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 commutative ring with unit, and let M, N and L be R-modules.

A pairing is any R-bilinear map

Equivalently, a pairing is an R-linear map where

denotes the tensor product of M and N. A pairing can also be considered as an R-linear map

, which matches the first definition by setting

A pairing is called perfect if the above map

is an isomorphism of R-modules and the other evaluation map

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

is called non-degenerate on the left if

A pairing is called alternating if

, while bilinearity shows

Thus, for an alternating pairing,

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

The Hopf map

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

be additive groups and

a multiplicative group, all of prime order

A pairing is a map:

for which the following holds: Note that it is also common in cryptographic literature for all groups to be written in multiplicative notation.

, the pairing is called symmetric.

is cyclic, the map

, there exist integers

The Weil pairing is an important concept in elliptic curve cryptography; e.g., it may be used to attack certain elliptic curves (see MOV attack).

It and other pairings have been used to develop identity-based encryption schemes.

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.