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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 ...
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]
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.
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 ...
A basis of linearly independent lattice vectors b 1, b 2, ..., b n can be defined by g(b j) = λ j.. The lower bound is proved by considering the convex polytope 2n with vertices at ±b j / λ j, which has an interior enclosed by K and a volume which is 2 n /n!λ 1 λ 2...λ n times an integer multiple of a primitive cell of the lattice (as seen by scaling the polytope by λ j along each basis ...
In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions and modular forms are defined. Fundamental parallelogram defined by a pair of vectors in the complex plane.
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, = (‖ ‖, ‖ ‖, …, ‖ ‖).