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On the secrecy gain of \(\ell\)-modular lattices. (English) Zbl 1447.11077
In a lattice code the codewords are elements of cosets in a quotient \(\Lambda/M\) where \(\Lambda\) is a lattice (namely a discrete additive subgroup of the additive structure of a real vector space) and \(M\) a sublattice. The decoding procedure consists of the calculation of the closest coset to the received codeword. This problem is related naturally to the sphere packing problem, in which the centers of the spheres are precisely the lattice points. On the other hand, the kissing number of a lattice packing is the number of balls touching the sphere centered at the origin. When considering the generating function of the number of lattice points for each integer, which is indeed the theta series of the lattice, the kissing number is given by its first non-zero coefficient.
The cubic lattice is the lattice consisting of points in the real space having just integer coordinates, it is \(\mathbb{Z}^n\). Belfiore and Oggier defined (see [F. Oggier et al., IEEE Trans. Inf. Theory 62, No. 10, 5690–5708 (2016; Zbl 1359.94149)]) the secrecy function of a lattice by comparing its theta series with the theta series of the cubic lattice, \(\forall y\in\mathbb{R}^+\), \(\Xi_{\Lambda}(y) = \Theta_{\mathbb{Z}^n}(y)/ \Theta_{\Lambda}(y)\) and it was observed that for unimodular lattices, namely lattices whose primitive cells have volume 1, this function attains its maximum at \(y=1\). An \(\ell\)-modular lattice is an integral lattice such that for an orthogonal map \(U\), \(\sqrt{\ell}U(\Lambda^*)\subset\Lambda\). In an \(\ell\)-modular lattice the primitive cell has volume \(\ell^{\frac{n}{4}}\) and the secrecy function is defined by comparison with the rescaled cubic lattice \((\ell^{\frac{n}{4}}\mathbb{Z})^n\). Hence, it was conjectured that its maximum is attained at \(y=\ell^{-\frac{1}{2}}\). The authors of the reviewed paper refuted this conjecture by a counterexample and they suggested an alternative secrecy function, considering the lattice \(D^{\ell}\) instead of the cubic lattice, and they re-asserted the above conjecture with respect to their proposed secrecy function. In this paper the authors propose a sufficient condition for the conjecture to hold. The techniques are rather technical but very illustrative of the involved geometrical notions, all of them relevant to these numerical lattices.

MSC:
11H71 Relations with coding theory
94A62 Authentication, digital signatures and secret sharing
11F20 Dedekind eta function, Dedekind sums
11F27 Theta series; Weil representation; theta correspondences
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