Pairing

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 $e:M\; imes\; N\; o\; L$. 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

:$M\; otimes\_R\; N\; o\; L$

where $M\; otimes\_R\; N$ denotes the tensor product of "M" and "N".

A pairing can also be considered as an R-linear map$Phi\; :\; M\; o\; operatorname\{Hom\}\_\{R\}\; (N,\; L)$, 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.

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

A pairing is called non-degenerate if for the above map we have $e(m,n)\; =\; 1$ 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 $k^2\; imes\; k^2\; o\; k$.

The Hopf map $S^3\; o\; S^2$ written as $h:S^2\; imes\; S^2\; o\; S^2$ is an example of a pairing. In [A nontrivial pairing of finite T0 spaces Authors: Hardie K.A.1; Vermeulen J.J.C.; Witbooi P.J.

Source: Topology and its Applications, Volume 125, Number 3, 20 November 2002 , pp. 533-542(10)

] 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 [Dan Boneh, Matthew K. Franklin, Identity-Based Encryption from the Weil Pairing "Advances in Cryptology - Proceedings of CRYPTO 2001" (2001)] :

Let $extstyle\; G\_1$ be an additive and $extstyle\; G\_2$ a multiplicative group both of prime order $extstyle\; p$. Let $extstyle\; P,\; Q$ be generators $extstyle\; in\; G\_1$.

A pairing is a
$e:\; G\_1\; imes\; G\_1\; ightarrow\; G\_2$

for which the following holds:
# Bilinearity: $extstyle\; forall\; P,Q\; in\; G\_1,,\; a,b\; in\; mathbb\{Z\}\_p^*:\; eleft(aP,\; bQ\; ight)\; =\; eleft(P,\; Q\; ight)^\{ab\}$
# Non-degeneracy: $extstyle\; forall\; P\; in\; G\_1,,P\; eq\; infty:\; eleft(P,\; P\; ight)\; eq\; 1$
# For practical purposes, $extstyle\; e$ 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

* [http://planeta.terra.com.br/informatica/paulobarreto/pblounge.html The Pairing-Based Crypto Lounge]

References