论文标题
四个晶格常数的尖锐界限
Sharper Bounds on Four Lattice Constants
论文作者
论文摘要
Korkine-Zolotareff（Kz）减少及其概括是通信和加密术中广泛使用的晶格策略。 KZ常数和Schnorr的常数由Schnorr在1987年定义。Kz常数可用于量化Kz还原矩阵的某些有用特性。 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.兰金（Rankin）在1953年引入了兰金（Rankin）的常数，这是对Hermite常数的概括。 Gama等人提出，它在表征块酶减少的输出质量方面起着重要作用。在2006年。在本文中，我们首先在Hermite的常数上开发了线性上限，然后使用它在Kz常数上开发上限。这些上限比作者最近获得的上限要尖锐，而新线性上限与1929年Blichfeldt在Hermite常数上开发的新线性上限与非线性上限的比率是渐近的1.0047。此外，我们在Schnorr的常数上发展了下层和上限。与Gama等人开发的最尖锐的局面相比，下限的改进。在渐近的1.7倍大约是1.7倍，并且在最尖锐的现有界限上的改进也是Gama等人也开发的。渐近差4倍。最后，我们在Rankin的常数上发展了下层和上限。在定义常数的参数中，对最清晰的现有界限的改进是指数的。
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 Blichfeldt in 1929, on Hermite's constant is asymptotically 1.0047. Furthermore, we develop lower and upper bounds on Schnorr's constant. The improvement to the lower bound over the sharpest existing one developed by Gama et al. is around 1.7 times asymptotically, and the improvement to the upper bound over the sharpest existing one which was also developed by Gama et al. is around 4 times asymptotically. Finally, we develop lower and upper bounds on Rankin's constant. The improvements of the bounds over the sharpest existing ones, also developed by Gama et al., are exponential in the parameter defining the constant.
