Bounding basis reduction properties

Neumaier, Arnold

doi:10.1007/s10623-016-0273-9

Cited by 14 publications

(10 citation statements)

References 33 publications

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“…In this paper, we have investigated some vital properties of KZ reduced matrices and developed an improved KZ reduction algorithm. We first developed a linear upper bound on the Hermit constant which is around 7 8 times of the upper bound given by [31,Thm. 3.4], and an upper bound on the KZ constant which is polynomially small than [32,Thm.…”

Section: Discussionmentioning

confidence: 99%

“…which is presented in [31]. It is stated in [31,Thm. 3.4] that this bound can be proved by combining (10) and the fact that the inequality holds for 3 ≤ n ≤ 36 [40].…”

Section: A a Linear Upper Bound On The Hermite Constantmentioning

confidence: 99%

“…• Some important properties of a KZ reduced matrix are studied in this paper. Specifically, we first propose a linear upper bound on the Hermit constant, which is around 7 8 times of the existing sharpest one that was recently presented in [31,Thm. 3.4].…”

Section: Introductionmentioning

confidence: 99%

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On the KZ Reduction

Wen

Chang

2019

IEEE Trans. Inform. Theory

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The Korkine-Zolotareff (KZ) reduction is one of the often used reduction strategies for lattice decoding. In this paper, we first investigate some important properties of KZ reduced matrices. Specifically, we present a linear upper bound on the Hermit constant which is around 7 8 times of the existing sharpest linear upper bound, and an upper bound on the KZ constant which is polynomially smaller than the existing sharpest one. We also propose upper bounds on the lengths of the columns of KZ reduced matrices, and an upper bound on the orthogonality defect of KZ reduced matrices which are even polynomially and exponentially smaller than those of boosted KZ reduced matrices, respectively. Then, we derive upper bounds on the magnitudes of the entries of any solution of a shortest vector problem (SVP) when its basis matrix is LLL reduced. These upper bounds are useful for analyzing the complexity and understanding numerical stability of the basis expansion in a KZ reduction algorithm. Finally, we propose a new KZ reduction algorithm by modifying the commonly used Schnorr-Euchner search strategy for solving SVPs and the basis expansion method proposed by Zhang et al.Simulation results show that the new KZ reduction algorithm is much faster and more numerically reliable than the KZ reduction algorithm proposed by Zhang et al., especially when the basis matrix is ill conditioned.

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Section: Discussionmentioning

confidence: 99%

“…which is presented in [31]. It is stated in [31,Thm. 3.4] that this bound can be proved by combining (10) and the fact that the inequality holds for 3 ≤ n ≤ 36 [40].…”

Section: A a Linear Upper Bound On The Hermite Constantmentioning

confidence: 99%

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On the KZ Reduction

Wen

Chang

2019

IEEE Trans. Inform. Theory

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“…As explained in Section I, linear upper bounds on γ n are very useful. There are several linear upper bounds: γ n ≤ 2 3 n (for n ≥ 2) [6]; γ n ≤ 1 + n 4 (for n ≥ 1) [14, p.35] and γ n ≤ n+6 7 (for n ≥ 2) [15]. The most recent linear upper bound on γ n is…”

Section: A Sharpermentioning

confidence: 99%

“…In the above applications, the Hermite constant's linear upper bounds play crucial roles. Hence, in addition to the nonlinear upper bound [13], several linear upper bounds on the Hermite constant have been proposed in [6], [14], [15]. The second main aim of this paper is to improve the sharpest available linear upper bound provided in [7].…”

Section: Introductionmentioning

confidence: 99%

Improved Upper Bounds on the Hermite and KZ Constants

Wen

Chang

Weng

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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The Korkine-Zolotareff (KZ) reduction is a widely used lattice reduction strategy in communications and cryptography. The Hermite constant, which is a vital constant of lattice, has many applications, such as bounding the length of the shortest nonzero lattice vector and orthogonality defect of lattices. The KZ constant can be used in quantifying some useful properties of KZ reduced matrices. In this paper, we first develop a linear upper bound on the Hermite constant and then use the bound to develop an upper bound on the KZ constant. These upper bounds are sharper than those obtained recently by the first two authors. Some examples on the applications of the improved upper bounds are also presented.

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