In mathematics, a pairing is an R-bilinear map of modules, where R is the underlying ring.
Contents
Definition
Let R be a commutative ring with unity, and let M, N and L be three R-modules.
A pairing is any R-bilinear map
for any
where
A pairing can also be considered as an R-linear map
A pairing is called perfect if the above map
A pairing is called non-degenerate if for the above map we have that
A pairing is called alternating if
Examples
Any scalar product on a real vector space V is a pairing (set M = N = V, R = R in the above definitions).
The determinant map (2 × 2 matrices over k) → k can be seen as a pairing
The Hopf map
Pairings in cryptography
In cryptography, often the following specialized definition is used:
Let
A pairing is a map:
for which the following holds:
- Bilinearity:
∀ a , b ∈ Z : e ( a P , b Q ) = e ( P , Q ) a b - Non-degeneracy:
e ( P , Q ) ≠ 1 - For practical purposes,
e has to be computable in an efficient manner
Note that is also common in cryptographic literature for all groups to be written in multiplicative notation.
In cases when
The Weil pairing is an important pairing in elliptic curve cryptography; e.g., it may be used to attack certain elliptic curves (see 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.