# Pairing

{{#invoke:Hatnote|hatnote}} 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

$e(rm,n)=e(m,rn)=re(m,n)$ ,
$e(m_{1}+m_{2},n)=e(m_{1},n)+e(m_{2},n)$ and $e(m,n_{1}+n_{2})=e(m,n_{1})+e(m,n_{2})$ $M\otimes _{R}N\to L$ A pairing can also be considered as an R-linear map $\Phi :M\to \operatorname {Hom} _{R}(N,L)$ , 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

{{#invoke:main|main}} In cryptography, often the following specialized definition is used:

for which the following holds:

Note that is also common in cryptographic literature for all groups to be written in multiplicative notation.

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.