A Note on Banaszczyk's Inequality

TL;DR

This paper improves Banaszczyk's inequality for lattice Gaussian measures under certain conditions, providing a better bound with implications for cryptographic attacks on LWE.

Contribution

It offers a refined version of Banaszczyk's inequality with a transparent proof, enhancing its applicability in lattice cryptography.

Findings

Achieves a significantly better bound for the inequality

Provides a transparent proof of the improved inequality

Enables improved analysis of dual attacks on LWE

Abstract

Banaszczyk's inequality establishes a tail estimate for the discrete Gaussian measure on a lattice in $\mathbb{R}^n$. This classic result has been influential and plays an important role in lattice-based cryptography. An improvement of the inequality with a transparent proof was given by Tian, Liu and Xu. In this note, we further improve this inequality by imposing an appropriate condition, obtaining a significantly better bound. This refined inequality can be used to investigate dual attacks against the Learning With Errors (LWE) problem.