Lattice

A lattice is a discrete set of points generated by all integer combinations of a finite set of linearly independent basis vectors. Its repeating structure supports geometric, algebraic, and computational analysis.

Linear independence: A set of vectors is linearly independent when no nontrivial linear combination of them equals zero. Independent basis vectors prevent duplicate coordinates and determine the lattice’s dimension.
Lattice basis: A lattice basis is a set of vectors whose integer combinations form a particular lattice. It provides a finite description of an otherwise infinite point set.
Lattice-based cryptography: Lattice-based cryptography builds security on computational problems involving high-dimensional lattices. Hard lattice problems underpin encryption and signatures designed to resist quantum attacks.
Abelian group: An Abelian group is a group whose operation is commutative. Under addition, a lattice is a free Abelian group generated by its basis vectors.
Quasicrystal: A quasicrystal has long-range order but lacks the translational periodicity of a crystal lattice. Quasicrystals show that ordered atomic patterns need not arise from a periodic lattice.
Integer: An integer is a whole number, positive, negative, or zero. Integer coefficients specify exactly which combinations of basis vectors belong to the lattice.
Lattice determinant: The determinant of a lattice is the volume of a fundamental parallelepiped formed by a basis. It gives the volume per lattice point and is unchanged by a basis change.
Closest vector problem: The closest vector problem asks for the lattice point nearest to a given target point. It models decoding and optimization tasks over lattice points.
Bravais lattice: A Bravais lattice is an infinite array of points generated by translations in a crystal’s space. It is the crystallographic use of periodic point lattices.
Dense subgroup: A dense subgroup has elements arbitrarily close to every point in its ambient space. Unlike a lattice, a dense subgroup has no isolated points.

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