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 mapEquivalently, a pairing is an R-linear map wheredenotes the tensor product of M and N. A pairing can also be considered as an R-linear map, which matches the first definition by settingA pairing is called perfect if the above mapis an isomorphism of R-modules and the other evaluation mapIn 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 thatis called non-degenerate on the left ifA pairing is called alternating if, while bilinearity showsThus, 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 pairingThe Hopf mapFor 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] Letbe additive groups anda multiplicative group, all of prime orderA 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 integersThe 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.