Pairing-based cryptography: Difference between revisions

Revision as of 21:58, 3 November 2023

Revision as of 19:45, 13 January 2024

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The following definition is commonly used in most academic papers.<ref>{{cite book|last1=Koblitz|first1=Neal|last2=Menezes|first2=Alfred|title=Cryptography and Coding |chapter=Pairing-Based cryptography at high security levels|series=Lecture Notes in Computer Science|date=2005|volume=3796|pages=13–36 |doi=10.1007/11586821_2|isbn=978-3-540-30276-6 }}</ref>

Let <math>F_q\mathbb{F}_q</math> be a [[Finite field]] over prime <math>q</math>, <math>G_1, G_2</math> two additive [[cyclic group]]s 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:

; [[Bilinear map|Bilinearity]]: <math> \forall a,b \in F_q\mathbb{F}_q^*, P\in G_1, Q\in G_2:\ e\left(aP, bQ\right) = e\left(P, Q\right)^{ab}</math>

; [[Degeneracy (mathematics)|Non-degeneracy]]: <math>e \neq 1</math>

; Computability: There exists an efficient [[algorithm]] to compute <math>e</math>.