Ideal Lattices

  • Sueli I. R. Costa
  • Frédérique Oggier
  • Antonio Campello
  • Jean-Claude Belfiore
  • Emanuele Viterbo
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

In Chapter 2, interesting lattices together with their parameters and applications were presented. In Chapter 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 view points on lattices because of the communities that studied them. We will explain the former technique 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. For the latter case, “ideal lattices” refer to a family of lattices recently used in cryptography.

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Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Sueli I. R. Costa
    • 1
  • Frédérique Oggier
    • 2
  • Antonio Campello
    • 3
  • Jean-Claude Belfiore
    • 4
  • Emanuele Viterbo
    • 5
  1. 1.Institute of Mathematics, Statistics and Computer ScienceUniversity of CampinasCampinasBrazil
  2. 2.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore
  3. 3.Department of Electrical and Electronic EngineeringImperial College LondonLondonUK
  4. 4.Communications and Electronics DepartmentTélécom ParisTechParisFrance
  5. 5.Department of Electrical and Computer Systems EngineeringMonash UniversityClaytonAustralia

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