# Ideal Lattices

## 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.

## References

- 9.E. Bayer-Fluckiger, F. Oggier, E. Viterbo, New algebraic constructions of rotated \(\mathbb {Z}^n\)-lattice constellations for the Rayleigh fading channel. IEEE Trans. Inf. Theory
**50**(4), 702–714 (2004)Google Scholar - 15.J. Boutros, E. Viterbo, C. Rastello, J.C. Belfiore, Good lattice constellations for both rayleigh fading and gaussian channels. IEEE Trans. Inf. Theory
**42**(2), 502–518 (1996)CrossRefzbMATHGoogle Scholar - 58.G.C. Jorge, A.A. de Andrade, S.I.R. Costa, J.E. Strapasson, Algebraic constructions of densest lattices. J. Algebra
**429**, 218–235 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 59.W. Kositwattanarerk, S.S. Ong, F. Oggier, Construction a of lattices over number fields and block fading (wiretap) coding. IEEE Trans. Inf. Theory
**61**(5), 2273–2282 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 71.D. Micciancio, Generalized compact knapsacks, cyclic lattices, and efficient one-way functions. Comput. Complex.
**16**, 365–411 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 73.D. Micciancio, O. Regev,
*Lattice-Based Cryptography*. Post-Quantum Cryptography (Springer, Berlin, 2009)CrossRefzbMATHGoogle Scholar - 79.F. Oggier, E. Viterbo, Algebraic number theory and code design for rayleigh fading channels. Found. Trends Commun. Inf. Theory
**1**(3), 333–415 (2004)CrossRefzbMATHGoogle Scholar - 84.W.W. Peterson, J.B. Nation, M.P. Fossorier, Reflection group codes and their decoding. IEEE Trans. Inf. Theory
**56**(12), 6273–6293 (2010)MathSciNetCrossRefzbMATHGoogle Scholar - 102.D. Stehlé, Ideal lattices. Talk given at Berkeley, 07 July 2015, https://simons.berkeley.edu/sites/default/files/docs/3472/stehle.pdf