Definitions

# Pairing

[pair-ing]
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

$e\left(rm,n\right)=e\left(m,rn\right)=re\left(m,n\right)$

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 $Phi : M to operatorname\left\{Hom\right\}_\left\{R\right\} \left(N, L\right)$, which matches the first definition by setting $Phi \left(m\right) \left(n\right) := e\left(m,n\right)$.

A pairing is called perfect if the above map $Phi$ is an isomorphism of R-modules.

A pairing is called alternating if for the above map we have $e\left(m,m\right) = 1$.

A pairing is called non-degenerate if for the above map we have $e\left(m,n\right) = 1$ for all $m$ implies $n=0$.

## 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: $textstyle forall P,Q in G_1,, a,b in mathbb\left\{Z\right\}_p^*: eleft\left(aP, bQright\right) = eleft\left(P, Qright\right)^\left\{ab\right\}$
2. Non-degeneracy: $textstyle forall P in G_1,,P neq infty: eleft\left(P, Pright\right) neq 1$
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.