A Time-Distance Trade-Off for GDD with Preprocessing - Instantiating the DLW Heuristic
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^{1-2 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 ||y-t|| <= 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 Stehlé (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 one-dimensional projections of this distribution behave very similarly to the continuous Gaussian. This lemma might be of independent interest.
Lattices
guaranteed distance decoding
GDD
GDDP
Theory of computation~Design and analysis of algorithms
11:1-11:8
Regular Paper
A full version of the paper is available at http://arxiv.org/abs/1902.08340. (Please read the full version!)
I thank Guillaume Hanrot, Thijs Laarhoven, Alice Pellet - Mary, Oded Regev, and Damien Stehlé for helpful discussions. I also thank Alice Pellet - Mary, Guillaume Hanrot, and Damien Stehlé for sharing early versions of their work with me. I am also grateful to the CCC 2019 reviewers for their very helpful comments, and Daniel Dadush for showing me how to obtain the stronger results to be written up in the full version.
Noah
Stephens-Davidowitz
Noah Stephens-Davidowitz
Massachusetts Institute of Technology, Cambridge, MA, USA
http://www.noahsd.com
10.4230/LIPIcs.CCC.2019.11
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Noah Stephens-Davidowitz
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