Pairing-based cryptography

Pairing-based cryptography is the use of a pairing between elements of two cryptographic groups to a third group with a mapping ${\displaystyle e:G_{1}\times G_{2}\to G_{T}}$ to construct or analyze cryptographic systems.

The following definition is commonly used in most academic papers.

Let ${\displaystyle G_{1},G_{2}}$ be two additive cyclic groups of prime order ${\displaystyle q}$, and ${\displaystyle G_{T}}$ another cyclic group of order ${\displaystyle q}$ written multiplicatively. A pairing is a map: ${\displaystyle e:G_{1}\times G_{2}\rightarrow G_{T}}$, which satisfies the following properties:

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