I have rather simplistic understanding of cryptography, so I have to beg a pardon in advance if the question has internal contradictions.
Imagine data $A$ with a hash $hashA$. This data is encrypted with private key $K$ to get data $E$ and the hash of the encrypted data is $hashE$.
Are there encryption method and hashing method, which will allow to verify, given $hashA$, $hashE$ and $K$, that $hashE$ is indeed a hash of “A encrypted with K”?
How secure could that algorithm be against attempted forgeries, i.e. someone having access to $A$ creating $E$ and $K$, so that $hashA$, $hashE$ and $K$ are correct in sense of $1$, but $E$ is not “A encrypted with K”?
Clearly the identity hash $(hash(x)=x)$ works, but I'm more interested in hashes which are significantly smaller than the original data.