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arXiv · 2305.19777

On the Shortest Lattice Vector vs. the Shortest Basis

Yael Eisenberg·Itamar Rot·Muli Safra
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Abstract

Given an arbitrary basis for a mathematical lattice, to find a ``good" basis for it is one of the classic and important algorithmic problems. In this note, we give a new and simpler proof of a theorem by Regavim (arXiv:2106.03183): we construct a 18-dimensional lattice that does not have a basis that satisfies the following two properties simultaneously: 1. The basis includes the shortest non-zero lattice vector. 2. The basis is shortest, that is, minimizes the longest basis vector (alternatively: the sum or the sum-of-squares of the basis vectors). The vectors' length can be measured in any $\ell^q$ norm, for $q\in \mathbb{N}_+$ (albeit, via another lattice, of a somewhat larger dimension).

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Yael Eisenberg, Itamar Rot, Muli Safra. 2023-05-31. On the Shortest Lattice Vector vs. the Shortest Basis. https://arxiv.org/abs/2305.19777

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