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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|doi=10.1016/j.dam.2007.12.010}}</ref>
==Usage in cryptography==
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