Pairing-based cryptography is the use of a pairing between elements of two cryptographic groups to a third group with a mapping ${\displaystyle e:G_{1}\times G_{2}\to G_{T}}$ to construct or analyze cryptographic systems.

The following definition is commonly used in most academic papers.

Let ${\displaystyle \mathbb {F} _{q}}$ be a finite field over prime ${\displaystyle q}$, ${\displaystyle G_{1},G_{2}}$ 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:

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:

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.

Let ${\displaystyle e}$ be a non-degenerate, efficiently computable, bilinear pairing. Let ${\displaystyle g}$ be a generator of ${\displaystyle G}$. Consider an instance of the CDH problem, ${\displaystyle g}$,${\displaystyle g^{x}}$, ${\displaystyle g^{y}}$. Intuitively, the pairing function ${\displaystyle e}$ does not help us compute ${\displaystyle g^{xy}}$, the solution to the CDH problem. It is conjectured that this instance of the CDH problem is intractable. Given ${\displaystyle g^{z}}$, we may check to see if ${\displaystyle g^{z}=g^{xy}}$ without knowledge of ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$, by testing whether ${\displaystyle e(g^{x},g^{y})=e(g,g^{z})}$ holds.