I know the basic definitions of bilinear groups. For example, there is a bilinear pairing that uses elliptic curves and has the following properties: For $G_1$, and $G_2$ are cyclic groups of prime order $p$ with $P_1$ is the generator of G_1 and $P_2$ is the generator of G_2. And there is $e : G_1 × G_1 → G_T$, meaning that (1) For all $P ∈ G_1$, $Q ∈ G_2$ it holds that $e(aP_1,bQ )=e(P,Q)^{ab}$ (2) $e(P_1,P_2)≠1$;

But I have just read this in a paper https://infoscience.epfl.ch/record/128718/files/CCS08.pdf, that is based on BLS signature https://www.iacr.org/archive/asiacrypt2001/22480516.pdf. And the paper stated that"

Let PG be a pairing group generator that on input $1^k$ outputs descriptions of multiplicative groups $G_1$ and $G_T$ of prime order $p$ where $|p| =k$. Let and let $g \in G_1^{*}$ and $G_{1}^{*}=G_{1} \backslash\{1\}$ .The generated groups are such that there exists an admissible bilinear map $e : G_1 × G_1 → G_T$, meaning that (1) for all $a, b ∈ Z_p$ it holds that $e(g^a,g^b )=e(g,g)^{ab}$ (2) $e(g,g)≠1$; and (3) the bilinear map is efficiently n.

What is the difference between the first pairing group that uses elliptic curves and the second one? How could I name the second pairing groups? Could anyone suggest examples for the two examples?


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