Modular Construction a Lattices from Cyclotomic Fields and their Applications in Information Security

Abstract

We present an overview of recent advances in the Area of information security using algebraic number fields. This overview indicates the importance of modular lattices in information security and in recently proposed methods for obtaining modular lattices using algebraic number fields. Obtaining Construction a unimodular lattices using cyclotomic number fields of prime orders have been addressed in the literature. Recently, a new lattice invariant called secrecy gain has been defined and it has been shown that it characterizes the confusion at the eavesdropper when using lattices in the Gaussian wiretap channels. There is a symmetry point, called weak secrecy gain, in the secrecy function of modular lattices. It is conjectured that the weak secrecy gain is the secrecy gain. It is known that d-modular lattices with high level d are more likely to have a large length for the shortest nonzero vector, which results in a higher weak secrecy gain. In search of such lattices, we prove that there is no modular lattices built using Construction A over cyclotomic fields of prime power order $p^{n}$, with $n > 1$. We also present a new framework based on Construction A lattices and cyclotomic number fields that gives a family of p-modular lattices with $p\equiv 1 (\mathrm {m}\mathrm {o}\mathrm {d}~4)$.