BLS hash as a group element exponent? In BLS short signatures paper, the authors describe a hash function $H\colon\ \{0, 1\}^∗ → G^∗$, where $G$ is a Gap-Diffie-Hellman group.
They present a structure where a standard hash is used on a field element, and then mapped to a subgroup of $G$, later to be used in the security proof as a random oracle.
How would the security proof change (or break) if we'll simply use an exponentiation of some generator of $G$? i.e., to have $H(m) := g^m$ for some generator $g$.
It is not a secure hash function. For example, one can easily break the 
collision resistance property: let $q$ be the order of $g$, then $m$ and $m+q$ have the same hash value, $H(m+q)=g^{m+q}=g^m=H(m)$.