Gruber P. Convex and Discrete Geometry

Shortest Lattice Vector Problem
28 Basis Reduction and Polynomial Algorithms 419
The formal statement of this homogeneous problem is as follows:
Problem 28.1. Given a lattice L in E d , find a point l ∈ L \{o}, such that:
�l� =min � �m� :m ∈ L \{o} � .
In the present chapter this problem appears in different versions:
Find a point v ∈ Z d \{o} such that q(v) = min � q(u) : u ∈ Z d \{o} � , where
q(·) is a positive definite quadratic form, see Corollary 22.1.
Determine λ1(B d , L) = 2ϱ(B d , L) or, more generally, determine λ1(C, L) =
2ϱ(C, L) where C is a o-symmetric convex body. In essence, the latter means
that the Euclidean norm is replaced by an arbitrary norm on E d . See Sects. 23.2
and 26.3.
The Minkowski fundamental theorem applied to B d shows that:
min � �m� :m ∈ L \{o} � = λ1(B d , L) ≤
�
2dd(L) � 1
d
,
V (C)
but it does not explicitly provide a point l ∈ L \{o}, such that �l� =λ1(B d , L). As
an immediate consequence of the Lenstra, Lenstra, Lovász theorem, we obtain the
following approximate answer to the shortest lattice vector problem.
Corollary 28.3. There is a polynomial time algorithm which, for any given basis
{a1,...,ad} of a rational lattice L in E d , finds a vector l ∈ L \{o}, such that:
�l� ≤2 1 2 (d−1) min � �m� :m ∈ L \{o} � .
Remark. For any fixed ε>0 there is a polynomial time algorithm due to Schnorr
[912] with the factor (1 + ε) 1 2 (d−1) instead of 2 1 2 (d−1) .
Remark. Using an approximate algorithmic version of John’s theorem 11.2, this
result can easily be extended to arbitrary norms instead of the Euclidean norm. This,
then, is an approximate algorithmic version of Minkowski’s fundamental theorem.
Nearest Lattice Point Problem
This is the following inhomogeneous problem.
Problem 28.2. Given a lattice L in E d , find for any x ∈ E d a point l ∈ L such that:
�x − l� =min � �x − m� :m ∈ L � .
Van Emde Boaz [1007] has shown that this problem is NP-hard. Of course, it
may be considered also for norms different from �·�. Using the LLL-basis reduction,
Babai [44] showed that an approximate answer to the nearest lattice point problem
is possible in polynomial time: