Gruber P. Convex and Discrete Geometry

414 Geometry of Numbers
LLL-Reduction is Polynomial
We now come to the main result, where a lattice in E d is rational if all its points have
rational coordinates.
Theorem 28.2. There is a polynomial time algorithm that finds, for any given basis
{a1,...,ad} of a rational lattice L in E d , an L L L-reduced basis.
Proof. Without loss of generality, we may assume that L ⊆ Z d , and let {a1,...,ad}
be a basis of L. We shall transform this basis until the conditions (2) and (3)
are satisfied. The algorithm starts with the basis {b1,...,bd} = {a1,...,ad}.
Let { ˆb1,..., ˆbd} be the corresponding Gram–Schmidt orthogonalization and let the
coefficients µ ji be as in (1).
Step I. For j = 1,...,d, and, given j, for i = 1,..., j − 1, replace b j by
b j −⌈µ ji⌋bi, where ⌈µ ji⌋ is the integer nearest to µ ji. (In case of ambiguity choose
the smaller integer.) Then go to Step II.
Step I does not change the Gram–Schmidt orthogonalization of the basis. Hence,
after executing (the substeps of) Step I for a certain j, the new µ ji will satisfy |µ ji|≤
1
2 for i = 1,..., j − 1. Executing (the substeps of) Step I for a larger j will not spoil
this property. Thus executing (all substeps of) Step I yields a basis which satisfies
condition (2).
Step II. If there is an index j violating condition (3), interchange b j and b j+1
and return to Step I.
We analyze this step. Let
c1 = b1,...,c j−1 = b j−1, c j = b j+1, c j+1 = b j, c j+2 = b j+2,...,cd = bd.
Then ĉi = ˆbi for i �= j, j + 1. Further, ĉ j = ˆb j+1 + µ j+1 j ˆb j and since condition
(3) is violated for j, we have that:
(6) �ĉ j� 2 < 3
4 � ˆb j� 2 .
The formula for ĉ j+1 is more complicated, but it is not needed. For our purpose it is
sufficient to note that from
�ĉ1� 2 ···�ĉd� 2 =�ˆb1� 2 ···�ˆbd� 2 = d(L) 2
it follows that �ĉ j � 2 �ĉ j+1� 2 =�ˆb j� 2 � ˆb j+1� 2 and thus by (6)
(7) �ĉ j� 2(d− j)+2 �ĉ j+1� 2(d− j) < 3
4 � ˆb j� 2(d− j)+2 � ˆb j+1� 2(d− j) .
After having described the algorithm we come to the proof of the Theorem. First,
the following will be shown:
(8) The total number of arithmetic operations in the algorithm is polynomial in
the encoding length of the input.