Pairing-based cryptography
Encyclopedia
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.
Pairing
The concept of pairing treated here occurs in mathematics.-Definition:Let R be a commutative ring with unity, and let M, N and L be three R-modules.A pairing is any R-bilinear map e:M \times N \to L...
between elements of two cryptographic groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
to a third group to construct cryptographic
Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...
systems. If the same group is used for the first two groups, the pairing is called symmetric and is a mapping
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
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
Weil pairing
In mathematics, the Weil pairing is a construction of roots of unity by means of functions on an elliptic curve E, in such a way as to constitute a pairing on the torsion subgroup of E...
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
Cryptanalysis
Cryptanalysis is the study of methods for obtaining the meaning of encrypted information, without access to the secret information that is normally required to do so. Typically, this involves knowing how the system works and finding a secret key...
, pairings have since been used to construct many cryptographic systems for which no other efficient implementation is known, such as identity based encryption.