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{{About|the mathematics concept|other uses|Pair (disambiguation)}}
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 [[Module (mathematics)|''R''-modules]].

A '''pairing''' is any ''R''-bilinear map <math>e:M \times N \to L</math>. That is, it satisfies

:<math>e(rm,n)=e(m,rn)=re(m,n)</math>,

:<math>e(m_1+m_2,n)=e(m_1,n)+e(m_2,n)</math> and <math>e(m,n_1+n_2)=e(m,n_1)+e(m,n_2)</math>

for any <math>r \in R</math> and any <math>m,m_1,m_2 \in M</math> and any <math>n,n_1,n_2 \in N </math>. Or equivalently, a pairing is an ''R''-linear map

:<math>M \otimes_R N \to L</math>

where <math>M \otimes_R N</math> denotes the [[tensor product]] of ''M'' and ''N''.

A pairing can also be considered as an R-linear map
<math>\Phi : M \to \operatorname{Hom}_{R} (N, L) </math>, which matches the first definition by setting
<math>\Phi (m) (n) := e(m,n) </math>.

A pairing is called '''perfect''' if the above map <math> \Phi </math> is an isomorphism of R-modules.

If <math> N=M </math> a pairing is called '''alternating''' if for the above map we have <math> e(m,m) = 0 </math>.

A pairing is called '''non-degenerate''' if for the above map we have that <math> e(m,n) = 0 </math> for all <math>m</math> implies <math> n=0 </math>.

==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 &times; 2 matrices over ''k'') → ''k''  can be seen as a pairing <math>k^2 \times k^2 \to k</math>.

The Hopf map <math>S^3 \to S^2</math> written as <math>h:S^2 \times S^2 \to S^2 </math> is an example of a pairing. In <ref>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)

</ref> 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:<ref>Dan Boneh, Matthew K. Franklin, Identity-Based Encryption from the Weil Pairing ''Advances in Cryptology - Proceedings of CRYPTO 2001'' (2001)</ref>

Let <math>\textstyle G_1, G_2</math> be additive groups and <math>\textstyle G_T</math> a multiplicative [[group (Mathematics)|group]], all of prime [[Order (group theory)|order]] <math>\textstyle p</math>. Let <math>\textstyle P \in G_1, Q \in G_2</math> be [[Generating set of a group|generators]] of <math>\textstyle G_1</math> and <math>\textstyle G_2</math> respectively.

A pairing is a map: <math> e: G_1 \times G_2 \rightarrow G_T </math>

for which the following holds:
# [[Bilinearity]]: <math>\textstyle \forall a,b \in \mathbb{Z}_p^*:\ e\left(P^a, Q^b\right) = e\left(P, Q\right)^{ab}</math>
# [[Degeneracy (mathematics)|Non-degeneracy]]: <math>\textstyle e\left(P, Q\right) \neq 1</math>
# For practical purposes, <math>\textstyle e</math> has to be [[computable]] in an efficient manner

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

In cases when <math>\textstyle G_1 = G_2 = G</math>, the pairing is called symmetric. If, furthermore, <math>\textstyle G</math> is [[Cyclic group|cyclic]], the map <math> e </math> will be [[Commutative property|commutative]]; that is, for any <math> P,Q \in G </math>, we have <math> e(P,Q) = e(Q,P) </math>. This is because for a generator <math> g \in G </math>, there exist integers <math> p </math>, <math> q </math> such that <math> P = g^p </math> and <math> Q=g^q </math>. Therefore <math> e(P,Q) = e(g^p,g^q) = e(g,g)^{pq} = e(g^q, g^p) = e(Q,P) </math>.

The [[Weil pairing]] is an important pairing in [[elliptic curve cryptography]]; e.g., it may be used to attack certain elliptic curves (see [http://crypto.stackexchange.com/q/1871/77 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'''.

==References==
<references/>

==External links==
* [http://www.larc.usp.br/~pbarreto/pblounge.html The Pairing-Based Crypto Lounge]
{{Use dmy dates|date=September 2010}}

[[Category:Linear algebra]]
[[Category:Module theory]]
[[Category:Pairing-based cryptography]]

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