Pairing - Pairings in Cryptography

Pairings in Cryptography

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:

1. Bilinearity:
2. Non-degeneracy:
3. For practical purposes, has to be computable in an efficient manner

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 a pairing important 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.