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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.
==Definition==
The following definition is commonly used in most academic papers.<ref>{{cite journal|last1=Koblitz|first1=Neal|last2=Menezes|first2=Alfred|title=Pairing-Based cryptography at high security levels|journal=LNCS|date=2005|volume=3796}}</ref>
Let <math>G_1, G_T</math> be two cyclic groups of prime order <math>q</math>. A pairing is a map: <math> e: G_1 \times G_1 \rightarrow G_T </math>, which satisfies the following properties:
# [[Bilinearity]]: <math> \forall a,b \in F_q^*,\ \forall P,Q \in G_1:\ e\left(P^a, Q^b\right) = e\left(P, Q\right)^{ab}</math>
# [[Degeneracy (mathematics)|Non-degeneracy]]: <math>e\left(P, Q\right) \neq 1</math>
# Computability: there exist an efficient algorithm to compute <math>e</math>
== Classification ==
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* '''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>.<ref name="pfc">{{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>
==Usage in cryptography==
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