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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.