Shortest Vector Problem

Related Concepts

Closest Vector Problem; Lattice; Lattice Reduction; Lattice-Based Cryptography; NTRU

Definition

Given k linearly independent (typically integer) vectors $\mathbf{ B} = [{\mathbf{b}}_{1},\ldots,{\mathbf{b}}_{k}]$ in n-dimensional Euclidean space ${\mathbb{R}}^{n}$, the Shortest Vector Problem (SVP) asks to find a nonzero integer linear combination $\mathbf{{B}{x}} ={ \sum \nolimits }_{i=1}^{k}{\mathbf{b}}_{i}{x}_{i}$ (with ${x} \in {\mathbb{Z}}^{k}\setminus \{{0}\}$) such that the norm $\|\mathbf{{B}{x}}\|$ is as small as possible. The problem can be defined with respect to any norm, but the Euclidean norm $\|\mathbf{v}\| = \sqrt{{\sum \nolimits }_{i=1}^{n}{v}_{i}^{2}}$ is the most common. The set of all integer linear combinations $\mathcal{L}(\mathbf{B}) =\{\mathbf{{B}{x}:{x}} \in {\mathbb{Z}}^{k}\}$ is a lattice. So, SVP can be concisely defined as the problem of finding the shortest nonzero vector in the lattice represented by $\mathbf{B}$. (Refer...