A short note on the kissing number of the lattice in Gaussian wiretap coding
This work addresses theoretical security improvements in lattice-based coding for cryptography, but appears incremental as it builds on existing conjectures and known lattice structures.
The paper investigates how the secrecy gain in Gaussian wiretap coding relates to lattice properties, showing that decreasing the number of vectors of a specific length increases secrecy gain, and providing an example where two lattices with identical shortest vector length and kissing number have different secrecy gains.
We show that on an $n=24m+8k$-dimensional even unimodular lattice, if the shortest vector length is $\geq 2m$, then as the number of vectors of length $2m$ decreases, the secrecy gain increases. We will also prove a similar result on general unimodular lattices. Furthermore, assuming the conjecture by Belfiore and Solé, we will calculate the difference between inverses of secrecy gains as the number of vectors varies. Finally, we will show by an example that there exist two lattices in the same dimension with the same shortest vector length and the same kissing number, but different secrecy gains.