Is it possible to create a Bilinear Function with Already Assigned "Multiplicative" Input Groups? Assume that we have an already assigned Multiplicative Cyclic Group $\mathbb Z_p^*$ with order $q=p-1$, and $p$ is a prime number, is it possible to create a bilinear function $\hat{e}: \mathbb Z_p^* \times \mathbb Z_p^* \rightarrow \mathbb G_2$ with the property that follows:

$$\hat{e}(g^{a},g^{sb})=\hat{e}(g^{b},g^{sa})$$

$g$ is a generator for $\mathbb Z_p^*$, and $\mathbb G_2$ is also a Multiplicative Cyclic Group with order $q$

it seems that if the creation of this function is practical in polynomial time then there should be new attacks on some cryptographic protocols with security property of anonymity.
If you can find such an efficiently computable function (other than the trivial solution $\hat{e}( x, y ) = 1\ \ $ for all $x$, $y$), then you have shown that the decisional Diffie-Hellman problem is easy.  That is, given $g$, $g^a$, $g^b$, $g^c$, you can check whether

$$ab = c$$

simply by testing:

$$\hat{e}( g^a, g^b ) = \hat{e}( g, g^c)$$

The decisional Diffie-Hellman problem is believed to be hard in appropriately sized Multiplicative Cyclic groups, hence we believe that there is no such efficiently computable $\hat{e}$