I know that the purpose of the blinding key is to make it difficult to obtain the hidden value. More formally: the Pedersen commitment is comprised of the blinding key $\alpha$, two generators $H$ and $G$ and the value we wish to commit to $v$.
I am familiar with two versions of the Pedersen commitment, the "regular" version where the commitment is $P=G^\alpha+H^v$, and the Eliptic curve version where $P = \alpha\cdot G + v\cdot H$. In both versions, assuming I obtained the blinding key $\alpha$, can I obtain $v$ ($H$ and $G$ are publicly known) in an efficient way?
For the regular commitment I can see that it is a hard problem (under the CDH assumption) so I assume there is no efficient way. However, for the version over eliptic curve, it seems that it is possible to obtain $v$ efficiently, but I am uncertain about it.