Let's say, I have a group $G$ of large prime order $p$. A set $S$ consists of $n$ random elements chosen from $G$. Without using a collision resistant hash function $H$, how can I map elements of $G$ to unique elements in $Z_p$?
I thought about it before posting here. But, finding discrete logarithm for an arbitrary group is hard, right? In theory, it'd satisfy the requirement of collision resistance, but inefficient.
I don't really know the structure of the group beforehand. What I am essentially trying to do is, I am trying to figure out a generic means to map public key ($PK$) of a Public Key Encryption system to $z \in Z_p$, such that a third party can be convinced about the co-relation between the $PK$ and $z$ without revealing $PK$ but $z$ and some form of proof-of-knowledge. Hash functions, due it inherent nature of pre-image resistance, seems unsuitable.