Pairing-based cryptography: Difference between revisions

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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:.

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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 researchers classify pairing instantiations into three (or more) basic types:

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>.<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>

==Usage in cryptography==