出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/05/02 17:24:42」(JST)
The concept of pairing treated here occurs in mathematics.
Let R be a commutative ring with unity, and let M, N and L be three R-modules.
A pairing is any R-bilinear map . That is, it satisfies
for any and any and any . Or 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.
If a pairing is called alternating if for the above map we have .
A pairing is called non-degenerate if for the above map we have that for all implies .
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 written as is an example of a pairing. In [1] for instance, Hardie et al. 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 . Let be generators of and respectively.
A pairing is a map:
for which the following holds:
Note that is also common in cryptographic literature for all groups to be written in multiplicative notation.
In cases when , the pairing is called symmetric. If, furthermore, 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 pairing 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.
全文を閲覧するには購読必要です。 To read the full text you will need to subscribe.
| リンク元 | 「対合」「対形成」 |
| 拡張検索 | 「mispairing」「base pairing」「chromosome pairing」 |