On the Secrecy Gain of Extremal Even l-modular Lattices

ABSTRACT

The secrecy gain is a lattice invariant that appears in the context of wiretap lattice coding. It has been studied for unimodular lattices, for 2 −, 3 −, and 5 −modular lattices. This paper studies the secrecy gain for extremal even l-modular lattices, for l ∈ {2, 3, 5, 6, 7, 11, 14, 15, 23}. We compute the highest secrecy gains as a function of the lattice dimension and the lattice level l. We show in particular that l = 2, 3, 6, 7, 11 are best for the respective ranges of dimensions {80, 76, 72}, {68, 64, 60, 56, 52, 48}, {44, 40, 36}, {34, 32, 30, 28, 26, 24, 22}, {18, 16, 14, 12, 10, 8}. This suggests that within a range of dimensions where different levels exist, the highest value of l tends to give the best secrecy gain. A lower bound computation on the maximal secrecy gain further shows that extremal lattices provide secrecy gains which are very close to this lower bound, thus confirming the good behavior of this class of lattices with respect to the secrecy gain.

Keywords:

• Lattices
• secrecy gain
• theta series

2010 AMS Subject Classification:

• 11H06

Acknowledgments

Part of this work was done while F. Oggier was visiting the Huawei France Research Center, whose hospitality is gratefully acknowledged. We also would like to thank the anonymous reviewer whose suggestions enriched this paper.

Notes

1 We provide the computations for the sake of demonstrating a method to compute a theta series of lattice using another one.