Pairings

Synonyms

Bilinear pairings

Related Concepts

Identity-Based Cryptosystems; Pairing-Based Key Exchange

Definition

Let n be a prime number, and let ${\mathbb{G}}_{1} =\langle {g}_{1}\rangle$, ${\mathbb{G}}_{2} =\langle {g}_{2}\rangle$, ${\mathbb{G}}_{3}$ be (multiplicatively written) cyclic groups of order n. A pairing on $({\mathbb{G}}_{1}, {\mathbb{G}}_{2}, {\mathbb{G}}_{3})$ is a map

$$\hat{e} : {\mathbb{G}}_{1} \times {\mathbb{G}}_{2} \rightarrow {\mathbb{G}}_{3}$$

that satisfies the following three conditions:

• 1. (Bilinearity) For all ${r}_{1},{r}_{2} \in {\mathbb{G}}_{1}$ and ${s}_{1},{s}_{2} \in {\mathbb{G}}_{2}$, $\hat{e}({r}_{1}{r}_{2},{s}_{1}) =\hat{ e}({r}_{1},{s}_{1}) \cdot \hat{ e}({r}_{2},{s}_{1})$ and $\hat{e}({r}_{1},{s}_{1}{s}_{2}) =\hat{ e}({r}_{1},{s}_{1}) \cdot \hat{ e}({r}_{1},{s}_{2})$.

• 2. (Non-degeneracy) $\hat{e}({g}_{1},{g}_{2})\neq 1$.

• 3. (Computability) $\hat{e}(r,s)$ can be efficiently computed for all $r \in {\mathbb{G}}_{1}$ and \(s \in...