A Note on Banaszczyk's Inequality

2026-05-11

AI Summary

This work addresses the problem of tail bounds for discrete Gaussian distributions over lattices by introducing novel constraints into Banaszczyk’s inequality, yielding significantly tighter tail estimates. The proposed approach maintains a clear and rigorous proof structure while improving upon existing theoretical bounds. These refined bounds directly enhance the precision of theoretical analyses for dual attacks on the Learning With Errors (LWE) problem, thereby providing stronger foundations for assessing the security of lattice-based cryptographic constructions.

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.

Problem

• Banaszczyk's inequality
• discrete Gaussian measure
• lattice-based cryptography
• Learning With Errors
• dual attacks

Innovation

• Banaszczyk's inequality
• discrete Gaussian measure
• lattice-based cryptography
• dual attack
• Learning With Errors