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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. ## Examples

Any scalar product on a real vector space V is a pairing (set M = N = V, R = R in the above definitions).## Pairings in Cryptography

In cryptography, often the following specialized definition is used :## 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

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 $Phi\; :\; M\; to\; operatorname\{Hom\}\_\{R\}\; (N,\; L)$, 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 $n=0$.

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.

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: $textstyle\; forall\; P,Q\; in\; G\_1,,\; a,b\; in\; mathbb\{Z\}\_p^*:\; eleft(aP,\; bQright)\; =\; eleft(P,\; Qright)^\{ab\}$
- Non-degeneracy: $textstyle\; forall\; P\; in\; G\_1,,P\; neq\; infty:\; eleft(P,\; Pright)\; neq\; 1$
- 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.

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Last updated on Saturday September 13, 2008 at 10:19:36 PDT (GMT -0700)

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Last updated on Saturday September 13, 2008 at 10:19:36 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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