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'''Pairing-based cryptography''' is the use of a [[pairing]] between elements of two cryptographic [[Group (mathematics)|groups]] to a third group <math>e :G_1 \times G_2 \to G_T</math> to construct [[cryptography|cryptographic]] systems
== Classification == If the same group is used for the first two groups (i.e. <math> G_1 = G_2</math>), the pairing is called ''symmetric'' and is a [[Map (mathematics)|mapping]] from two elements of one group to an element from a second group. Some researches separates different possibile pairing instantiations into three basic types:<ref>{{cite journal|last1=Galbraith|first1=Steven|last2=Paterson|first2=Kenneth|last3=Smart|first3=Nigel|title=Pairings for Cryptographers|journal=Discrete Applied Mathematics|date=2008|volume=156|issue=16|pages=3113–3121}}</ref>
* '''Type 1''': <math> G_1 = G_2</math>;
* '''Type 2''': <math> G_1 \ne G_2</math> but there is an ''efficiently computable'' [[homomorphism]] <math>\phi : G_2 \to G_1</math>;
* '''Type 3''': <math> G_1 \ne G_2</math> and there are no ''efficiently computable'' homomorphisms between <math>G_1</math> and <math>G_2</math>.
==Usage in cryptography==
If symmetric, pairings can be used to reduce a hard problem in one group to a different, usually easier problem in another group.
For example, in groups equipped with a [[bilinear mapping]] such as the [[Weil pairing]] or [[Tate pairing]], generalizations of the [[Diffie–Hellman problem|computational Diffie–Hellman problem]] are believed to be infeasible while the simpler [[decisional Diffie–Hellman assumption|decisional Diffie–Hellman problem]] can be easily solved using the pairing function. The first group is sometimes referred to as a '''Gap Group''' because of the assumed difference in difficulty between these two problems in the group.
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