The concept of pairing
treated here occurs in mathematics
be a commutative ring with unity, and let M
be three R
A pairing is any R-bilinear map . That is, it satisfies
for any . Or equivalently, a pairing is an R-linear map
where denotes the tensor product of M and N.
A pairing can also be considered as an R-linear map
, which matches the first definition by setting
A pairing is called perfect if the above map is an isomorphism of R-modules.
A pairing is called alternating if for the above map we have .
A pairing is called non-degenerate if for the above map we have for all implies .
Any scalar product
on a real
vector space V is a pairing (set M
, R = R
in the above definitions).
The determinant map (2 × 2 matrices over k) → k can be seen as a pairing .
The Hopf map written as is an example of a pairing. In for instance, Hardie et. al present an explicit construction of the map using poset models.
Pairings in Cryptography
, often the following specialized definition is used :
Let be an additive and a multiplicative group both of prime order . Let be generators .
A pairing is a map:
for which the following holds:
- For practical purposes, has to be computable in an efficient manner
The Weil pairing is a pairing important in elliptic curve cryptography to avoid the MOV attack. It and other pairings have been used to develop identity-based encryption schemes.
Slightly different usages of the notion of pairing
Scalar products on complex
vector spaces are sometimes called pairings, although they are not bilinear.
For example, in representation theory
, one has a scalar product on the characters of complex representations of a finite group which is frequently called character pairing