Pairing-based cryptography

The following definition is commonly used in most academic papers.[1]

Let ${\displaystyle G_{1},G_{2}}$ be two additive cyclic groups of prime order ${\displaystyle q}$, and ${\displaystyle G_{T}}$ another cyclic group of order ${\displaystyle q}$ written multiplicatively. A pairing is a map: ${\displaystyle e:G_{1}\times G_{2}\rightarrow G_{T}}$, which satisfies the following properties:

Bilinearity

${\displaystyle \forall a,b\in F_{q}^{*},\ \forall P\in G_{1},Q\in G_{2}:\ e\left(aP,bQ\right)=e\left(P,Q\right)^{ab}}$

Non-degeneracy

${\displaystyle e\neq 1}$

Computability

There exists an efficient algorithm to compute ${\displaystyle e}$.

If the same group is used for the first two groups (i.e. ${\displaystyle G_{1}=G_{2}}$), the pairing is called symmetric and is a mapping from two elements of one group to an element from a second group.

Some researchers classify pairing instantiations into three (or more) basic types:

1. ${\displaystyle G_{1}=G_{2}}$;
2. ${\displaystyle G_{1}\neq G_{2}}$ but there is an efficiently computable homomorphism ${\displaystyle \phi :G_{2}\to G_{1}}$;
3. ${\displaystyle G_{1}\neq G_{2}}$ and there are no efficiently computable homomorphisms between ${\displaystyle G_{1}}$ and ${\displaystyle G_{2}}$.[2]

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,[3] pairings have also been used to construct many cryptographic systems for which no other efficient implementation is known, such as identity based encryption or attribute based encryption schemes.

A contemporary example of using bilinear pairings is exemplified in the Boneh-Lynn-Shacham signature scheme.

Pairing-based cryptography relies on hardness assumptions separate from e.g. the Elliptic Curve Discrete Logarithm Problem, which is older and has been studied for a longer time.

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.[4]

• Lecture on Pairing-Based Cryptography
• Ben Lynn's PBC Library