Miller primality test
E735950
The Miller primality test is a randomized algorithm used to determine whether a number is prime with high confidence, forming the basis of the widely used Miller–Rabin primality test in computational number theory and cryptography.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Miller primality test canonical | 1 |
| Miller–Rabin primality test | 1 |
| Probabilistic Algorithm for Testing Primality | 1 |
| Rabin–Miller primality test | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8448944 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Miller primality test Context triple: [Adleman–Pomerance–Rumely primality test, relatedTo, Miller primality test]
-
A.
Fermat primality test
The Fermat primality test is a probabilistic algorithm that checks whether a number is prime by verifying congruences derived from Fermat's little theorem.
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B.
AKS primality test
The AKS primality test is a landmark deterministic polynomial-time algorithm that can conclusively determine whether a number is prime without relying on unproven assumptions.
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C.
Adleman–Pomerance–Rumely primality test
The Adleman–Pomerance–Rumely primality test is an early deterministic algorithm in computational number theory used to determine whether a given number is prime, notable for its theoretical importance in the development of modern primality testing methods.
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D.
Selfridge–Conway primality test
The Selfridge–Conway primality test is a probabilistic algorithm in number theory used to determine whether a given integer is prime.
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E.
Fermat pseudoprime
A Fermat pseudoprime is a composite number that nevertheless satisfies Fermat's little theorem for a given base, making it appear prime under that specific primality test.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Miller primality test Target entity description: The Miller primality test is a randomized algorithm used to determine whether a number is prime with high confidence, forming the basis of the widely used Miller–Rabin primality test in computational number theory and cryptography.
-
A.
Fermat primality test
The Fermat primality test is a probabilistic algorithm that checks whether a number is prime by verifying congruences derived from Fermat's little theorem.
-
B.
AKS primality test
The AKS primality test is a landmark deterministic polynomial-time algorithm that can conclusively determine whether a number is prime without relying on unproven assumptions.
-
C.
Adleman–Pomerance–Rumely primality test
The Adleman–Pomerance–Rumely primality test is an early deterministic algorithm in computational number theory used to determine whether a given number is prime, notable for its theoretical importance in the development of modern primality testing methods.
-
D.
Selfridge–Conway primality test
The Selfridge–Conway primality test is a probabilistic algorithm in number theory used to determine whether a given integer is prime.
-
E.
Fermat pseudoprime
A Fermat pseudoprime is a composite number that nevertheless satisfies Fermat's little theorem for a given base, making it appear prime under that specific primality test.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
algorithm in number theory
ⓘ
primality test ⓘ randomized algorithm ⓘ |
| application |
cryptographic parameter generation
ⓘ
key generation in public-key cryptography ⓘ testing large integers for primality ⓘ |
| assumption | generalized Riemann hypothesis for certain error bounds ⓘ |
| author | Gary L. Miller NERFINISHED ⓘ |
| basedOn | Riemann hypothesis NERFINISHED ⓘ |
| classification | Monte Carlo algorithm NERFINISHED ⓘ |
| compositenessWitnessCondition | x ≠ 1 and x ≠ n−1 and no square equals n−1 modulo n ⓘ |
| coreIdea |
search for nontrivial square roots of 1 modulo n
ⓘ
use repeated squaring of a^d mod n ⓘ |
| deterministicUnder | generalized Riemann hypothesis ⓘ |
| errorDirection |
may classify some composite numbers as probably prime
ⓘ
never classifies a prime as composite ⓘ |
| errorType | one-sided error ⓘ |
| field |
computational number theory
ⓘ
cryptography ⓘ |
| generalizationOf | Fermat primality test ⓘ |
| guarantee |
deterministic polynomial-time primality test under GRH
ⓘ
never declares a composite number prime if the generalized Riemann hypothesis holds ⓘ |
| improvesOn | Fermat primality test NERFINISHED ⓘ |
| influenced | practical primality testing algorithms ⓘ |
| input | odd integer n > 2 ⓘ |
| inspired | Miller–Rabin primality test NERFINISHED ⓘ |
| language | number-theoretic algorithm ⓘ |
| output | "composite" or "probably prime" ⓘ |
| relatedAlgorithm | AKS primality test NERFINISHED ⓘ |
| relatedConcept |
Carmichael number
ⓘ
strong pseudoprime ⓘ witness for compositeness ⓘ |
| relation | forms the theoretical basis of the Miller–Rabin primality test ⓘ |
| step |
check if x = 1 or x = n−1
ⓘ
choose random base a with 1 < a < n−1 ⓘ compute x = a^d mod n ⓘ square x repeatedly up to s−1 times ⓘ write n−1 = 2^s·d with d odd ⓘ |
| timeComplexity | polynomial in log n under GRH ⓘ |
| typicalImplementationLanguage |
C
ⓘ
C++ NERFINISHED ⓘ Java NERFINISHED ⓘ Python NERFINISHED ⓘ |
| usedWith | modular exponentiation by repeated squaring ⓘ |
| uses |
decomposition of n−1 as 2^s·d
ⓘ
modular exponentiation ⓘ witnesses for compositeness ⓘ |
| yearProposed | 1976 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Miller primality test Description of subject: The Miller primality test is a randomized algorithm used to determine whether a number is prime with high confidence, forming the basis of the widely used Miller–Rabin primality test in computational number theory and cryptography.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Miller–Rabin primality test
subject surface form:
Michael O. Rabin
this entity surface form:
Rabin–Miller primality test
subject surface form:
Michael O. Rabin
this entity surface form:
Probabilistic Algorithm for Testing Primality