Hensel Lifting and Factorisation over the Integers

Executive summary

Factor modulo a small prime, lift, then recombine — the last step is the hard one

A factorisation modulo p lifts to a factorisation modulo any power of p, provided the factors are coprime modulo p. Lifting far enough that the coefficients exceed Mignotte's bound means the true integer factors can be read off — if the correct grouping of modular factors is known. Finding that grouping is the recombination problem, and its naive solution is exponential. The Lenstra–Lenstra–Lovász paper solved it, which is why LLL exists at all.

Learning objectives

State Hensel's lemma and its coprimality hypothesis.
Implement quadratic lifting and state its cost.
Use Mignotte's bound to determine the required lifting height.
Explain why recombination is the bottleneck.
Outline the LLL and van Hoeij approaches to recombination.

Section 01Hensel's lemma

Suppose f ≡ gh (mod p) with g, h coprime modulo p. Then the factorisation lifts uniquely to modulo p^k for every k.

AlgorithmQuadratic Hensel liftingin: f, factors mod p, target height → out: factors mod p^k

Compute s, t with sg + th ≡ 1 (mod p) by the extended Euclidean algorithm in F_p[x]. Coprimality is exactly what makes this possible.
Given g, h correct modulo p^k, set e ← f − gh, which is divisible by p^k.
Solve for corrections: δg ← te mod g and δh ← se mod h, both reduced modulo p^k.
Set g ← g + δg and h ← h + δh; the factorisation is now correct modulo p^2k. The exponent doubles each round.
Update s and t to the new modulus and repeat until p^k exceeds the target.

Coprimality is the whole hypothesis

If the modular factors share a root, the lift is not unique and the algorithm fails. This is why f must be squarefree and why p must not divide the discriminant — and why the choice of p includes a check that the factorisation modulo p is squarefree.

Section 02How far to lift: Mignotte's bound

The coefficients of any divisor of f over ℤ are bounded in terms of the degree and the norm of f:

‖g‖_∞ ≤ 2^{deg g} ‖f‖₂

Lifting until p^k exceeds twice this bound guarantees that each true factor is determined by its residue modulo p^k, taken in the symmetric range.

The bound is pessimistic and the cost is real

Mignotte's bound is worst-case and usually far above the actual coefficient sizes, so the lifting height — and therefore the cost of every subsequent operation — is larger than strictly necessary. Practical implementations lift incrementally, attempt recombination early, and lift further only if it fails.

Section 03The recombination problem

Suppose f factors into r irreducible factors modulo p^k. Each true factor over ℤ is a product of some subset of them. Testing all subsets costs 2^r trial divisions.

2^rsubsets in the naive search

Polynomialcost of the LLL-based method

≈ r²typical lattice dimension in van Hoeij's method

Where the difficulty actually is

The hard case is a polynomial that is irreducible over ℤ but splits into many factors modulo every prime — the Swinnerton-Dyer polynomials are the standard family. Every subset must be rejected, so the naive search runs its full exponential course before concluding that no factorisation exists.

Recombination strategies
Strategy	Cost	Notes
Exhaustive subsets (Zassenhaus)	O(2^r)	Fast when r is small, which is the common case in practice
LLL on a coefficient lattice	Polynomial	The original polynomial-time result; the lattice dimension makes it slow in practice
van Hoeij's knapsack method	Polynomial, and fast	Reduces a much smaller lattice built from power sums — the current standard
Multiple primes	Heuristic	Factor modulo several primes; the true degrees must be consistent across all of them

Section 04The complete pipeline

Stage 01PreprocessRemove content, make primitive, take the squarefree part. Handle degree 0 and 1 directly.
Stage 02Choose a primeSelect p not dividing the leading coefficient or the discriminant, so that f stays squarefree modulo p. Try several and keep the one giving the fewest factors.
Stage 03Factor modulo pApply the finite-field pipeline: squarefree, distinct-degree, equal-degree.
Stage 04Hensel liftLift to p^k beyond Mignotte's bound, using quadratic lifting.
Stage 05RecombineFind the subsets of modular factors forming true integer factors; verify each by exact division.

Choosing the prime is worth effort

The number of modular factors varies substantially with p, and recombination cost depends on it directly. Trying several candidate primes and keeping the one with the fewest factors is cheap and frequently pays for itself several times over.

ReferenceFrequently asked questions

Why lift quadratically rather than linearly?

Because the number of rounds drops from O(k) to O(log k). Each quadratic round is more expensive, but the reduction in round count dominates once the target height is more than a few powers of p.

Does this extend to multivariate polynomials?

Yes — lift with respect to one variable, treating the others as parameters, using the ideal generated by (y − a) in place of p. The recombination problem reappears in the same form, and the same techniques apply.

Is factoring over a number field similar?

Structurally yes: reduce modulo a prime ideal of good reduction, factor over the residue field, lift and recombine. The extra difficulty is choosing a prime ideal of degree 1 that does not divide the relevant discriminants, and handling the denominators introduced by the integral basis.

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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 Hensel Lifting and Factorisation over the Integers. 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 Hensel Lifting and Factorisation over the Integers as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—section, hensel, lifting, lemma, factorisation—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 Hensel Lifting and Factorisation over the Integers?