# 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

#### Derived terms

### Verb

pairing- present participle of pair

# Extensive Definition

- This article is about the mathematics concept. For other uses, see pair.

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:

- Bilinearity: \textstyle \forall P,Q \in G_1,\, a,b \in \mathbb_p^*:\ e\left(aP, bQ\right) = e\left(P, Q\right)^
- Non-degeneracy: \textstyle \forall P \in G_1,\,P \neq \infty:\ e\left(P, P\right) \neq 1
- 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.## External links

## References

# Synonyms, Antonyms and Related Words

Janus,
agglomeration,
agglutination,
aggregation,
ambiguity, ambivalence, articulation, biformity, bifurcation, bond, bracketing, clustering, combination, communication, concatenation, concourse, concurrence, confluence, congeries, conglomeration, conjugation, conjunction, connection, convergence, copulation, coupling, dichotomy, doubleness, doublethink, doubling, dualism, duality, duplexity, duplication, duplicity, equivocality, gathering, halving, hookup, intercommunication,
intercourse,
interlinking,
irony, joinder, joining, jointure, junction, knotting, liaison, linkage, linking, marriage, meeting, merger, merging, polarity, splice, symbiosis, tie, tie-in, tie-up, twinning, two-facedness,
twoness, unification, union, yoking