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Wikipedia / Pairing

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

Definition

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

A pairing is any R. That is, it satisfies

  1. ,

. Equivalently, a pairing is an R-linear map

denotes the tensor product of M and N.

A pairing can also be considered as an R-linear map.

A pairing is called perfect is an isomorphism of R 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 is called non-degenerate on the left.

, a pairing is called symmetric for all m, n, and alternating for all m.

Examples

is also called a bilinear form. Examples include any scalar product on a real vector space V, or the determinant map (2 2 matrices over k) → k.

For the dual module is known as the natural pairing.

Other examples of pairings occur in various duality theories, such as Serre duality or Poincaré duality. See also Dual system.

Another example is the Yoneda product between Ext groups.

The Hopf map is an example of a pairing. 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:

a multiplicative group, all of prime order be generators respectively.

for which the following holds:

  1. Bilinearity

  2. Non-degeneracy

  3. 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.

is cyclic will be commutative.

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.

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