Smoothing Codes and Lattices: Systematic Study and New Bounds

TL;DR

This paper systematically studies smoothing bounds for lattices and codes, introducing new techniques and bounds that improve upon previous results, especially for uniform noise distributions, with implications for cryptographic security.

Contribution

It provides a unified framework for deriving smoothing bounds for both lattices and codes, utilizing diverse noise distributions and improving existing bounds with novel decomposition methods.

Findings

Optimal bounds combine Parseval's Identity, Cauchy-Schwarz, and linear programming.

Uniform ball noise yields better bounds than Gaussian or Bernoulli noise.

Decomposition of distributions enhances bounds for Gaussian and Bernoulli noise.

Abstract

In this article we revisit smoothing bounds in parallel between lattices $and$ codes. Initially introduced by Micciancio and Regev, these bounds were instantiated with Gaussian distributions and were crucial for arguing the security of many lattice-based cryptosystems. Unencumbered by direct application concerns, we provide a systematic study of how these bounds are obtained for both lattices $and$ codes, transferring techniques between both areas. We also consider multiple choices of spherically symmetric noise distribution. We found that the best strategy for a worst-case bound combines Parseval's Identity, the Cauchy-Schwarz inequality, and the second linear programming bound, and this holds for both codes and lattices and all noise distributions at hand. For an average-case analysis, the linear programming bound can be replaced by a tight average count. This alone gives optimal results for spherically uniform noise over random codes and random lattices. This also improves previous Gaussian smoothing bound for worst-case lattices, but surprisingly this provides even better results with uniform ball noise than for Gaussian (or Bernoulli noise for codes). This counter-intuitive situation can be resolved by adequate decomposition and truncation of Gaussian and Bernoulli distributions into a superposition of uniform noise, giving further improvement for those cases, and putting them on par with the uniform cases.