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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Notes
- 1.
Such suitable structures are orders (rings with a \(\mathbb {Z}\)-basis) and their ideals, which explains the terminology ideal lattice.
- 2.
Writing \(\sigma (\sqrt {\alpha }\mathbb {Z}[\theta ])\) is a slight abuse of notation, since σ cannot really be applied to \(\sqrt {\alpha }\) when it does not belong to \(\mathbb {Z}[\theta ]\).
- 3.
We voluntarily skip the definition of compositum of two fields with coprime discriminants here, which would be the proper way to describe the suitable field extension.
- 4.
A reader familiar with the theory of cyclic codes will notice the analogy between cyclic codes and cyclic lattices and their characterization.
- 5.
For linear codes, we would call these pseudo-cyclic codes.
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Costa, S.I.R., Oggier, F., Campello, A., Belfiore, JC., Viterbo, E. (2017). Ideal Lattices. In: Lattices Applied to Coding for Reliable and Secure Communications. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-67882-5_4
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DOI: https://doi.org/10.1007/978-3-319-67882-5_4
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