# A time-distance trade-off for GDD with preprocessing---Instantiating the DLW heuristic

## TL;DR

This paper presents a non-heuristic preprocessing method for a lattice-based algorithm that improves the reliability of solving approximate closest vector problems, with implications for cryptographic algorithms like IdealSVP.

## Contribution

It introduces a specific preprocessing approach that guarantees correctness of the DLW heuristic algorithm without heuristic assumptions, enhancing its applicability.

## Key findings

- Preprocessing enables heuristic-free correctness proof.
- The approach reduces heuristic assumptions in IdealSVP algorithms.
- A new lemma on Gaussian projections over lattices may be of independent interest.

## Abstract

For $0 \leq \alpha \leq 1/2$, we show an algorithm that does the following. Given appropriate preprocessing $P(\mathcal{L})$ consisting of $N_\alpha := 2^{O(n^{1-2\alpha} + \log n)}$ vectors in some lattice $\mathcal{L} \subset \mathbb{R}^n$ and a target vector $\boldsymbol{t}\in \mathbb{R}^n$, the algorithm finds $\boldsymbol{y} \in \mathcal{L}$ such that $\|\boldsymbol{y}- \boldsymbol{t}\| \leq n^{1/2 + \alpha} \eta(\mathcal{L})$ in time $\mathrm{poly}(n) \cdot N_\alpha$, where $\eta(\mathcal{L})$ 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 $P(\mathcal{L})$ and target $\boldsymbol{t}$. 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 Stehl\'e (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 $\mathcal{L}$, 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.