Legendre, Jacobi and Kronecker Symbols

Executive summary

A residuacity test that costs no more than a GCD

The Legendre symbol records whether an integer is a square modulo an odd prime. Euler's criterion evaluates it by modular exponentiation, but the Jacobi extension to composite moduli can be evaluated far faster using quadratic reciprocity — in time comparable to a GCD, and crucially without factoring the modulus. The extension comes at a cost in meaning: a Jacobi symbol of +1 does not imply that the argument is a quadratic residue.

Learning objectives

Define the Legendre symbol and evaluate it by Euler's criterion.
State quadratic reciprocity and its supplementary laws.
Implement the binary reciprocity algorithm for the Jacobi symbol.
Explain exactly what a Jacobi value of +1 does and does not mean.
Identify where the Kronecker extension is required.

Section 01Residues and the Legendre symbol

For an odd prime p the group (ℤ/pℤ)* is cyclic of order p−1, so exactly half its elements are squares. The Legendre symbol is

(a/p) = 0 if p | a; +1 if a is a non-zero square mod p; −1 otherwise

Euler's criterion gives a direct evaluation:

(a/p) ≡ a^(p−1)/2 (mod p)

This costs a modular exponentiation, roughly O(log p) modular multiplications. The reciprocity algorithm below is asymptotically better, and it does not require p to be prime.

Section 02Extensions and their meaning

The three symbols compared
Symbol	Denominator	Definition	Meaning of +1
Legendre	Odd prime p	Euler's criterion	a is a quadratic residue mod p — exact
Jacobi	Odd positive n	Product of Legendre symbols over the prime factorisation of n, with multiplicity	Only that the number of prime factors with −1 is even — not residuacity
Kronecker	Any integer, including 0, −1 and 2	Jacobi extended by convention on the factors 0, −1 and 2	As Jacobi; used mainly for discriminants and character evaluation

The classic misreading

If the Jacobi symbol is −1, the argument is definitely a non-residue. If it is +1, nothing follows: a may be a non-residue modulo every prime factor, with the signs cancelling. Only the −1 outcome is conclusive. This asymmetry is exactly what the Solovay–Strassen primality test exploits, and misunderstanding it is a common source of incorrect residuacity code.

Section 03Evaluation by reciprocity

Quadratic reciprocity relates the symbol to its reverse, allowing the arguments to be reduced alternately in a manner closely parallel to the Euclidean algorithm:

(m/n)(n/m) = (−1)^{((m−1)/2)((n−1)/2)} for odd coprime m, n > 0

(−1/n) = (−1)^(n−1)/2, (2/n) = (−1)^(n²−1)/8

AlgorithmBinary Jacobi symbolin: a ∈ ℤ, n odd > 0 → out: (a/n) ∈ {−1, 0, 1}

If n is even or n ≤ 0, reject — the Jacobi symbol requires odd positive n.
Set a ← a mod n and s ← 1.
While a ≠ 0:
While a is even: set a ← a/2, and if n ≡ 3 or 5 (mod 8) set s ← −s. Supplementary law for the factor 2.
Swap a and n; if both are ≡ 3 (mod 4), set s ← −s. Main reciprocity law.
Set a ← a mod n.
If n = 1 return s, otherwise return 0. n ≠ 1 means gcd(a, n) > 1.

The strategic point

The Jacobi symbol is computable without knowing the factorisation of the modulus. That single property is what makes it usable in primality testing, in the quadratic sieve's factor base selection, and in the definition of genus characters — all settings where the factorisation is precisely what is unknown.

ReferenceFrequently asked questions

When do I need the Kronecker symbol rather than the Jacobi symbol?

Whenever the lower argument may be even, negative or zero — which happens constantly when the argument is a field discriminant. Discriminants are congruent to 0 or 1 modulo 4 and are frequently negative, so quadratic field work uses Kronecker throughout.

Is Euler's criterion ever preferable?

Only when the modular power is needed anyway — for instance inside Tonelli–Shanks, where the exponentiation both decides residuacity and initialises the square-root search. As a standalone residuacity test, reciprocity is faster.

How does this relate to primality testing?

The Solovay–Strassen test compares the Jacobi symbol, computed by reciprocity, against Euler's criterion, computed by exponentiation. For a prime the two must agree; a disagreement proves compositeness. Miller–Rabin has since superseded it, being strictly stronger for the same cost.

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Curated next steps from this page. The site also surfaces algorithmically related reading below.

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 Legendre, Jacobi and Kronecker Symbols. 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 Legendre, Jacobi and Kronecker Symbols as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—symbol, legendre, jacobi, kronecker, section—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 Legendre, Jacobi and Kronecker Symbols?