computer science

Lenstra–Lenstra–Lovász lattice basis reduction algorithm

Lenstra–Lenstra–Lovász lattice basis reduction algorithm is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Lenstra–Lenstra–Lovász lattice basis reduction algorithm rather than just read about it. In short: The Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and László Lovász in 1982. Given a basis B = { b 1 , b 2 , … , b d } {\displaystyle \mathbf {B} =\{\mathbf {b} _{1},\mathbf {b} _{2},\dots ,\mathbf {b} _{d}\}} with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rn) with d ≤ n {\dis…

Key takeaways

Lenstra–Lenstra–Lovász lattice basis reduction algorithm belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Lenstra–Lenstra–Lovász lattice basis reduction algorithm to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Lenstra–Lenstra–Lovász lattice basis reduction algorithm from memory before moving on to harder problems.

Reference excerpt

The Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and László Lovász in 1982. Given a basis B = { b 1 , b 2 , … , b d } {\displaystyle \mathbf {B} =\{\mathbf {b} _{1},\mathbf {b} _{2},\dots ,\mathbf {b} _{d}\}} with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rn) with d ≤ n {\displaystyle d\leq n} , the LLL algorithm calculates an LLL-reduced (short, nearly orthogonal) lattice basis in time O ( d 5 n log 3 ⁡ B ) {\displaystyle {\mathcal {O}}(d^{5}n\log ^{3}B)} where B {\displaystyle B} is the largest length of b i {\displaystyle \mathbf {b} _{i}} under the Euclidean norm, that is, B = max ( ‖ b 1 ‖ 2 , ‖ b 2 ‖ 2 , … , ‖ b d ‖ 2 ) {\displaystyle B=\max \left(\|\mathbf {b} _{1}\|_{2},\|\mathbf {b} _{2}\|_{2},\dots ,\|\mathbf {b} _{d}\|_{2}\right)} . The original applications were to give polynomial-time algorithms for factorizing polynomials with rational coefficients, for finding simultaneous rational approximations to real numbers, and for solving the integer linear programming problem in fixed dimensions.

LLL reduction The precise definition of LLL-reduced is as follows: Given a basis

B = { b 1 , b 2 , … , b n } , {\displaystyle \mathbf {B} =\{\mathbf {b} _{1},\mathbf {b} _{2},\dots ,\mathbf {b} _{n}\},}

define its Gram–Schmidt process orthogonal basis

B ∗ = { b 1 ∗ , b 2 ∗ , … , b n ∗ } , {\displaystyle \mathbf {B} ^{*}=\{\mathbf {b} _{1}^{*},\mathbf {b} _{2}^{*},\dots ,\mathbf {b} _{n}^{*}\},}

and the Gram-Schmidt coefficients

μ i , j = ⟨ b i , b j ∗ ⟩ ⟨ b j ∗ , b j ∗ ⟩ , {\displaystyle \mu _{i,j}={\frac {\langle \mathbf {b} _{i},\mathbf {b} _{j}^{*}\rangle }{\langle \mathbf {b} _{j}^{*},\mathbf {b} _{j}^{*}\rangle }},} for any 1 ≤ j < i ≤ n {\displaystyle 1\leq j<i\leq n} . Then the basis B {\displaystyle B} is LLL-reduced if there exists a parameter δ {\displaystyle \delta } in (0.25, 1] such that the following holds:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Lenstra–Lenstra–Lovász lattice basis reduction algorithm

Start with the simplest possible case. Write down what Lenstra–Lenstra–Lovász lattice basis reduction algorithm claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Lenstra–Lenstra–Lovász lattice basis reduction algorithm before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Lenstra–Lenstra–Lovász lattice basis reduction algorithm ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Lenstra–Lenstra–Lovász lattice basis reduction algorithm

In research: Lenstra–Lenstra–Lovász lattice basis reduction algorithm appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Lenstra–Lenstra–Lovász lattice basis reduction algorithm in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Lenstra–Lenstra–Lovász lattice basis reduction algorithm is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational number theory, Lattice points, Theory of cryptography, so understanding it makes those chapters shorter.
In everyday life: Look for Lenstra–Lenstra–Lovász lattice basis reduction algorithm outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Lenstra–Lenstra–Lovász lattice basis reduction algorithm” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Lenstra–Lenstra–Lovász lattice basis reduction algorithm in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lenstra–Lenstra–Lovász lattice basis reduction algorithm means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Lenstra–Lenstra–Lovász lattice basis reduction algorithm out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lenstra–Lenstra–Lovász lattice basis reduction algorithm in simple terms?

Why does Lenstra–Lenstra–Lovász lattice basis reduction algorithm matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Lenstra–Lenstra–Lovász lattice basis reduction algorithm?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Lenstra–Lenstra–Lovász lattice basis reduction algorithm.