Weil pairing
E244846
The Weil pairing is a bilinear, alternating, non-degenerate pairing on the torsion points of an elliptic curve, fundamental in number theory and modern cryptography.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Weil pairing canonical | 1 |
| Weil pairing on abelian varieties | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2228057 — 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: Weil pairing Context triple: [André Weil, notableConcept, Weil pairing]
-
A.
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.
-
B.
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.
-
C.
Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
-
D.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
E.
Deuring reduction theorem
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
- 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: Weil pairing Target entity description: The Weil pairing is a bilinear, alternating, non-degenerate pairing on the torsion points of an elliptic curve, fundamental in number theory and modern cryptography.
-
A.
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.
-
B.
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.
-
C.
Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
-
D.
Weierstrass elliptic functions
Weierstrass elliptic functions are a class of doubly periodic meromorphic functions that play a central role in the theory of elliptic curves and complex analysis.
-
E.
Deuring reduction theorem
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
alternating form
ⓘ
bilinear form ⓘ mathematical concept ⓘ non-degenerate pairing ⓘ pairing ⓘ |
| appearsIn | theory of abelian varieties ⓘ |
| codomain |
multiplicative group of the base field extension
ⓘ
roots of unity ⓘ |
| definedOn | E[n] × E[n] for an elliptic curve E and integer n ⓘ |
| definedOver | algebraic closure of the base field ⓘ |
| domain | torsion points of an elliptic curve ⓘ |
| field |
arithmetic geometry
ⓘ
cryptography ⓘ elliptic curve theory ⓘ number theory ⓘ |
| generalizedBy |
Weil pairing
self-linksurface differs
ⓘ
surface form:
Weil pairing on abelian varieties
|
| hasApplication |
attacks on certain elliptic curve cryptosystems via MOV reduction
ⓘ
construction of bilinear maps for cryptographic pairings ⓘ |
| introducedIn | 20th century ⓘ |
| invariantUnder | isogenies of elliptic curves up to compatibility ⓘ |
| isAlternating | true ⓘ |
| isBilinear | true ⓘ |
| isGaloisEquivariant | true ⓘ |
| isNondegenerate | true ⓘ |
| isPerfectPairing | true ⓘ |
| isSkewSymmetric | true ⓘ |
| namedAfter | André Weil ⓘ |
| relatedTo |
Miller algorithm
ⓘ
Tate pairing ⓘ elliptic curve isogenies ⓘ |
| relates | n-torsion subgroup of an elliptic curve to roots of unity ⓘ |
| requires | choice of integer n relatively prime to the characteristic ⓘ |
| satisfiesProperty |
e_n(P+P',Q)=e_n(P,Q)·e_n(P',Q)
ⓘ
e_n(P,P)=1 for all P in E[n] ⓘ e_n(P,Q)=e_n(Q,P)^{-1} ⓘ e_n(P,Q+Q')=e_n(P,Q)·e_n(P,Q') ⓘ e_n(P,kQ)=e_n(P,Q)^k ⓘ e_n(kP,Q)=e_n(P,Q)^k ⓘ |
| usedIn |
cryptographic protocol design
ⓘ
elliptic curve cryptography ⓘ identity-based encryption ⓘ pairing-based cryptography ⓘ tripartite Diffie–Hellman key exchange ⓘ |
| usedTo |
construct pairings with cryptographic hardness assumptions
ⓘ
prove properties of torsion points on elliptic curves ⓘ study Galois representations attached to elliptic curves ⓘ transfer discrete logarithm problems from elliptic curves to finite fields ⓘ |
| valuesIn | group of n-th roots of unity ⓘ |
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: Weil pairing Description of subject: The Weil pairing is a bilinear, alternating, non-degenerate pairing on the torsion points of an elliptic curve, fundamental in number theory and modern cryptography.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Weil pairing on abelian varieties