Improved Upper Bounds on the Hermite and KZ Constants

Wen, Jinming; Chang, Xiao-Wen; Weng, Jian

doi:10.1109/isit.2019.8849452

Cited by 3 publications

(1 citation statement)

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“…In summary, this paper will improve upper bounds on the KZ and Hermite's constants, and upper and lower bounds on both Schnorr's constant and Rankin's constant. More exactly, the main contributions of this paper are summarized as follows, where the first two contributions were published in the conference paper [19].…”

Section: Introductionmentioning

confidence: 99%

Sharper Bounds on Four Lattice Constants

Wen¹,

Chang²

2022

Preprint

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The Korkine-Zolotareff (KZ) reduction, and its generalisations, are widely used lattice reduction strategies in communications and cryptography. The KZ constant and Schnorr's constant were defined by Schnorr in 1987. The KZ constant can be used to quantify some useful properties of KZ reduced matrices. Schnorr's constant can be used to characterize the output quality of his block 2k-reduction and is used to define his semi block 2k-reduction, which was also developed in 1987. Hermite's constant, which is a fundamental constant lattices, has many applications, such as bounding the length of the shortest nonzero lattice vector and the orthogonality defect of lattices. Rankin's constant was introduced by Rankin in 1953 as a generalization of Hermite's constant. It plays an important role in characterizing the output quality of block-Rankin reduction, proposed by Gama et al. in 2006. In this paper, we first develop a linear upper bound on Hermite's constant and then use it to develop an upper bound on the KZ constant. These upper bounds are sharper than those obtained recently by the authors, and the ratio of the new linear upper bound to the nonlinear upper bound, developed by

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Section: Introductionmentioning

confidence: 99%