- Pairing
The concept of pairing treated here occurs in
mathematics .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 imes N o 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 o L
where M otimes_R N denotes the
tensor product of "M" and "N".A pairing can also be considered as an R-linear mapPhi : M o operatorname{Hom}_{R} (N, L) , which matches the first definition by setting Phi (m) (n) := e(m,n) .
A pairing is called perfect if the above map Phi is an isomorphism of R-modules.
A pairing is called alternating if for the above map we have e(m,m) = 1 .
A pairing is called non-degenerate if for the above map we have e(m,n) = 1 for all m implies n=0 .
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 imes k^2 o k.
The Hopf map S^3 o S^2 written as h:S^2 imes S^2 o S^2 is an example of a pairing. In [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)
] 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 [Dan Boneh, Matthew K. Franklin, Identity-Based Encryption from the Weil Pairing "Advances in Cryptology - Proceedings of CRYPTO 2001" (2001)] :Let extstyle G_1 be an additive and extstyle G_2 a multiplicative group both of prime order extstyle p. Let extstyle P, Q be generators extstyle in G_1.
A pairing is a
e: G_1 imes G_1 ightarrow G_2for which the following holds:
#Bilinearity : extstyle forall P,Q in G_1,, a,b in mathbb{Z}_p^*: eleft(aP, bQ ight) = eleft(P, Q ight)^{ab}
#Non-degeneracy : extstyle forall P in G_1,,P eq infty: eleft(P, P ight) eq 1
# For practical purposes, extstyle e has to becomputable in an efficient mannerThe
Weil pairing is a pairing important inelliptic curve cryptography to avoid theMOV attack . It and other pairings have been used to developidentity-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.External links
* [http://planeta.terra.com.br/informatica/paulobarreto/pblounge.html The Pairing-Based Crypto Lounge]
References
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