Pairing-based cryptography: Difference between revisions

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Let <math>G_1, G_2</math> be two additive cyclic groups of prime order <math>q</math>, and <math>G_T</math> another cyclic group of order <math>q</math> written multiplicatively. A pairing is a map: <math> e: G_1 \times G_2 \rightarrow G_T </math>, which satisfies the following properties:

# [[Bilinearity]]: <math> \forall a,b \in F_q^*,\ \forall P\in G_1, Q\in G_2:\ e\left(a P, b Q\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>.