Abstract
For 0 <= alpha <= 1/2, we show an algorithm that does the following. Given appropriate preprocessing P(L) consisting of N_alpha := 2^{O(n^{12 alpha} + log n)} vectors in some lattice L subset {R}^n and a target vector t in R^n, the algorithm finds y in L such that yt <= n^{1/2 + alpha} eta(L) in time poly(n) * N_alpha, where eta(L) is the smoothing parameter of the lattice.
The algorithm itself is very simple and was originally studied by Doulgerakis, Laarhoven, and de Weger (to appear in PQCrypto, 2019), who proved its correctness under certain reasonable heuristic assumptions on the preprocessing P(L) and target t. Our primary contribution is a choice of preprocessing that allows us to prove correctness without any heuristic assumptions.
Our main motivation for studying this is the recent breakthrough algorithm for IdealSVP due to Hanrot, Pellet  Mary, and StehlĂ© (to appear in Eurocrypt, 2019), which uses the DLW algorithm as a key subprocedure. In particular, our result implies that the HPS IdealSVP algorithm can be made to work with fewer heuristic assumptions.
Our only technical tool is the discrete Gaussian distribution over L, and in particular, a lemma showing that the onedimensional projections of this distribution behave very similarly to the continuous Gaussian. This lemma might be of independent interest.
BibTeX  Entry
@InProceedings{stephensdavidowitz:LIPIcs:2019:10833,
author = {Noah StephensDavidowitz},
title = {{A TimeDistance TradeOff for GDD with Preprocessing  Instantiating the DLW Heuristic}},
booktitle = {34th Computational Complexity Conference (CCC 2019)},
pages = {11:111:8},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771160},
ISSN = {18688969},
year = {2019},
volume = {137},
editor = {Amir Shpilka},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10833},
URN = {urn:nbn:de:0030drops108331},
doi = {10.4230/LIPIcs.CCC.2019.11},
annote = {Keywords: Lattices, guaranteed distance decoding, GDD, GDDP}
}
Keywords: 

Lattices, guaranteed distance decoding, GDD, GDDP 
Collection: 

34th Computational Complexity Conference (CCC 2019) 
Issue Date: 

2019 
Date of publication: 

16.07.2019 