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

For each step t > p of the algorithm, the integers (ũt+1, ...., ũs) are fixed.
All left integers ut−1, for which Ft( �k i=t ũibi) < Fj(bj), are enumerated.
Since Ft(−x) = Ft(x) for all x ∈ Rn if ũk = .... = ũt = 0 we
could limit the search to only nonnegative integers. Every time a coefficient
is computed it is organized in a search tree with depth k − j + 2.
Currently the Depth-First-Search is considered most appropriate for the
enumeration. With respect to the Euclidean norm and z ∈ R, the distance
function Ft can be efficiently computed with the help of the Gram-Schmidt
orthogonal system. From
˜ct := � πt( �k i=t ũibi) �2 2 and ct := � ˆ bt �2 2 = � πt(bt) �2 2 , for 1 ≤ t ≤ k
we get
F 2 t ( � k
i=t ũibi) = ˜ct = ˜ct+1 + (ũt + � k
i=t+1 ũiµi,t) 2 ct
−yt is the minimal real point for yt := � k
i=t+1 ũiµi,t. We get the whole
minimum through the symmetry
ct(−yt + x, ũt+1, ...., ũk) = ct(−yt − x, ũt+1, ...., ũk)
at the point νt := ⌈−yt⌋. Depending on whether νt > −yt holds true or
not, we get a sequence of non-declining values ˜ct either in order (νt, νt − 1,
νt + 1, νt − 2, ...) or in order (νt, νt + 1, νt − 1, νt + 2, ....).
Algorithm 3.2 ENUM with Euclidean Norm
INPUT: j, k, b1, ...., bm
1. • s := t := j, ¯cj := cj, ũj := uj := 1
• νj := yj := ∆j := 0, δj := 1
• FOR i = j + 1, ...., k + 1
– ˜ci := ui := ũi := νi := ∆i := 0
– δi := 1
2. WHILE t ≤ k
• ˜ct := ˜ct+1 + (yt + ũt) 2 ct
• IF ˜ct < ¯cj
• THEN
– IF t > j
– THEN
∗ t := t − 1, yt := � s
i=t+1 ũiµi,t
∗ ũt := νt := ⌈−yt⌋, ∆t := 0
∗ IF ũt > −yt
∗ THEN δt := −1
∗ ELSE δt := 1
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