Lenstra elliptic-curve factorization method
E824095
The Lenstra elliptic-curve factorization method is an integer factorization algorithm that uses properties of elliptic curves over finite fields to efficiently find nontrivial factors of large numbers, especially those with relatively small prime divisors.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Lenstra elliptic-curve factorization method canonical | 2 |
| Factoring integers with elliptic curves | 1 |
| Lenstra ECM | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9838958 — 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: Lenstra elliptic-curve factorization method Context triple: [Hendrik Lenstra, knownFor, Lenstra elliptic-curve factorization method]
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A.
Berlekamp’s algorithm for factoring polynomials over finite fields
Berlekamp’s algorithm for factoring polynomials over finite fields is a foundational deterministic method in computational algebra that efficiently decomposes polynomials into irreducible factors over finite fields and underpins many modern algorithms in coding theory and cryptography.
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B.
Cantor–Zassenhaus algorithm
The Cantor–Zassenhaus algorithm is a probabilistic method used to factor polynomials over finite fields efficiently, widely employed in computational algebra and cryptography.
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C.
Schoof–Elkies–Atkin (SEA) point-counting algorithm
The Schoof–Elkies–Atkin (SEA) point-counting algorithm is an efficient method in computational number theory and elliptic curve cryptography for determining the number of points on an elliptic curve over a finite field.
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D.
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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E.
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.
- 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: Lenstra elliptic-curve factorization method Target entity description: The Lenstra elliptic-curve factorization method is an integer factorization algorithm that uses properties of elliptic curves over finite fields to efficiently find nontrivial factors of large numbers, especially those with relatively small prime divisors.
-
A.
Berlekamp’s algorithm for factoring polynomials over finite fields
Berlekamp’s algorithm for factoring polynomials over finite fields is a foundational deterministic method in computational algebra that efficiently decomposes polynomials into irreducible factors over finite fields and underpins many modern algorithms in coding theory and cryptography.
-
B.
Cantor–Zassenhaus algorithm
The Cantor–Zassenhaus algorithm is a probabilistic method used to factor polynomials over finite fields efficiently, widely employed in computational algebra and cryptography.
-
C.
Schoof–Elkies–Atkin (SEA) point-counting algorithm
The Schoof–Elkies–Atkin (SEA) point-counting algorithm is an efficient method in computational number theory and elliptic curve cryptography for determining the number of points on an elliptic curve over a finite field.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
computational number theory algorithm
ⓘ
elliptic-curve method ⓘ integer factorization algorithm ⓘ |
| alsoKnownAs |
Lenstra ECM
ⓘ
linked to:
Lenstra elliptic-curve factorization method
elliptic curve method (ECM) for factorization ⓘ |
| basedOn | properties of elliptic curves modulo n ⓘ |
| betterThan | Pollard p-1 method for many inputs ⓘ |
| category | public-key cryptanalysis tool ⓘ |
| comparedTo |
Pollard p-1 factorization method
ⓘ
general number field sieve ⓘ quadratic sieve ⓘ |
| complexityDependsOn | size of the smallest prime factor of n ⓘ |
| coreOperation |
elliptic-curve point multiplication
ⓘ
modular arithmetic ⓘ |
| field |
computational number theory
ⓘ
cryptography ⓘ number theory ⓘ |
| goal | integer factorization ⓘ |
| hasParameter |
number of random curves to try
ⓘ
second-stage bound B2 ⓘ smoothness bound B1 ⓘ |
| hasVariant | stage-2 ECM ⓘ |
| implementedIn |
GMP-ECM
ⓘ
PARI/GP ⓘ SageMath ⓘ |
| influenced | elliptic curve method (ECM) implementations in cryptographic libraries ⓘ |
| input | composite integer n ⓘ |
| introducedIn | 1985 ⓘ |
| inventor |
Hendrik W. Lenstra Jr.
ⓘ
linked to:
Hendrik Lenstra
|
| namedAfter |
Hendrik Willem Lenstra Jr.
ⓘ
linked to:
Hendrik Lenstra
|
| optimizedFor | integers with relatively small prime factors ⓘ |
| output |
failure indication if no factor found in given bounds
ⓘ
nontrivial factor of n ⓘ |
| probabilistic | true ⓘ |
| publication |
Factoring integers with elliptic curves
ⓘ
linked to:
Lenstra elliptic-curve factorization method
|
| publishedIn | Annals of Mathematics ⓘ |
| purpose | to find nontrivial factors of composite integers ⓘ |
| randomized | true ⓘ |
| relatedTo |
RSA cryptosystem security
ⓘ
elliptic-curve cryptography ⓘ |
| reliesOn | failure of modular inversion revealing a nontrivial gcd ⓘ |
| step |
choose random elliptic curve modulo n
ⓘ
choose random point on the elliptic curve modulo n ⓘ compute greatest common divisor when inversion fails ⓘ compute scalar multiples of the point ⓘ perform group operations modulo n ⓘ |
| usedFor |
factoring RSA moduli with a relatively small prime factor
ⓘ
finding medium-size prime factors in GNFS precomputation ⓘ |
| uses |
elliptic curves over finite fields
ⓘ
group law on elliptic curves ⓘ |
| yearOfPublication | 1987 ⓘ |
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: Lenstra elliptic-curve factorization method Description of subject: The Lenstra elliptic-curve factorization method is an integer factorization algorithm that uses properties of elliptic curves over finite fields to efficiently find nontrivial factors of large numbers, especially those with relatively small prime divisors.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
Lenstra elliptic-curve factorization method
→
publication
→
Factoring integers with elliptic curves
ⓘ
linked to: Lenstra elliptic-curve factorization method
linked to: Lenstra elliptic-curve factorization method