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A basis of a Euclidean lattice $\Lambda \subset {\mathbb{R}}^n$ is a list of linearly independent vectors $b_1,\dots ,b_n$ such thatThe coefficients must be integers: a real basis spanning the same ambient vector space need not generate the same Euclidean lattice.

For ${\mathbb{Z}}^2$, the vectors $(1,0),(1,1)$ form a lattice basis, because $(a,b)=(a-b)(1,0)+b(1,1)$. The independent pair $(2,0),(0,1)$ instead generates only points with even first coordinate. Writing basis vectors as columns of $B$, another matrix $BU$ is a basis of the same lattice exactly when $U$ is an integer unimodular matrix, with determinant $\pm 1$. The covolume is $|\det B|$; basis changes preserve it. This separates changing a description from changing the discrete set itself.

László Lovász is a mathematician whose work connects combinatorics, graph theory and efficient algorithms. With Arjen K. Lenstra and Hendrik W. Lenstra, he developed the LLL algorithm, which replaces a basis of a Euclidean lattice by a more geometrically controlled basis of the same lattice. Their 1982 paper uses lattice reduction to factor polynomials over the rational numbers in polynomial time.^[ref]

The point of reduction is not to change the lattice: integer basis changes preserve it. Instead, shorter, better organized vectors expose relations hidden by a poor basis. This is a striking bridge from geometric reasoning to exact algebraic computation. Lovász shared the 2021 Abel Prize with Avi Wigderson for contributions linking discrete mathematics and theoretical computer science.^[ref]

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.