# Is the commitment $g^u\cdot x$ with $x\in\langle g\rangle$ and $u \gets \mathbb{Z}_n$ hiding and binding?

Consider the following commitment scheme, where $x$ belongs to $\langle g\rangle$ and $u$ is uniformly chosen from $\mathbb{Z}_n$:

$$\mathsf{commit}(u,x) = g^u\cdot x$$

Is it binding and hiding?

The scheme is clearly hiding: the value $g^u\cdot x$ distributes uniformly at random when $u$ is.
However, it is not at all binding: for a given commitment $c$, the sender, assuming he can compute inverse of an element in the group, can find many pairs, more accurately, as many as the size of the group, $u,x$ for which $c=g^u\cdot x$.