OBSERVATION CHAMBER / ACTIVE
Lattice Reduction Forge
Reduce a 2D integer lattice basis while keeping the lattice, determinant, and fundamental area unchanged.
01 · DEFINE THE BASIS
Current basis
CURRENT BASIS
Total squared length —
RESULT
Awaiting reduction- BEFORE
- not calculated
- AFTER
- not calculated
- BASIS LENGTH
- pending
- SAME LATTICE
- pending
Advanced steps and diagnostics
Detailed result
- BEFORE
- —
- AFTER
- —
- det(B) before / after
- —
- area before / after
- —
- U
- —
- det(U)
- —
- B′ = BU
- —
- invariants
- —
- shortest vector
- —
- steps
- —
Drag endpoints; coordinates snap to integers.
- Current μ
- —
- det(B)
- —
- area
- —
- det(U)
- —
- ‖b1‖²
- —
- ‖b2‖²
- —
- U
- —
- reduced
- —
- step
- step 0
μ is the nearest-integer projection coefficient in b₂ ← b₂ − μb₁.
Choose a basis, then REDUCE BASIS.
Lattice view
original currentThe lattice points stay fixed; the basis vectors are the moving description.
The viewport scale is visual only. Coordinates and invariants come from exact integer arithmetic.
Reduction trace
invariants pending- No operation yet.
REPORT
No export yetRun REDUCE BASIS to generate the exact report.Download PNG
2D Gauss reduction · exact integer arithmetic · not a proof or cryptanalysis tool.
OBSERVATION GUIDE
Touchpoints for Observation
A long integer basis becomes a shorter, easier-to-read basis while the generated lattice remains the same.
- Press REDUCE BASIS on the Fibonacci basis, then open Advanced to inspect each exact step, μ, U, and invariant.
- The model uses exact BigInt arithmetic and a finite viewport.
- Equivalent bases may differ by sign or column order.
- This is 2D Gauss reduction, not high-dimensional LLL or a general optimality proof.
It is not a cryptanalysis or engineering-design tool, and it does not claim a universal shortest-basis theorem.
Runs inside the browser with no upload and no registration.
SYSTEM NOTE
For B=[b₁ b₂], Gauss reduction uses μ as the nearest integer to <b₁,b₂>/<b₁,b₁>, then replaces b₂ by b₂−μb₁ and swaps by norm when needed. Each step is B′=BU with U∈GL(2,Z), det(U)=±1, so det(B′)=det(B) and the fundamental area is unchanged.
OBSERVATION POLICY
This lab runs in your browser. Basis coordinates, calculations, reports, and images are not uploaded by this lab.
Route
This chamber belongs to Structures & Symmetries.
STRUCTURE Back to Structures & Symmetries Route.