Shortest Vector Problem

Years and Authors of Summarized Original Work

• 1982; Lenstra, Lenstra, Lovasz

Problem Definition

A point lattice is the set of all integer linear combinations

$$ \mathcal{L}({\mathbf b_1, \dots,\mathbf b_n}) = \left\{ \sum_{i=1}^{n} x_i \mathbf b_i \colon x_1, \dots,x_n \in \mathbb{Z} \right\} $$

of n linearly independent vectors ${ \mathbf b_1, \dots,\mathbf b_n \!\in \mathbb{R}^m }$ in m-dimensional Euclidean space. For computational purposes, the lattice vectors ${ \mathbf b_1, \dots,\mathbf b_n }$ are often assumed to have integer (or rational) entries, so that the lattice can be represented by an integer matrix ${ \mathbf B = [\mathbf b_1, \dots,\mathbf b_n] \in \mathbb{Z}^{m\times n} }$ (called basis) having the generating vectors as columns. Using matrix notation, lattice points in ${ \mathcal{L}({\mathbf B}) }$ can be conveniently represented as ${ \mathbf{B} \mathbf x }$ where ${ \mathbf x }$ is an integer vector. The integers m and n are called the dimension...