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On the secrecy gain of extremal even \(l\)-modular lattices. (English) Zbl 07136204
Summary: 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 \(l \in\{2, 3, 5, 6, 7, 11, 14, 15, 23\}\). 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.
MSC:
11H06 Lattices and convex bodies (number-theoretic aspects)
Software:
SageMath
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