Lattices and Quadratic Forms

Executive summary

One object, two languages

A lattice is a discrete subgroup of ℝⁿ of full rank: the set of integer combinations of a basis. Its determinant — the covolume of a fundamental domain — is independent of the basis chosen, while the basis itself is defined only up to unimodular transformation. Attaching the quadratic form given by the Gram matrix translates every lattice question into a question about forms, and back again. The central computational problem is finding short vectors.

Learning objectives

Define a lattice, its basis and its determinant.
Relate lattices to positive definite quadratic forms via the Gram matrix.
State Minkowski's theorem and the meaning of the successive minima.
Explain why an arbitrary basis is usually a poor one.
Identify where lattices arise in number theory.

Section 01Lattices and bases

Given linearly independent b₁, …, b_n in ℝⁿ, the lattice they generate is

L = { ∑ x_ib_i : x_i ∈ ℤ }

Two bases generate the same lattice exactly when they differ by a unimodular matrix. The determinant, defined as |det B| for a basis matrix B, is therefore an invariant of the lattice — a unimodular change of basis has determinant ±1 and cannot alter it.

Every lattice has infinitely many bases, most of them bad

A basis obtained from a natural description of a problem is typically skewed: its vectors are long and nearly parallel. The lattice is unchanged, but computations become hopeless. Reduction theory exists to replace such a basis with a short, near-orthogonal one.

Section 02The Gram matrix and quadratic forms

The Gram matrix G = BB^t has entries ⟨b_i, b_j⟩ and defines a positive definite quadratic form

q(x) = x^tGx = ‖∑x_ib_i‖²

The correspondence is exact: lattices up to isometry correspond to positive definite quadratic forms up to unimodular equivalence, and det G = (det L)². Every question about short lattice vectors becomes a question about small values of a form.

Dictionary between the two languages
Lattice language	Form language
Basis change by unimodular U	Equivalence of forms G → U^tGU
Determinant of the lattice	Square root of the discriminant of the form
Shortest non-zero vector	Minimum of the form
Reduced basis	Reduced form
Sublattice of finite index	Form of larger discriminant representing a subset of values

Why the form language persists

In dimension 2 the theory of binary quadratic forms predates lattice reduction by a century and remains the natural language for quadratic fields, where forms correspond to ideal classes. Higher-dimensional work generally prefers the lattice language.

Section 03Minima, Minkowski and Hermite

The successive minima λ₁ ≤ … ≤ λ_n record the smallest radii within which the lattice contains 1, 2, …, n linearly independent vectors. Minkowski's convex body theorem bounds the first of them.

λ₁ ≤ γ_n^1/2 (det L)^1/n

Here γ_n is Hermite's constant, known exactly only in low dimensions. The bound is existential: it guarantees a short vector exists but gives no procedure for finding one.

NP-hardshortest vector, under randomised reductions

2^O(n)approximation factor achieved by LLL

PolynomialLLL running time

The hardness gap is the whole subject

Finding a genuinely shortest vector is hard in high dimension. LLL settles for a vector within an exponential factor of the minimum — which sounds weak, and is entirely sufficient for most number-theoretic applications, because the lattices arising there have an unusually large gap between the shortest vector and the rest.

Section 04Where lattices arise in number theory

Source

Rings of integers

ℤ_K embeds as a lattice of rank n in ℝⁿ via the archimedean embeddings; its determinant is essentially the field discriminant.

Source

Ideals

Every non-zero ideal is a full-rank sublattice of ℤ_K, with index equal to its norm.

Source

Unit groups

The logarithmic embedding maps the units onto a lattice of rank r₁+r₂−1 whose covolume is the regulator.

Source

Polynomial factorisation

Recombining modular factors is a short-vector problem in a lattice built from the coefficients.

Source

Integer relations

Detecting a linear relation among real numbers is a short-vector problem in a lattice with a scaled last coordinate.

Source

Sieving

The number field sieve enumerates lattice points in a sieving region — its dominant cost.

ReferenceFrequently asked questions

Must a lattice have full rank?

Not necessarily — a lattice of rank k in dimension n is perfectly well defined. But most algorithms assume full rank, and a lower-rank lattice should be re-expressed in the span it actually occupies before reduction.

Is the Gram matrix enough to recover the lattice?

It determines the lattice up to isometry, which is all that matters for questions about lengths and inner products. It does not determine the embedding in ℝⁿ, so the actual coordinates are lost.

Why is the shortest vector problem hard when Minkowski guarantees one exists?

Because the guarantee is non-constructive. It bounds the length of the shortest vector from the volume, but locating it requires searching a region containing exponentially many candidates in high dimension.

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ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Lattices and Quadratic Forms. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Lattices and Quadratic Forms as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—lattices, quadratic, section, forms, gram—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Lattices and Quadratic Forms?