Schoof–Elkies–Atkin (SEA) point-counting algorithm
E753155
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Schoof–Elkies–Atkin (SEA) point-counting algorithm canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T8733512 — 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: Schoof–Elkies–Atkin (SEA) point-counting algorithm Context triple: [Hasse bound for elliptic curves, usedIn, Schoof–Elkies–Atkin (SEA) point-counting algorithm]
-
A.
Koblitz curves
Koblitz curves are a special class of elliptic curves defined over binary fields that enable particularly efficient and fast implementations of elliptic curve 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.
Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
-
D.
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.
-
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: Schoof–Elkies–Atkin (SEA) point-counting algorithm Target entity description: 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.
-
A.
Koblitz curves
Koblitz curves are a special class of elliptic curves defined over binary fields that enable particularly efficient and fast implementations of elliptic curve 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.
Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
-
D.
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.
-
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 (47)
| Predicate | Object |
|---|---|
| instanceOf |
algorithm
ⓘ
computational number theory method ⓘ elliptic curve algorithm ⓘ point-counting algorithm ⓘ |
| advantageOver | exponential-time point-counting methods ⓘ |
| appliesTo |
elliptic curves over finite fields of large characteristic
ⓘ
elliptic curves used in cryptography ⓘ |
| basedOn | Schoof algorithm ⓘ |
| comparedTo | brute-force point counting ⓘ |
| computes |
number of rational points on an elliptic curve over a finite field
ⓘ
trace of Frobenius of an elliptic curve over a finite field ⓘ |
| developedInContextOf | elliptic curve cryptography ⓘ |
| extends | Schoof algorithm ⓘ |
| field |
computational number theory
ⓘ
elliptic curve cryptography ⓘ
linked to:
Elliptic Curve Cryptography
|
| goal | efficient point counting on large elliptic curves ⓘ |
| hasPart |
Atkin prime stage
ⓘ
Chinese remainder reconstruction stage ⓘ Elkies prime stage ⓘ |
| improvesOn | Schoof algorithm ⓘ |
| input | elliptic curve over a finite field ⓘ |
| namedAfter |
A. O. L. Atkin
ⓘ
linked to:
W. R. Atkin
Noam Elkies ⓘ René Schoof ⓘ |
| output | cardinality of the elliptic curve group over the finite field ⓘ |
| property |
deterministic
ⓘ
uses Chinese remainder theorem ⓘ
linked to:
Chinese remainder theorem
uses classification of primes into Elkies and Atkin primes ⓘ uses isogenies of elliptic curves ⓘ uses modular polynomials ⓘ |
| relatedTo |
Frobenius endomorphism
ⓘ
Hasse bound for elliptic curves ⓘ isogeny volcanoes ⓘ modular polynomials Φ_l ⓘ |
| requires |
arithmetic in finite fields
ⓘ
computation of isogenies of small prime degree ⓘ computation of modular polynomials ⓘ |
| timeComplexity | polynomial in log(q) where q is the size of the finite field ⓘ |
| typicalUseCase |
selection of secure elliptic curves for public-key cryptography
ⓘ
verification of claimed curve orders in standards ⓘ |
| usedFor |
computing the order of an elliptic curve over a finite field
ⓘ
constructing elliptic curves with prescribed order ⓘ counting points on elliptic curves over finite fields ⓘ elliptic curve cryptosystem parameter generation ⓘ |
| usedIn |
CM (complex multiplication) method for generating elliptic curves
ⓘ
construction of cryptographically secure elliptic curves ⓘ elliptic curve cryptography parameter validation ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Schoof–Elkies–Atkin (SEA) point-counting algorithm Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.