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Consider a bilinear pairing $e: G_1 times G_2 rightarrow G_T$. Let’s assume, $G_1 = G_2 = G_T = (mathbb{Z}_n,+)$, i.e. additive group of integer modulo $n$ and $e(x,y) = xy$ mod $n$. Isn’t it an example of bilinear map?

$G_1 = G_2 = G_T = (mathbb{Z}_n,ast)$, i.e. multiplicative group of integer modulo $n$ and $e(x,y) = y^x$ mod $n$, is it still a bilinear map?

Can there be other bilinear map defined over $mathbb{Z}_n$?

EDIT:

Actually all my question were to understand Groth-Sahai proof system. Here, they have recast the general equation to fit the form of a quadratic equation. (See teh highlighted part below). But, how can they remark that all maps with $f(x,y) = xy$ mod $n$ will satisfy properties of bilinear pairing?