Pairing

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This article is about the mathematics concept. For other uses, see pair.

The concept of pairing treated here occurs in mathematics.

Contents

[edit] Definition

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 e:M \times N \to L. That is, it satisfies

e(rm,n) = e(m,rn) = re(m,n)

for any r \in R. Or equivalently, a pairing is an R-linear map

M \otimes_R N \to L

where M \otimes_R N denotes the tensor product of M and N.

A pairing can also be considered as an R-linear map \Phi : M \to \operatorname{Hom}_{R} (N, L) , which matches the first definition by setting Φ(m)(n): = e(m,n).

A pairing is called perfect if the above map Φ is an isomorphism of R-modules.

[edit] Examples

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 k^2 \times k^2 \to k.

The Hopf map S^3 \to S^2 written as h:S^2 \times S^2 \to S^2 is an example of a pairing. In [1] for instance, Hardie et. al present an explicit construction of the map using poset models.

[edit] Pairings in Cryptography

In cryptography, often the following specialized definition is used [2]:

Let \textstyle G_1 be an additive and \textstyle G_2 a multiplicative group both of prime order \textstyle p. Let \textstyle P, Q be generators \textstyle \in G_1.

A pairing is a map: e: G_1 \times G_1 \rightarrow G_2

for which the following holds:

  1. Bilinearity: \textstyle \forall P,Q \in G_1,\, a,b \in \mathbb{Z}_p^*:\ e\left(aP, bQ\right) = e\left(P, Q\right)^{ab}
  2. Non-degeneracy: \textstyle \forall P \in G_1,\,P \neq \infty:\ e\left(P, P\right) \neq 1
  3. For practical purposes, \textstyle e has to be computable in an efficient manner

The Weil pairing is a pairing important in elliptic curve cryptography to avoid the MOV attack. It and other pairings have been used to develop identity-based encryption schemes.

[edit] 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.

[edit] External links

[edit] References

  1. ^ A nontrivial pairing of finite T0 spaces Authors: Hardie K.A.1; Vermeulen J.J.C.; Witbooi P.J. Source: Topology and its Applications, Volume 125, Number 3, 20 November 2002 , pp. 533-542(10)
  2. ^ Dan Boneh, Matthew K. Franklin, Identity-Based Encryption from the Weil Pairing Advances in Cryptology - Proceedings of CRYPTO 2001 (2001)