Lattice Basis Reduction in Infinity Norm - Technische Universität ...

Figure 1: First Successive Minimum
An example of a first successive minimum λ1(L(a, b)) = x is the figure
above.
Since for every lattice L ⊆ R n � x �∞ ≤ � x �2 ≤ √ m � x �∞ for all
x ∈ R n , it holds for the successive minima that
λ1,∞(L) ≤ λ1,2(L) ≤ √ mλ1,∞(L)
Theorem 2.9. For each lattice L ⊆ R n of rank m, � L �∞ ≤ (det L) 1
m
2.3 Distance Functions
The term ”Distance Function” was integrated into the lattice reduction theory
by Lovász and Scarf.
Definition 2.10. Let b1, b2, ...., bm ∈ R n be linearly independent vectors
for all m ≤ n and let � · �p be an arbitrary norm on R n . The functions
Fi : R n → R with
Fi(x) := minξ1,....,ξi−1∈R � x + � i−1
j=1 ξjbj �p for 1 ≤ i ≤ m + 1
are called distance functions.
The distance functions determine the distance between a vector bi and
span(b1, b2, ...., bi−1) with respect to an arbitrary norm. The functions
Fi(x) are actually the norm of span(b1, b2, ...., bi−1) ⊥ . With respect to
the Euclidean norm they are also Euclidean norms of the subspace. The
distance functions are very useful because the length of the shortest lattice
vector could be restricted by them.
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