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Lattice reduction in two dimensions: the black vectors are the given basis for the lattice (represented by blue dots), the red vectors are the reduced basis. In mathematics, the goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This is realized using different ...
Given a basis = {,, …,} with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of R n) with , the LLL algorithm calculates an LLL-reduced (short, nearly orthogonal) lattice basis in time () where is the largest length of under the Euclidean norm, that is, = (‖ ‖, ‖ ‖, …, ‖ ‖).
In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.
In a fractional coordinate system the basis vectors of the coordinate system are chosen to be lattice vectors and the basis is then termed a crystallographic basis (or lattice basis). In a lattice basis, any lattice vector t {\displaystyle \mathbf {t} } can be represented as,
A crystal is made up of one or more atoms, called the basis or motif, at each lattice point. The basis may consist of atoms, molecules, or polymer strings of solid matter, and the lattice provides the locations of the basis. Two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 5 ...
Lattice reduction algorithms aim, given a basis for a lattice, to output a new basis consisting of relatively short, nearly orthogonal vectors. The Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL) was an early efficient algorithm for this problem which could output an almost reduced lattice basis in polynomial time. [33]
In either case, one needs to choose the three lattice vectors a 1, a 2, and a 3 that define the unit cell (note that the conventional unit cell may be larger than the primitive cell of the Bravais lattice, as the examples below illustrate). Given these, the three primitive reciprocal lattice vectors are also determined (denoted b 1, b 2, and b 3).
For example, the vectors (,,), (,,), and (,,) form an alternative basis for . The most important lattice-based computational problem is the shortest vector problem (SVP or sometimes GapSVP), which asks us to approximate the minimal Euclidean length of a non-zero lattice vector.