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**2018/786**

**Title:**

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

**Authors:**
Zhongxiang Zheng, Guangwu Xu, 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}). ]

**ePrint:**
https://eprint.iacr.org/2018/786

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