# New bounds of the smoothing parameter for lattices

## TL;DR

This paper improves the upper bound of the smoothing parameter in lattice-based cryptography, making it more efficient for practical use.

## Contribution

A new upper bound for the smoothing parameter is established, improving upon classical results for both low and high-dimensional lattices.

## Key findings

- A more precise upper bound is achieved for one-dimensional integer lattices under optimized conditions.
- A new upper bound is derived for high-dimensional lattices with large dimensions and specific error parameters.
- The new bound allows for a smaller and more natural error parameter setting in practical cryptographic applications.

## Abstract

The smoothing parameter on lattices is crucial for lattice-based cryptographic design. In this study, we establish a new upper bound for the lattice smoothing parameter, which represents an improvement over several significant classical findings. For one-dimensional integer lattices, under specific and optimized conditions, we have achieved a more precise upper bound compared to previous research. Regarding general high-dimensional lattices, when the lattice dimension is large enough and the error parameter is within a particular range, we have derived a new upper bound. In the practical applications of lattice-based cryptography, where the lattice dimension is typically large, our new bound enables a more natural and smaller setting for the error parameter, thereby improving the upper bounds on all known smoothing parameters.