A New Secrecy Function for Modular Lattices
Y. Diop *
Department of Mathematics, Laboratoire d'Algebre de Cryptographie de Geometrie Algebrique et Applications (LACGAA), UCAD, Dakar, Senegal
Cheikh Thiecoumba Gueye
Department of Mathematics, Laboratoire d'Algebre de Cryptographie de Geometrie Algebrique et Applications (LACGAA), UCAD, Dakar, Senegal
Patrick Sole
CNRS, LTCI, Telecom Paris Tech, Paris, France
*Author to whom correspondence should be addressed.
Abstract
A recent line of work to improve the secrecy capacity within wiretap gaussian channel has introduced a new lattice invariant called secrecy gain. Belfiore and Sol´e made a conjecture about the point at which the the secrecy gain is maximum. Verified by most unimodular lattices, this conjecture does not hold in general for l-modular lattices (l ≥ 2). Ernvall-Hytönen modified the secrecy function and proved that it satisfies the conjecture for 2-odd modular lattices. In this paper, the authors introduce a new secrecy function for 2-modular lattices. They show that, by using the lattice D4 instead of Dl = Z⊕√lZ , the conjecture holds for both 2-even and odd modular lattices in dimension n ≥ 4. Using that result, they further prove that the modified secrecy function of A.-M. Ernvall-Hytönen holds for both 2-even and odd modular lattices.
Keywords: Gaussian wiretap channel, lattice codes, secrecy gain, modular lattices, theta series