Gruber P. Convex and Discrete Geometry

28 Basis Reduction and Polynomial Algorithms 411
28 Basis Reduction and Polynomial Algorithms
The basic problem of reduction theory can be stated in two equivalent ways. First,
given a lattice, determine a (not necessarily unique) basis having nice geometric
properties, a reduced basis. Here, nice may mean that the basis vectors are short or
almost orthogonal. Second, given a positive definite quadratic form, find a (not necessarily
unique) equivalent form having nice arithmetic properties, a reduced form.
There are several classical reduction methods for positive definite quadratic
forms, going back at least to Lagrange [628]. Rather geometric are the reduction
methods of Korkin and Zolotarev [610] and Lenstra, Lenstra and Lovász [646]. The
latter provides polynomial time algorithms for geometric problems dealing with the
shortest lattice vector problem and thus with Minkowski’s fundamental theorem and
with the nearest lattice point problem. In addition, it yields polynomial time algorithms
for a multitude of other problems. These include the factoring of polynomials,
Diophantine approximation, integer programming and cryptography. It was used for
the disproof of Mertens’s conjecture on the Riemann zeta-function and had a strong
impact on complexity problems in algorithmic geometry. The Lenstra, Lenstra and
Lovász algorithm was inspired by the 2-dimensional reduction method of Gauss.
In the following we first present the LLL-basis reduction algorithm and then
describe its applications to Diophantine approximation and the nearest lattice point
problem.
For additional information and references on reduction and geometric algorithms,
see Gruber and Lekkerkerker [447], Kannan [563], Grötschel, Lovász and
Schrijver [409], Gruber [430], Koy and Schnorr [613] and Micciancio and Goldwasser
[721]. We are not aware of a comprehensive and unified exposition of the various reduction
methods.
28.1 LLL-Basis Reduction
We will describe the Lenstra, Lenstra and Lovász [646] basis reduction algorithm
and show that it is polynomial in input size. Our presentation follows the exposition
of Grötschel, Lovász and Schrijver [409].
Definition of LLL-Reduced Bases
Let {b1,...,bd} be an (ordered) basis of a lattice L in E d . Using the Gram–
Schmidt orthogonalization method, we assign to {b1,...,bd} an (ordered) system
{ ˆb1,..., ˆbd} of d orthogonal vectors such that:
(1)
b1 = ˆb1
b2 = µ21 ˆb1 + ˆb2
..................................
bd = µd1 ˆb1 +···+µdd−1 ˆbd−1 + ˆbd
where µ ji = b j · ˆbi
.
� ˆb 2
j�