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In the geometry of numbers, a lattice algorithm works with a Euclidean lattice, usually specified by linearly independent vectors $b_1,\dots ,b_m$ whose integer linear combinations formThe basis of a Euclidean lattice is not unique. For example, $(1,0)$ and $(100,1)$ generate the same Euclidean lattice as $(1,0)$ and $(0,1)$: replace the second vector by itself minus 100 times the first. This integer linear combination is reversible, so it changes the representation without changing the set of Euclidean lattice points.

Lattice reduction seeks a more useful basis of a Euclidean lattice, with shorter and less nearly dependent vectors. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm combines integer basis changes with Gram-Schmidt orthogonalization and comparisons controlling the quality of the result. It is a polynomial-time example, but it does not generally return an exactly shortest vector.^[ref] Better bases make geometric searches and lattice-based cryptography analysis more practical. The geometry and the sizes of the integer coordinates both matter: a small number of arithmetic steps need not imply a small bit complexity.