Pairing-based cryptography
Pairing-based cryptography is the use of a pairing between elements of two cryptographic groups to a third group to construct cryptographic systems. If the same group is used for the first two groups, the pairing is called symmetric and is a mapping from two elements of 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 computational Diffie–Hellman problem are believed to be infeasible while the simpler 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 or attribute based encryption.
Cryptanalysis
In June 2012 the National Institute of Information and Communications Technology (NICT), Kyushu University, and Fujitsu Laboratories Limited improved the previous bound for successfully computing a discrete logarithm on a supersingular elliptic curve from 676 bits to 923 bits.[3]