Gruber P. Convex and Discrete Geometry

412 Geometry of Numbers
ˆb j+1 is the orthogonal projection of b j+1 onto the subspace lin{b1,...,b j } ⊥ =
lin{ ˆb1,..., ˆb j } ⊥ . The following result exhibits a connection between the orthogonal
system { ˆb1,..., ˆbd} and the shortest (non-zero) vector problem in L.
Lemma 28.1. Let {b1,...,bd} be a basis of a lattice L and { ˆb1,..., ˆbd} its Gram–
Schmidt orthogonalization. Then
Proof. Let l ∈ L. Then
�l� ≥min{� ˆb1�,...,� ˆbd�} for any l ∈ L \{o}.
l = �
uibi where ui ∈ Z.
i
Let k be the largest index with uk �= 0. Replace b1,...,bk by their expressions from
(1). This gives
k�
l = αi ˆbi where αk = uk ∈ Z \{0}.
i=1
Since ˆb1,..., ˆbk are pairwise orthogonal, this yields the desired inequality:
�l� 2 =
k�
α 2 i � ˆbi� 2 ≥ α 2 k � ˆbk� 2 ≥�ˆbk� 2 . ⊓⊔
i=1
After these preparations, the definition of an LLL-reduced basis is as follows:
Let {b1,...,bd} be a basis of a lattice L, { ˆb1,..., ˆbd} its Gram–Schmidt orthogonalization
and the numbers µ ji as in (1). Then {b1,...,bd} is an LLL-reduced basis
of L if it satisfies the following conditions:
(2) |µ ji|≤ 1
for i, j = 1,...,d, i < j,
2
(3) � ˆb j+1 + µ j+1 j ˆb j� 2 ≥ 3
4 � ˆb j� 2 for j = 1,...,d − 1.
Roughly speaking, the first condition means that the basis vectors {b1,...,bd} are
pairwise almost orthogonal. The vectors ˆb j and ˆb j+1 + µ j+1 j ˆb j are the projections
of b j and b j+1 onto lin{ ˆb1,..., ˆb j−1} ⊥ . Thus, the second condition says that the
length of the projection of b j+1 cannot be much smaller than the length of the projection
of b j.
Properties of LLL-Reduced Bases
We collect some basic properties of LLL-reduced bases which will be needed later:
Theorem 28.1. Let {b1,...,bd} be an LLL-reduced basis of a lattice L in E d . Then
the following hold:
(i) �b1� ≤2 1 4 (d−1) d(L) 1 d