On the Secrecy Gain of Formally Unimodular Construction $\text{A}_4$ Lattices

TL;DR

This paper analyzes the secrecy gain of Construction A4 lattices derived from codes over Z4, showing they can outperform existing lattices in secrecy, and introduces a new code construction for self-dual Z4-linear codes.

Contribution

It derives the theta series of Construction A4 lattices, provides a universal method to compute their secrecy gains, and introduces a novel construction of formally self-dual Z4-linear codes.

Findings

Construction A4 lattices can achieve higher secrecy gain than existing formally unimodular lattices.

A universal approach to determine secrecy gains of these lattices is proposed.

A new code construction for formally self-dual Z4-linear codes is introduced.

Abstract

Lattice coding for the Gaussian wiretap channel is considered, where the goal is to ensure reliable communication between two authorized parties while preventing an eavesdropper from learning the transmitted messages. Recently, a measure called secrecy gain was proposed as a design criterion to quantify the secrecy-goodness of the applied lattice code. In this paper, the theta series of the so-called formally unimodular lattices obtained by Construction $\text{A}_4$ from codes over $\mathbb{Z}_4$ is derived, and we provide a universal approach to determine their secrecy gains. Initial results indicate that Construction $\text{A}_4$ lattices can achieve a higher secrecy gain than the best-known formally unimodular lattices from the literature. Furthermore, a new code construction of formally self-dual $\mathbb{Z}_4$-linear codes is presented.