Pairing In mathematics , a pairing is an R -bilinear map from a product of two R -modules to the underlying ring R . When R is a field and the two modules are equal, this gives a bilinear form . Thus, bilinear pairings give a generalization of inner products .Definition Let R be a commutative ring with unit , and let M , N and L be R -modules. A pairing is any R -bilinear map. That is, it satisfiesfor any and any and any. 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. A pairing is called non-degenerate on the right if for the above map we have that for all implies ; similarly, is called non-degenerate on the left if for all implies. A pairing is called alternating if and for all m . In particular, this implies, while bilinearity shows. Thus, for an alternating pairing,, which justifies the name.Examples Any scalar product on a real vector space V is a pairing. The determinant map → k can be seen as a pairing. The Hopf map written as is an example of a pairing. For instance, Hardie et al . present an explicit construction of the map using poset models.In cryptography, often the following specialized definition is used: Let be additive groups and a multiplicative group , all of prime order . Let be generators of and respectively. A pairing is a map: for which the following holds: Bilinearity: Non-degeneracy: For practical purposes, 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, the pairing is called symmetric. As is cyclic , the map will be commutative ; that is, for any, we have. This is because for a generator , there exist integers, such that and. Therefore. The Weil pairing is an important concept in elliptic curve cryptography ; e.g., it may be used to attack certain elliptic curves . 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 .
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