Gitterbasen

Gitterbasen (literally lattice bases) is a term used in mathematics and related disciplines to refer to a basis of a lattice. A lattice L in R^n is a discrete subgroup formed by all integer combinations of n linearly independent vectors b1, ..., bn; these vectors constitute a Gitterbasen. Every point of L can be written uniquely as z1 b1 + ... + zn bn with integers z i.

If B is the n×n matrix whose columns are the basis vectors, then L = B Z^n. The determinant det(B) equals the volume of the fundamental parallelepiped and is an invariant of the lattice up to sign. A change of basis via an integer unimodular matrix U yields a new basis B' = B U that spans the same lattice.

Basis reduction: Algorithms such as the LLL algorithm produce a short, nearly orthogonal Gitterbasen, which simplifies solving the closest vector problem or SVP and improves numerical stability in computations.

Applications: Lattice bases are central in crystallography and solid-state physics for modeling crystal structures, in computer graphics and digital geometry for grid-based representations, and in cryptography, where the hardness of lattice problems underpins many schemes.

Examples: The standard lattice Z^n uses the basis e1, ..., en with e i as unit vectors. A different basis for Z^2 is b1 = (2, 1), b2 = (1, -1), which spans the same lattice.

See also: lattice basis; change of basis; Gram matrix; unimodular matrix; LLL algorithm.