# Closest Vectors, Successive Minima, and Dual HKZ-Bases of Lattices

## Abstract

In this paper we introduce a new technique to solve lattice problems. The technique is based on dual HKZ-bases. Using this technique we show how to solve the closest vector problem in lattices with rank n in time n! · *s* ^{O(1)}, where *s* is the input size of the problem. This is an exponential improvement over an algorithm due to Kannan and Helfrich [16,15]. Based on the new technique we also show how to compute the successive minima of a lattice in time *n*! · 3^{n} · *s* ^{O(1)}, where *n* is the rank of the lattice and *s* is the input size of the lattice. The problem of computing the successive minima plays an important role in Ajtai’s worst-case to average-case reduction for lattice problems. Our results reveal a close connection between the closest vector problem and the problem of computing the successive minima of a lattice.

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