Pairing

In mathematics, a pairing is an R-bilinear map from the Cartesian product of two R-modules, where the underlying ring R is commutative.

Let R be a

.

A pairing is any R-bilinear map ${\displaystyle e:M\times N\to L}$. That is, it satisfies

${\displaystyle e(r\cdot m,n)=e(m,r\cdot n)=r\cdot e(m,n)}$,

${\displaystyle e(m_{1}+m_{2},n)=e(m_{1},n)+e(m_{2},n)}$ and ${\displaystyle e(m,n_{1}+n_{2})=e(m,n_{1})+e(m,n_{2})}$

for any ${\displaystyle r\in R}$ and any ${\displaystyle m,m_{1},m_{2}\in M}$ and any ${\displaystyle n,n_{1},n_{2}\in N}$. Equivalently, a pairing is an R-linear map

${\displaystyle M\otimes _{R}N\to L}$

where ${\displaystyle M\otimes _{R}N}$ denotes the tensor product of M and N.

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

A pairing is called perfect if the above map ${\displaystyle \Phi }$ is an isomorphism of R-modules and the other evaluation map ${\displaystyle \Phi '\colon N\to \operatorname {Hom} _{R}(M,L)}$ is an isomorphism also. In nice cases, it suffices that just one of these be an isomorphism, e.g. when R is a field, M,N are finite dimensional vector spaces and L=R.

A pairing is called non-degenerate on the right if for the above map we have that ${\displaystyle e(m,n)=0}$ for all ${\displaystyle m}$ implies ${\displaystyle n=0}$; similarly, ${\displaystyle e}$ is called non-degenerate on the left if ${\displaystyle e(m,n)=0}$ for all ${\displaystyle n}$ implies ${\displaystyle m=0}$.

A pairing is called alternating if ${\displaystyle N=M}$ and ${\displaystyle e(m,m)=0}$ for all m. In particular, this implies ${\displaystyle e(m+n,m+n)=0}$, while bilinearity shows ${\displaystyle e(m+n,m+n)=e(m,m)+e(m,n)+e(n,m)+e(n,n)=e(m,n)+e(n,m)}$. Thus, for an alternating pairing, ${\displaystyle e(m,n)=-e(n,m)}$.

Any

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 ${\displaystyle k^{2}\times k^{2}\to k}$.

The

${\displaystyle S^{3}\to S^{2}}$ written as ${\displaystyle h:S^{2}\times S^{2}\to S^{2}}$ is an example of a pairing. For instance, Hardie et al.^[1] present an explicit construction of the map using poset models.

In cryptography, often the following specialized definition is used:^[2]

Let ${\displaystyle \textstyle G_{1},G_{2}}$ be additive groups and ${\displaystyle \textstyle G_{T}}$ a multiplicative

${\displaystyle \textstyle p}$. Let ${\displaystyle \textstyle P\in G_{1},Q\in G_{2}}$ be generators of ${\displaystyle \textstyle G_{1}}$ and ${\displaystyle \textstyle G_{2}}$ respectively.

A pairing is a map: ${\displaystyle e:G_{1}\times G_{2}\rightarrow G_{T}}$

for which the following holds:

1. Bilinearity
   : ${\displaystyle \textstyle \forall a,b\in \mathbb {Z} :\ e\left(aP,bQ\right)=e\left(P,Q\right)^{ab}}$
2. Non-degeneracy: ${\displaystyle \textstyle e\left(P,Q\right)\neq 1}$
3. For practical purposes, ${\displaystyle \textstyle e}$ has to be
   computable
   in an efficient manner

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

In cases when ${\displaystyle \textstyle G_{1}=G_{2}=G}$, the pairing is called symmetric. As ${\displaystyle \textstyle G}$ is cyclic, the map ${\displaystyle e}$ will be commutative; that is, for any ${\displaystyle P,Q\in G}$, we have ${\displaystyle e(P,Q)=e(Q,P)}$. This is because for a generator ${\displaystyle g\in G}$, there exist integers ${\displaystyle p}$, ${\displaystyle q}$ such that ${\displaystyle P=g^{p}}$ and ${\displaystyle Q=g^{q}}$. Therefore ${\displaystyle e(P,Q)=e(g^{p},g^{q})=e(g,g)^{pq}=e(g^{q},g^{p})=e(Q,P)}$.

The

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.

• Dual system
• Yoneda product

• The Pairing-Based Crypto Library