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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 to construct [[cryptography|cryptographic]] systems. Usually the same group is used for the first two groups, making the pairing in fact a [[Map_(mathematics)|mapping]] from two elements from one group to an element from a second group. In this way, pairings can be used to reduce a hard problem in one group to a different, usually easier problem in another group.
For example, in groups equipped with a [[bilinear mapping]] such as the [[Weil pairing]] or [[Tate pairing]], generalizations of the [[Diffie–Hellman problem|computational Diffie–Hellman problem]] are believed to be computationally infeasible while the simpler [[decisional Diffie–Hellman assumption|decisional Diffie–Hellman problem]] can be easily solved using the pairing function. The first group is sometimes referred to as a '''Gap Group''' because of the assumed difference in difficulty between these two problems in the group.
While first used for [[cryptanalysis]], pairings have since been used to construct many cryptographic systems for which no other efficient implementation is known, such as [[identity based encryption]] .
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