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.