Discrete Gaussian Measures and New Bounds of the Smoothing Parameter for Lattices

Zhongxiang Zheng, Guangwu Xu, and Chunhuan Zhao

Abstract

In this paper, we start with a discussion of discrete Gaussian measures on lattices. Several results of Banaszczyk are analyzed, different approaches are suggested. In the second part of the paper we prove two new bounds for the smoothing parameter of lattices. Under the natural assumption that $\varepsilon$ is suitably small, we obtain two estimations of the smoothing parameter: 1. $\eta_{\varepsilon}(\mathbb{Z}) \le \sqrt{\frac{\ln \big(\frac{\varepsilon}{44}+\frac{2}{\varepsilon}\big)}{\pi}}.$ 2. For a lattice ${\cal L}\subset \mathbb{R}^n$ of dimension $n$, $\eta_{\varepsilon}({\cal L}) \le \sqrt{\frac{\ln \big(n-1+\frac{2n}{\varepsilon}\big)}{\pi}}\tilde{bl}({\cal L}).$

Available format(s)
Publication info
Preprint. MINOR revision.
Keywords
Latticesdiscrete Gaussian measuresmoothing parameter
Contact author(s)
gxu4uwm @ uwm edu
History
Short URL
https://ia.cr/2018/786

CC BY

BibTeX

@misc{cryptoeprint:2018/786,
author = {Zhongxiang Zheng and Guangwu Xu and Chunhuan Zhao},
title = {Discrete Gaussian Measures and New Bounds of the Smoothing Parameter for Lattices},
howpublished = {Cryptology ePrint Archive, Paper 2018/786},
year = {2018},
note = {\url{https://eprint.iacr.org/2018/786}},
url = {https://eprint.iacr.org/2018/786}
}
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