Tate pairing
E860120
The Tate pairing is a bilinear, non-degenerate pairing on the points of an elliptic curve (or abelian variety) over a finite field, fundamental in number theory and widely used in pairing-based cryptography.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Tate pairing canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10389270 — 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: Tate pairing Context triple: [Weil pairing, relatedTo, Tate pairing]
-
A.
Weil pairing
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.
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B.
Cassels–Tate pairing
The Cassels–Tate pairing is a bilinear pairing on the Tate–Shafarevich group of an abelian variety over a number field that plays a central role in arithmetic geometry and the study of rational points.
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C.
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.
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D.
Montgomery ladder
The Montgomery ladder is a scalar multiplication algorithm on elliptic curves that provides efficient, uniform, and side-channel-resistant computation for cryptographic protocols such as those based on Curve25519.
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E.
Twisted Edwards curve
A Twisted Edwards curve is a type of elliptic curve with a specific algebraic form that enables especially fast and secure implementations of cryptographic operations such as digital signatures and key exchange.
- 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: Tate pairing Target entity description: The Tate pairing is a bilinear, non-degenerate pairing on the points of an elliptic curve (or abelian variety) over a finite field, fundamental in number theory and widely used in pairing-based cryptography.
-
A.
Weil pairing
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.
-
B.
Cassels–Tate pairing
The Cassels–Tate pairing is a bilinear pairing on the Tate–Shafarevich group of an abelian variety over a number field that plays a central role in arithmetic geometry and the study of rational points.
-
C.
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.
-
D.
Montgomery ladder
The Montgomery ladder is a scalar multiplication algorithm on elliptic curves that provides efficient, uniform, and side-channel-resistant computation for cryptographic protocols such as those based on Curve25519.
-
E.
Twisted Edwards curve
A Twisted Edwards curve is a type of elliptic curve with a specific algebraic form that enables especially fast and secure implementations of cryptographic operations such as digital signatures and key exchange.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
Weil–Tate pairing
ⓘ
bilinear pairing ⓘ mathematical concept ⓘ |
| codomain |
group of roots of unity
ⓘ
multiplicative group of the finite field modulo r-th powers ⓘ |
| computableBy | Miller algorithm NERFINISHED ⓘ |
| definedOn |
abelian varieties over finite fields
ⓘ
elliptic curves over finite fields ⓘ |
| definedUsing |
Galois cohomology
ⓘ
Kummer theory NERFINISHED ⓘ rational functions on elliptic curves ⓘ |
| dependsOn |
choice of integer r coprime to the characteristic
ⓘ
finite extension of the base field ⓘ |
| domain |
r-torsion points modulo r-th powers
ⓘ
torsion subgroup of an elliptic curve ⓘ |
| field |
arithmetic geometry
ⓘ
cryptography ⓘ number theory ⓘ pairing-based cryptography ⓘ |
| generalizationOf | Weil pairing on elliptic curves ⓘ |
| hasVariant |
ate pairing
ⓘ
eta pairing ⓘ optimal ate pairing ⓘ reduced Tate pairing NERFINISHED ⓘ |
| introducedBy | John Tate NERFINISHED ⓘ |
| invariantUnder | isogenies up to isomorphism ⓘ |
| isAlternating | often true up to normalization ⓘ |
| isBilinear | true ⓘ |
| isGaloisEquivariant | true ⓘ |
| isNondegenerate | true ⓘ |
| maps | pairs of points to elements of a finite multiplicative group ⓘ |
| nontrivialWhen | elliptic curve has nontrivial r-torsion over an extension field ⓘ |
| property |
bilinear in each argument modulo r-th powers
ⓘ
non-degenerate on appropriate quotient groups ⓘ |
| relatedTo | Weil pairing NERFINISHED ⓘ |
| securityDependsOn |
bilinear Diffie–Hellman problem
ⓘ
discrete logarithm problem on elliptic curves ⓘ |
| usedIn |
attribute-based encryption
ⓘ
broadcast encryption ⓘ cryptographic accumulators ⓘ group signatures ⓘ identity-based encryption ⓘ key agreement protocols ⓘ short signature schemes ⓘ succinct non-interactive arguments of knowledge ⓘ tripartite Diffie–Hellman key exchange ⓘ verifiable random functions ⓘ zero-knowledge proofs ⓘ |
| usedWith |
ordinary pairing-friendly curves
ⓘ
pairing-friendly elliptic curves ⓘ supersingular elliptic curves ⓘ |
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: Tate pairing Description of subject: The Tate pairing is a bilinear, non-degenerate pairing on the points of an elliptic curve (or abelian variety) over a finite field, fundamental in number theory and widely used in pairing-based cryptography.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.