# Approximate Voronoi cells for lattices, revisited

## TL;DR

This paper refines the understanding of approximate Voronoi cells for lattices, providing new asymptotic volume results and improved time-memory trade-offs for solving the closest vector problem efficiently.

## Contribution

It settles open problems on the volume asymptotics of Voronoi cells and introduces improved bounds and trade-offs for lattice CVPP algorithms.

## Key findings

- Established exact asymptotics for Voronoi cell volume under Gaussian heuristic.
- Improved upper bounds on the time complexity of the randomized iterative slicer.
- Achieved a continuous trade-off between advice size and query time complexity.

## Abstract

We revisit the approximate Voronoi cells approach for solving the closest vector problem with preprocessing (CVPP) on high-dimensional lattices, and settle the open problem of Doulgerakis-Laarhoven-De Weger [PQCrypto, 2019] of determining exact asymptotics on the volume of these Voronoi cells under the Gaussian heuristic. As a result, we obtain improved upper bounds on the time complexity of the randomized iterative slicer when using less than $2^{0.076d + o(d)}$ memory, and we show how to obtain time-memory trade-offs even when using less than $2^{0.048d + o(d)}$ memory. We also settle the open problem of obtaining a continuous trade-off between the size of the advice and the query time complexity, as the time complexity with subexponential advice in our approach scales as $d^{d/2 + o(d)}$, matching worst-case enumeration bounds, and achieving the same asymptotic scaling as average-case enumeration algorithms for the closest vector problem.

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Source: https://tomesphere.com/paper/1907.04630