Miller primality test

E735950

algorithm in number theory primality test randomized algorithm

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.

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All labels observed (4)

Label	Occurrences
Miller–Rabin primality test	6
Rabin–Miller primality test	3
Miller primality test canonical	2
Probabilistic Algorithm for Testing Primality	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.
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
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 ⓘ
basedOn	Riemann hypothesis ⓘ
classification	Monte Carlo algorithm ⓘ linked to: Monte Carlo method
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 ⓘ
influenced	practical primality testing algorithms ⓘ
input	odd integer n > 2 ⓘ
inspired	Miller–Rabin primality test ⓘ linked to: Miller primality test
language	number-theoretic algorithm ⓘ
output	"composite" or "probably prime" ⓘ
relatedAlgorithm	AKS primality test ⓘ
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++ ⓘ Java ⓘ Python ⓘ
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