### Smoothing Codes and Lattices: Systematic Study and New Bounds

##### 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 Bernouilli noise for codes). This counter-intuitive situation can be resolved by adequate decomposition and truncation of Gaussian and Bernouilli distributions into a superposition of uniform noise, giving further improvement for those cases, and putting them on par with the uniform cases.

Available format(s)
Category
Foundations
Publication info
Preprint.
Keywords
Code-based cryptography Lattice-based cryptography Smoothing parameter
Contact author(s)
thomas debris @ inria fr
L Ducas @ cwi nl
n a resch @ uva nl
jean-pierre tillich @ inria fr
History
2022-09-08: revised
See all versions
Short URL
https://ia.cr/2022/615

CC BY

BibTeX

@misc{cryptoeprint:2022/615,
author = {Thomas Debris and Léo Ducas and Nicolas Resch and Jean-Pierre Tillich},
title = {Smoothing Codes and Lattices: Systematic Study and New Bounds},
howpublished = {Cryptology ePrint Archive, Paper 2022/615},
year = {2022},
note = {\url{https://eprint.iacr.org/2022/615}},
url = {https://eprint.iacr.org/2022/615}
}

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