Lattice Reduction Over Imaginary Quadratic Fields

Lyu, Shanxiang; Porter, Christian; Ling, Cong

doi:10.1109/tsp.2020.3036647

Cited by 8 publications

(10 citation statements)

References 41 publications

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“…Hence, in general, the first successive minimum is expected to be smaller. Noteworthy, (55) is a special variant of the bound for lattices over imaginary quadratic fields [61]. Given quaternion-valued lattices, the bound for H is lowered by a factor of 1 √ 2 ≈ 0.707 in comparison to the bound for L.…”

Section: A Bounds On the Normsmentioning

confidence: 99%

Algorithms and Bounds for Complex and Quaternionic Lattices With Application to MIMO Transmission

Stern

Ling

Fischer

2022

IEEE Trans. Inform. Theory

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Lattices are a popular field of study in mathematical research, but also in more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In mathematical theory, most often lattices over real numbers are considered. However, in communications, complex-valued processing is usually of interest. Besides, by the use of dual-polarized transmission as well as the by the combination of two time slots or frequencies, four-dimensional (quaternion-valued) approaches become more and more important. Hence, in this paper, well-known lattice algorithms and related concepts are generalized to the complex and quaternion-valued case. To this end, a brief review of complex arithmetic, including the sets of Gaussian and Eisenstein integers, and an introduction into quaternion-valued numbers, including the sets of Lipschitz and Hurwitz integers, are given. On that basis, generalized variants of two important algorithms are derived: first, of the polynomial-time LLL algorithm, resulting in a reduced basis of a lattice, and second, of an algorithm to calculate the successive minima-the norms of the shortest independent vectors of a lattice-and its related lattice points. Generalized bounds for the quality of the particular results are established and the asymptotic complexities of the algorithms are assessed. These findings are extensively compared to conventional real-valued processing. It is shown that the generalized

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Section: A Bounds On the Normsmentioning

confidence: 99%

Algorithms and Bounds for Complex and Quaternionic Lattices With Application to MIMO Transmission

Stern

Ling

Fischer

2022

IEEE Trans. Inform. Theory

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“…Of course, for many fields it will hold that this bound is not optimal. For example, the authors have shown in previous work that for Euclidean imaginary quadratic fields, the vectors corresponding to the first two successive minima are always extendable to a basis for the lattice [14], and as such the exponent k − 1 may be replaced with k − l, for some l > 1. In fact, the bound can be improved for both imaginary quadratic and rational quaternion fields.…”

Section: The Quadratic Normmentioning

confidence: 99%

“…Proof. The values for g follows from the fact that the first g vectors must have lengths corresponding to the first g successive minima of the lattice (this is proved in [14], [29], and the author's PhD thesis). As in the proof of the previous theorem, we take the sublattice Λ k−1 spanned by the basis {b 1 , .…”

mentioning

confidence: 99%

Reduction Theory of Algebraic Modules and their Successive Minima

Porter¹,

Ling²

2021

Preprint

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Lattices defined as modules over algebraic rings or orders have garnered interest recently, particularly in the fields of cryptography and coding theory. Whilst there exist many attempts to generalise the conditions for LLL reduction to such lattices, there do not seem to be any attempts so far to generalise stronger notions of reduction such as Minkowski, HKZ and BKZ reduction. Moreover, most lattice reduction methods for modules over algebraic rings involve applying traditional techniques to the embedding of the module into real space, which distorts the structure of the algebra. In this paper, we generalise some classical notions of reduction theory to that of free modules defined over an order. Moreover, we extend the definitions of Minkowski, HKZ and BKZ reduction to that of such modules and show that bases reduced in this manner have vector lengths that can be bounded above by the successive minima of the lattice multiplied by a constant that depends on the algebra and the dimension of the module. In particular, we show that HKZ reduced bases are polynomially close to the successive minima of the lattice in terms of the module dimension. None of our definitions require the module to be embedded and thus preserve the structure of the module.

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“…In [41], an algorithm operating over the set of Eisenstein integers has been employed which is based on a further adaption of the quantization operation. Recently, in [42], LLL reduction over (other) imaginary quadratic (i.e., complex-valued) fields has been studied, again based on the adjustment of the corresponding quantization operation in the reduction algorithm. However, all those publications cover special cases rather than providing generalized criteria and algorithms.…”

Section: Introductionmentioning

confidence: 99%

Algorithms and Bounds for Complex and Quaternionic Lattices With Application to MIMO Transmission

Stern,

Ling,

Fischer

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Lattices are a popular field of study in mathematical research, but also in more practical areas like cryptology or multiple-input/multiple-output (MIMO) transmission. In mathematical theory, most often lattices over real numbers are considered. However, in communications, complex-valued processing is usually of interest. Besides, by the use of dualpolarized transmission as well as by the combination of two time slots or frequencies, four-dimensional (quaternion-valued) approaches become more and more important. Hence, to account for this fact, well-known lattice algorithms and related concepts are generalized in this work. To this end, a brief review of complex arithmetic, including the sets of Gaussian and Eisenstein integers, and an introduction to quaternion-valued numbers, including the sets of Lipschitz and Hurwitz integers, are given. On that basis, generalized variants of two important algorithms are derived: first, of the polynomial-time LLL algorithm, resulting in a reduced basis of a lattice by performing a special variant of the Euclidean algorithm defined for matrices, and second, of an algorithm to calculate the successive minima-the norms of the shortest independent vectors of a lattice-and its related lattice points. Generalized bounds for the quality of the particular results are established and the asymptotic complexities of the algorithms are assessed. These findings are extensively compared to conventional real-valued processing. It is shown that the generalized approaches outperform their real-valued counterparts in complexity and/or quality aspects. Moreover, the application of the generalized algorithms to MIMO communications is studied, particularly in the field of lattice-reduction-aided and integerforcing equalization.

show abstract

Lattice Reduction Over Imaginary Quadratic Fields

Cited by 8 publications

References 41 publications

Algorithms and Bounds for Complex and Quaternionic Lattices With Application to MIMO Transmission

Algorithms and Bounds for Complex and Quaternionic Lattices With Application to MIMO Transmission

Reduction Theory of Algebraic Modules and their Successive Minima

Algorithms and Bounds for Complex and Quaternionic Lattices With Application to MIMO Transmission

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