Is the concept of provably secure hash the same as entropy smoothing hash functions?
In the tutorial Sequences of Games: A Tool for Taming Complexity in Security Proofs V. Shoup shows us a proof of semantic security of a hashed ElGamal encryption.
In order to get that proof (pag. 9) he put the following condition: we need a family of keyed "hash" functions $\mathcal{H} := \{H_k\}_{k\in K}$ where each $H_k$ is a function mapping $G$ to $\{0,1\}^l$ ($l$ a positive integer and $G$ is a group) such that this family $\mathcal{H}$ is "entropy smoothing".
He tells us that "entropy smoothing" gives us some kind of guarantee that makes it hard to distinguish $(k,H_k(\delta))$ from $(k,h)$, where $k$ is a random element of $K$, $\delta$ is a random element of $G$, and $h$ is a random element of $\{0,1\}^l$.
The author said there are ways to construct entropy smoothing hash functions, but it's just a conjecture that ad-hoc hash functions own this property.
