Lattices Applied to Coding for Reliable and Secure Communications by Sueli I.R. Costa Frédérique Oggier Antonio Campello Jean-Claude Belfiore & Emanuele Viterbo

Author:Sueli I.R. Costa, Frédérique Oggier, Antonio Campello, Jean-Claude Belfiore & Emanuele Viterbo
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(2)Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

(3)Department of Electrical and Electronic Engineering, Imperial College London, London, UK

(4)Communications and Electronics Department, Télécom ParisTech, Paris, France

(5)Department of Electrical and Computer Systems Engineering, Monash University, Clayton, VIC, Australia

In Chap. 2, interesting lattices together with their parameters and applications were presented. In Chap. 3, one method to build such lattices was discussed, which consists of obtaining lattices from linear codes. This chapter presents two other methods to construct lattices, both called ideal lattices, because they both rely on the structure of ideals in rings. We recall that given a commutative ring R, an ideal of R is an additive subgroup of R which is also closed under multiplication by elements of R. The same terminology is used for two different viewpoints on lattices because of the communities that studied them. We will explain the first method using quadratic fields and refer to [79] for general number field constructions. We note that such a lattice construction from number fields can in turn be combined with Construction A to obtain further lattices, e.g., [59] and references therein. In the second method, “ideal lattices” refer to a family of lattices recently used in cryptography.

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