Algebraic constructions of densest lattices

Jorge, Grasiele C.; Andrade, Antonio Aparecido de; Costa, Sueli I. R.; Strapasson, João E.

doi:10.1016/j.jalgebra.2014.12.044

Cited by 20 publications

(18 citation statements)

References 21 publications

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“…Constructions of rotated D 3 , D 5 and E 7 -lattices via ideals and free Z-modules that are not ideals are also presented. The same lattices are also constructed in [15,16], through a different approach, where the authors construct these lattices by shifting ideal lattices constructed over cyclotomic fields via ideal or module in the maximal totally real subfields of cyclotomic fields. 180 ] and let M 1 be a submodule of O K generated by the linearly independent vectors In this case, σ α (M 1 ) is a lattice of rank 10 in R 24 and forall x ∈ M 1 we have that σ α (x) = γT A, where γ = (a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , a 9 , a 10 ),…”

Section: Constructions Of Algebraic Latticesmentioning

confidence: 99%

Trace forms of certain subfields of cyclotomic fields and applications

Ferrari

Andrade

Araujo

et al. 2020

Journal of Algebra Combinatorics Discrete Structures and Applications

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In this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in Euclidean space with optimal center density. We also obtain a closed formula for the Gram matrix of algebraic lattices obtained from these subfields. The obtained lattices are rotated versions of the lattices Λ9, Λ10 and Λ11 and they are images of Z-submodules of rings of integers under the twisted homomorphism, and these constructions, as algebraic lattices, are new in the literature. We also obtain algebraic lattices in odd dimensions up to 7 over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity.

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Section: Constructions Of Algebraic Latticesmentioning

confidence: 99%

Trace forms of certain subfields of cyclotomic fields and applications

Ferrari

Andrade

Araujo

et al. 2020

Journal of Algebra Combinatorics Discrete Structures and Applications

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“…In the next examples we have implemented algorithms following the ideas of [24], [25]. We have used thein the Mathematica software for the Gram matrix LLL reduction and the SAGE [26] for the computation of minimum distance.…”

Section: Families Of Lattices In Dimensionmentioning

confidence: 99%

Lattices Associated with Octonion Algebras

Brasil¹,

Benedito

Costa³

2018

JCIS

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Abstract-In this paper we propose a construction of lattices in dimension 8n via octonion orders over a totally real number field. To show the potential of this construction, we present rotated versions of E 8 , Λ 16 , the densest known lattices in these dimensions and a lattice with the same density of the Barnes-Wall lattice in dimension 32. Perspectives applications of these lattices based on octonion algebras are in lattice based cryptography and lattice coding for MIMO and MISO channels.

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“…In [16], [17], some of the best known sphere packings in low dimensions were constructed from real number fields. In our simulations we will consider the construction of D 4 from [16] and E 8 from [17], which we explain in more detail in the next section.…”

Section: B Well-rounded Lattices From Number Fieldsmentioning

confidence: 99%

“…Hence, we can use it to get E 8 as an embedding of a Z-module in K 17 (see [20]). However, this construction has a smaller product distance than another version of E 8 from [17] constructed from an ideal in O K60 , which we use for our simulations. As a sublattice, we choose 2E 8 having nesting index 256.…”

Section: From Number Field Extensionsmentioning

confidence: 99%

Analysis of Some Well-Rounded Lattices in Wiretap Channels

Damir

Gnilke

Amorós

et al. 2018

2018 IEEE 19th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

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Recently, various criteria for constructing wiretap lattice coset codes have been proposed, most prominently the minimization of the so-called flatness factor. However, these criteria are not constructive per se. As explicit constructions, well-rounded lattices have been proposed as possible minimizers of the flatness factor, but no rigorous proof has been given. In this paper, we study various well-rounded lattices, including the best sphere packings, and analyze their shortest vector lengths, minimum product distances, and flatness factors, with the goal of acquiring a better understanding of the role of these invariants regarding secure communications. Simulations are carried out in dimensions four and eight, yielding the conclusion that the best sphere packing does not necessarily yield the best performance, not even when compared to other well-rounded lattices having the same superlattice. This motivates further study and construction of well-rounded lattices for physical layer security. Index Terms-Algebraic number fields, coset coding, ideals, single-input single-output (SISO) channels, sphere packings, wellrounded lattices, wiretap channels.

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Algebraic constructions of densest lattices

Cited by 20 publications

References 21 publications

Trace forms of certain subfields of cyclotomic fields and applications

Trace forms of certain subfields of cyclotomic fields and applications

Lattices Associated with Octonion Algebras

Analysis of Some Well-Rounded Lattices in Wiretap Channels

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