# Dictionary Definition

pairing

### Noun

1 the act of pairing a male and female for reproductive purposes; "the casual couplings of adolescents"; "the mating of some species occurs only in the spring" [syn: coupling, mating, conjugation, union, sexual union]
2 the act of grouping things or people in pairs

# User Contributed Dictionary

## English

### Noun

1. The action of the verb to pair.

### Verb

pairing
1. present participle of pair

# Extensive Definition

The concept of pairing treated here occurs in mathematics.

## 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 e:M \times N \to L. That is, it satisfies
e(rm,n)=e(m,rn)=re(m,n)
for any r \in R. Or equivalently, a pairing is an R-linear map
M \otimes_R N \to L
where M \otimes_R N denotes the tensor product of M and N.
A pairing can also be considered as an R-linear map \Phi : M \to \operatorname_ (N, L) , which matches the first definition by setting \Phi (m) (n) := e(m,n) .
A pairing is called perfect if the above map \Phi is an isomorphism of R-modules.

## 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 k^2 \times k^2 \to k.
The Hopf map S^3 \to S^2 written as h:S^2 \times S^2 \to S^2 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

In cryptography, often the following specialized definition is used :
Let \textstyle G_1 be an additive and \textstyle G_2 a multiplicative group both of prime order \textstyle p. Let \textstyle P, Q be generators \textstyle \in G_1.
A pairing is a map: e: G_1 \times G_1 \rightarrow G_2
for which the following holds:
1. Bilinearity: \textstyle \forall P,Q \in G_1,\, a,b \in \mathbb_p^*:\ e\left(aP, bQ\right) = e\left(P, Q\right)^
2. Non-degeneracy: \textstyle \forall P \in G_1,\,P \neq \infty:\ e\left(P, P\right) \neq 1
3. For practical purposes, \textstyle e 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.