Smoothing Codes and Lattices: Systematic Study and New Bounds

arXiv (Cornell University)

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.

Authors 4

1. Thomas Debris-Alazard

   Institut national de recherche en sciences et technologies du numérique

   Affiliation as printed

   Inria

2. Léo Ducas Aachen Mathematical Institute

   Leiden University · Centrum Wiskunde & Informatica

   Affiliation as printed

   CWI , Amsterdam , The Netherlands

   Mathematical Institute , Leiden University

3. Nicolas Resch

   Institute of Informatics of the Slovak Academy of Sciences · University of Amsterdam

   Affiliation as printed

   Informatics' Institute , University of Amsterdam

4. Jean–Pierre Tillich

   Institut national de recherche en sciences et technologies du numérique

   Affiliation as printed

   Inria