comprehensive understanding of the best codes in dimensions 9 to 23 and linking them to error correction codes.

Contribution

It derives a formula for the secrecy gain of unimodular lattices and classifies the optimal codes in dimensions 9 to 23, including extremal and nonextremal cases.

Findings

Computed secrecy gains for all unimodular lattices in dimensions 16 to 23.

Classified the best wiretap lattice codes based on secrecy gain.

Connected lattice codes to error correction codes via Construction A.

Abstract

Lattice coding over a Gaussian wiretap channel, where an eavesdropper listens to transmissions between a transmitter and a legitimate receiver, is considered. A new lattice invariant called the secrecy gain is used as a code design criterion for wiretap lattice codes since it was shown to characterize the confusion that a chosen lattice can cause at the eavesdropper: the higher the secrecy gain of the lattice, the more confusion. In this paper, a formula for the secrecy gain of unimodular lattices is derived. Secrecy gains of extremal odd unimodular lattices as well as unimodular lattices in dimension n, 16 \leq n \leq 23 are computed, covering the 4 extremal odd unimodular lattices and all the 111 nonextremal unimodular lattices (both odd and even) providing thus a classification of the best wiretap lattice codes coming from unimodular lattices in dimension n, 8 < n \leq 23. Finally, to permit lattice encoding via Construction A, the corresponding error correction codes are determined.