# Pairing-based cryptography

Pairing-based cryptography is the use of a pairing between elements of two cryptographic groups to a third group $e :G_1 \times G_2 \to G_T$ to construct cryptographic systems:

## Classification

If the same group is used for the first two groups (i.e. $G_1 = G_2$), the pairing is called symmetric and is a mapping from two elements of one group to an element from a second group.

Some researches separates different possibile pairing instantiations into three basic types:[1]

• Type 1: $G_1 = G_2$;
• Type 2: $G_1 \ne G_2$ but there is an efficiently computable homomorphism $\phi : G_2 \to G_1$;
• Type 3: $G_1 \ne G_2$ and there are no efficiently computable homomorphisms between $G_1$ and $G_2$.

## Usage in cryptography

If symmetric, 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.[2]

## References

1. ^ Galbraith, Steven; Paterson, Kenneth; Smart, Nigel (2008). "Pairings for Cryptographers". Discrete Applied Mathematics 156 (16): 3113–3121.
2. ^ "NICT, Kyushu University and Fujitsu Laboratories Achieve World Record Cryptanalysis of Next-Generation Cryptography". Press release from NICT. June 18, 2012.